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Electroceramics Overview

Electro Ceramics Web Course (NPTEL)

Contact information of the course instructor

Ashish Garg Associate ProfessorDepartment of Materials science and EngineeringIndian Institute of Technology KanpurKanpur 208016 IndiaTelephone 0512-259-7904Email ashishgiitkacinWeb httphomeiitkacin~ashishg

Introduction

Electro-ceramics or broadly speaking electronic optical and magnetic ceramics are useful in a varietyof technological applications such as sensors actuators transducers data storage devices etc Some of the examples are

Dielectric materials such as SiO2 are used as data storage elements in random access

memories or RAMsFerroelectrics such as BaTiO3 and PbTiO3 are used as sensors and actuators

Magnetic oxides such as iron oxides are used for data storage in magnetic headsZnO is used circuit protection materials in devices named as varistorsZrO2 stabilized with other oxides is used in fuel cells and batteries

Hence to understand these materials better and to engineer them as per our needs we need tounderstand their science viz their structure defects in these materials phenomenon of conductionfundamentals of various functional properties A sound understanding of these would (hopefully)enable us tailor the structure and properties of these material with good degree of control

Pre-requisites

Basic courses on structure of materials thermodynamics and solid state physicsSuited for final year undergraduate students of most disciplines and fresh graduate students

List of Topics

Module Topics Equivalent Lectures (50-60 meach)

1 Structure of Ceramic Materials 5

2 Defect Chemistry and Equilibria 7

3 Diffusion and Conduction in Ceramics 7

4 Linear Dielectric Ceramics 8

5 Nonlinear Dielectric Ceramics 6

6 Magnetism and Magnetic Ceramics 5

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7 Superconducting Ceramics 1

8 Multiferroic and Magnetoelectric Ceramics 1

9 Synthesis Methods 1

Total number of equivalent lectures 41

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7 Superconducting Ceramics 1

8 Multiferroic and Magnetoelectric Ceramics 1

9 Synthesis Methods 1

Total number of equivalent lectures 41

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Electroceramics Table of Contents

Table of Contents

1 Structure of Ceramic Materials

11 Brief Review of Structure of Materials12 A Brief Review of Bonding in Materials13 Packing of atoms in metals14 Interstices in Structures15 Structure of Covalent Ceramics16 Ionically Bonded Ceramic Structures17 Compounds based on FCC Packing of ions18 Other cubic structures19 Orthogonal Structures110 Structures based on HCP packing of ions111 Summary

2 Defect Chemistry and Defect Equilibria

21 Point Defects22 KroumlgerndashVink notation in a metal oxide MO23 Defect Reactions24 Defect Structures in Stoichiometric Oxides25 Defect Structures in Non-stoichiometric Oxides26 Dissolution of foreign cations in an oxide27 Concentration of Intrinsic Defects28 Intrinsic and Extrinsic Defects29 Units for defect Concentration210 Defect Equilibria211 Defect Equilibria in Stoichiometric Oxides212 Defect Equilibria in Non-Stoichiometric Oxides213 Defect Structures involving Oxygen vacancies and interstitials214 Defect Equilibrium Diagram215 A Simple General Procedure for constructing at Brouwerrsquos Diagram216 Extent of non-stoichiometry217 Example Comparative behaviour of TiO2 and MgO vis-agrave-vis oxygen pressure

218 Electronic Disorder219 Examples220 Summary

3 Defects Diffusion and Conduction in Ceramics

31 Diffusion 32 Diffusion Kinetics 33 Examples of Diffusion in Ceramics 34 Mobility and Diffusivity

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35 Analogue to the electrical properties 36 Conduction in Ceramics vis-agrave-vis metallic conductors General Information37 Ionic Conduction Basic Facts 38 Ionic and Electronic Conductivity39 Characteristics of Ionic Conduction 310 Theory of Ionic Conduction Conduction in Glasses 311 Conduction in Glasses312 Fast Ion Conductors313 Examples of Ionic Conduction314 Electrochemical Potential315 Nernst Equation and Application of Ionic Conductors316 Examples of Ionic Conductors in Engineering Applications317 Summary

4 Dielectric Ceramics Basic Principles

41 Basic Properties Dielectrics in DC electric field 42 Mechanisms of Polarization43 Microscopic Approach 44 Determination of Local Field 45 Analytical treatment of Polarizability 46 Effect of alternating field on the behavior of a dielectric material 47 Frequency dependence of dielectric properties Resonance 48 Dipolar Relaxation ie Debye Relaxation is Polar Solids 49 Circuit Representation of a Dielectric and Impedance Analysis 410 Impedance Spectroscopy 411Dielectric Breakdown 412 Summary

5 Nonlinear Dielectrics

51 Introduction 52 Classification based on Crystal Classes 53 Ferroelectric Ceramics 531 Permanent Dipole Moment and Polarization 532 Principle of Ferroelectricity Energetics 533 Proof of Curie-Weiss Law 534 Thermodynamic Basis of Ferroelectric Phase Transitions 535 Case I Second order Transition 536 Case ndash II First Order Transition 537 Ferroelectric Domains 538 Analytical treatment of domain wall energy 539 Ferroelectric Switching and Domains 5310 Measurement of Hysteresis Loop 5311 Structural change and ferroelectricity in Barium Titanate (BaTiO3)

5312 Applications of Ferroelectrics

54 Piezoelectric Ceramics 541 Direct Piezoelectric Effect 542 Reverse or Converse Piezoelectric Effect

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543 Poling of Piezoelectric Materials 544 Depolarization of Piezoelectrics 545 Common Piezoelectric Materials 546 Measurement of Piezoelectric Properties 547 Applications of Piezoelectric Ceramics

55 Pyroelectric Ceramics 551 Difference between and pyroelectric and ferroelectric material 552 Theory of Pyroelectric Materials 553 Measurement of Pyroelectric coefficient 554 Direct and Indirect effect 555 Common Pyroelectric Materials 556 Common Applications

56 Summary

6 Magnetic Ceramics

61 Magnetic Moments62 Macroscopic view of Magnetization 63 Classification of Magnetism 64 Diamagnetism65 Paramagnetism66 Ferromagnetism67 Antiferromagnetism 68 Ferrimagnetism69 A Comparison 610 Magnetic Losses and Frequency Dependence 611 Magnetic Ferrites 612 Summary

7 High temperature Superconductors

71 Background72 Meissner Effect73 The critical field Hc74 Theory of Superconductivity75 Discovery of high temperature superconductivity76 Mechanism of high temperature superconductivity77 Applications78 Summary

8 Multiferroic and Magnetoelectric Ceramics

81 Introduction82 Historical Perspective83 Requirements of a magnetoelectric and multiferroic material84 Magnetoelectric Coupling85 Type I Multiferroics86 Type II Multiferroics87 Two Phase Materials88 Summary

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9 Synthesis Methods

91 Bulk Preparation Methods92 Thin Film Preparation Methods93 Thin film deposition Issues94 Summary

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Electroceramics General Bibliography

General Bibliography

The following are the books which can be referred for general reading More references are providedin each module

Recommended Reading

1 Physical Ceramics Principles for Ceramic Science and Engineering Y-M Chiang D PBirnie and W D Kingery Wiley-VCH

2 Introduction to Ceramics 2nd Edition W D Kingery H K Bowen D R Uhlmann Wiley3 Principles of Electronic Ceramics by L L Hench and J K West Wiley4 Electroceramics Materials Properties Applications by A J Moulson and J M Herbert Wiley5 Nonstoichiometry Diffusion and Electrical Conductivity in Binary Metal Oxides (Science amp

Technology of Materials) PK Kofstad John Wiley and Sons Inc

Supplementary Reading

6 Introduction to Solid State Physics C Kittel Wiley7 Electrical Properties of Materials L Solymer and D Walsh Oxford University Press8 Introduction of Solid State Physics NW Ashcroft and ND Mermin Brooks Cole9 Solid State Physics AJ Dekker Prentice-Hall

10 Transition Metal Oxides An Introduction to Their Electronic Structure and Properties PA CoxOxford University Press

11 Basic Solid State Chemistry AR West Wiley12 Non-stoichiometric Oxides O Toft Soslashrensen Academic Press13 Dielectrics and Waves AR von Hippel John Wiley and Sons14 Feynman Lectures on Physics Volume 1-3 RP Feynman Addison Wesley Longman15 Materials Science and Engineering A first course V Raghavan Prentice Hall of India16 Materials Science And Engineering An Introduction WD Callister Wiley

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Module 1 Structure of Ceramic Materials Introduction

In this module we will first review the structure and bonding in the materials in general followed by abrief discussion on how atoms pack together in the solids and what are the types of intersticespresent in various structures Then we would briefly delve into the types of bonding with reference tothe nature of materials Together this information will form the basis for structures in ceramicmaterials which are typically bonded with a mix of ionic and covalent bonding Subsequently wewould discuss the structure of ceramic materials with purely covalent bonding followed by ratherdetailed description of ceramic materials with ionic bonding These are essentially based on packing ofanions closed packed forms where cations fill the interstices

The Module contains

Brief Review of Structure of Materials

A Brief Review of Bonding in Materials

Packing of Atoms in Metals

Interstices in Structures

Ionically Bonded Ceramic Structures

Compounds based on FCC Packing of ions

Other Cubic Structures

Orthogonal Structures

Structures based on HCP packing of ions

Summary

Suggested Reading

Materials Science and Engineering WD Callister Jr Wiley

Physical Ceramics Principles for Ceramic Science and Engineering Y-MChiang D P Birnie and W D Kingery Wiley-VCH

Introduction to Ceramics W D Kingery H K Bowen D R Uhlmann Wiley

Fundamentals of Ceramics Michael Barsoum McGraw Hill

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

11 Brief Review of Structure of Materials

In the following sections we will quickly look at the concepts of lattice unit-cell and crystalstructures which will be useful to understand the crystal structures of common ceramic compounds

111 Point Lattice

In a point lattice the following characteristics are obeyed

There is a periodic arrangement of points in space (Figure 11(a))

In addition each point must have identical neighbourhood(Figure 11(b))

Lack of regular arrangementNo periodicityNon-identical neighbourhood

Regular arrangementPeriodicityIdentical neighbourhood of each point

Figure 11(a) Unit-cellrepresentation

>

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Figure 11(b) Schematics of arrangement of points inspace

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

112 Unit Cell

A unit cell is the smallest repeatable unit in a point lattice (Figure 12)

Choice of unit cell shape is not unique

Figure 12 Representation of a point lattice and anunit cell

Unit-cell parameters for a 3-D unit cells

axis lengths a b and c

anglesa szlig and γ

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

113 Motif and Crystal Structure

Crystal structure a combination of motif and point lattice

Motif is defined as a unit or a pattern For a crystal it can be an atom an ion or a group ofatoms or ions or a formula unit or formula units Often it is also called as Basis

When motif replaces points in a periodic point lattice it gives rise to what is called as acrystal with a defined structure

Figure 13 Formation of a periodic crystal structure

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

114 Types of Lattice

Lattice can further be classfied into two types

Primitive lattice having one formula unit or one lattice point or one unit of motif perunit cell and Non-primitive lattices having more than one lattice points or more than one unitof motif per unit cell

Figure 14 Primitive and Non-primitivelattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

115 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation

Symmetry elements underlying a point lattice (see the figure) Reflection reflection across a mirror plane

Rotation rotation around a crystallographic axis by an angle θ such as 360degθ isan integer of value 1 2 3 4 and 6 and is referred to as n -fold rotationInversion a point at xyz becomes its equivalent at (ndashx-y-z)Rotation-Inversion Rotation followed by inversion OR Rotation-ReflectionRotation followed by reflection

Figure 15 Basic symmetry operations in crystals

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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35 Analogue to the electrical properties 36 Conduction in Ceramics vis-agrave-vis metallic conductors General Information37 Ionic Conduction Basic Facts 38 Ionic and Electronic Conductivity39 Characteristics of Ionic Conduction 310 Theory of Ionic Conduction Conduction in Glasses 311 Conduction in Glasses312 Fast Ion Conductors313 Examples of Ionic Conduction314 Electrochemical Potential315 Nernst Equation and Application of Ionic Conductors316 Examples of Ionic Conductors in Engineering Applications317 Summary

4 Dielectric Ceramics Basic Principles

41 Basic Properties Dielectrics in DC electric field 42 Mechanisms of Polarization43 Microscopic Approach 44 Determination of Local Field 45 Analytical treatment of Polarizability 46 Effect of alternating field on the behavior of a dielectric material 47 Frequency dependence of dielectric properties Resonance 48 Dipolar Relaxation ie Debye Relaxation is Polar Solids 49 Circuit Representation of a Dielectric and Impedance Analysis 410 Impedance Spectroscopy 411Dielectric Breakdown 412 Summary

5 Nonlinear Dielectrics

51 Introduction 52 Classification based on Crystal Classes 53 Ferroelectric Ceramics 531 Permanent Dipole Moment and Polarization 532 Principle of Ferroelectricity Energetics 533 Proof of Curie-Weiss Law 534 Thermodynamic Basis of Ferroelectric Phase Transitions 535 Case I Second order Transition 536 Case ndash II First Order Transition 537 Ferroelectric Domains 538 Analytical treatment of domain wall energy 539 Ferroelectric Switching and Domains 5310 Measurement of Hysteresis Loop 5311 Structural change and ferroelectricity in Barium Titanate (BaTiO3)

5312 Applications of Ferroelectrics

54 Piezoelectric Ceramics 541 Direct Piezoelectric Effect 542 Reverse or Converse Piezoelectric Effect

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543 Poling of Piezoelectric Materials 544 Depolarization of Piezoelectrics 545 Common Piezoelectric Materials 546 Measurement of Piezoelectric Properties 547 Applications of Piezoelectric Ceramics

55 Pyroelectric Ceramics 551 Difference between and pyroelectric and ferroelectric material 552 Theory of Pyroelectric Materials 553 Measurement of Pyroelectric coefficient 554 Direct and Indirect effect 555 Common Pyroelectric Materials 556 Common Applications

56 Summary

6 Magnetic Ceramics

61 Magnetic Moments62 Macroscopic view of Magnetization 63 Classification of Magnetism 64 Diamagnetism65 Paramagnetism66 Ferromagnetism67 Antiferromagnetism 68 Ferrimagnetism69 A Comparison 610 Magnetic Losses and Frequency Dependence 611 Magnetic Ferrites 612 Summary

7 High temperature Superconductors

71 Background72 Meissner Effect73 The critical field Hc74 Theory of Superconductivity75 Discovery of high temperature superconductivity76 Mechanism of high temperature superconductivity77 Applications78 Summary

8 Multiferroic and Magnetoelectric Ceramics

81 Introduction82 Historical Perspective83 Requirements of a magnetoelectric and multiferroic material84 Magnetoelectric Coupling85 Type I Multiferroics86 Type II Multiferroics87 Two Phase Materials88 Summary

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9 Synthesis Methods

91 Bulk Preparation Methods92 Thin Film Preparation Methods93 Thin film deposition Issues94 Summary

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Electroceramics General Bibliography

General Bibliography

The following are the books which can be referred for general reading More references are providedin each module

Recommended Reading

1 Physical Ceramics Principles for Ceramic Science and Engineering Y-M Chiang D PBirnie and W D Kingery Wiley-VCH

2 Introduction to Ceramics 2nd Edition W D Kingery H K Bowen D R Uhlmann Wiley3 Principles of Electronic Ceramics by L L Hench and J K West Wiley4 Electroceramics Materials Properties Applications by A J Moulson and J M Herbert Wiley5 Nonstoichiometry Diffusion and Electrical Conductivity in Binary Metal Oxides (Science amp

Technology of Materials) PK Kofstad John Wiley and Sons Inc

Supplementary Reading

6 Introduction to Solid State Physics C Kittel Wiley7 Electrical Properties of Materials L Solymer and D Walsh Oxford University Press8 Introduction of Solid State Physics NW Ashcroft and ND Mermin Brooks Cole9 Solid State Physics AJ Dekker Prentice-Hall

10 Transition Metal Oxides An Introduction to Their Electronic Structure and Properties PA CoxOxford University Press

11 Basic Solid State Chemistry AR West Wiley12 Non-stoichiometric Oxides O Toft Soslashrensen Academic Press13 Dielectrics and Waves AR von Hippel John Wiley and Sons14 Feynman Lectures on Physics Volume 1-3 RP Feynman Addison Wesley Longman15 Materials Science and Engineering A first course V Raghavan Prentice Hall of India16 Materials Science And Engineering An Introduction WD Callister Wiley

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Module 1 Structure of Ceramic Materials Introduction

In this module we will first review the structure and bonding in the materials in general followed by abrief discussion on how atoms pack together in the solids and what are the types of intersticespresent in various structures Then we would briefly delve into the types of bonding with reference tothe nature of materials Together this information will form the basis for structures in ceramicmaterials which are typically bonded with a mix of ionic and covalent bonding Subsequently wewould discuss the structure of ceramic materials with purely covalent bonding followed by ratherdetailed description of ceramic materials with ionic bonding These are essentially based on packing ofanions closed packed forms where cations fill the interstices

The Module contains

Brief Review of Structure of Materials

A Brief Review of Bonding in Materials

Packing of Atoms in Metals

Interstices in Structures

Ionically Bonded Ceramic Structures

Compounds based on FCC Packing of ions

Other Cubic Structures

Orthogonal Structures

Structures based on HCP packing of ions

Summary

Suggested Reading

Materials Science and Engineering WD Callister Jr Wiley

Physical Ceramics Principles for Ceramic Science and Engineering Y-MChiang D P Birnie and W D Kingery Wiley-VCH

Introduction to Ceramics W D Kingery H K Bowen D R Uhlmann Wiley

Fundamentals of Ceramics Michael Barsoum McGraw Hill

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

11 Brief Review of Structure of Materials

In the following sections we will quickly look at the concepts of lattice unit-cell and crystalstructures which will be useful to understand the crystal structures of common ceramic compounds

111 Point Lattice

In a point lattice the following characteristics are obeyed

There is a periodic arrangement of points in space (Figure 11(a))

In addition each point must have identical neighbourhood(Figure 11(b))

Lack of regular arrangementNo periodicityNon-identical neighbourhood

Regular arrangementPeriodicityIdentical neighbourhood of each point

Figure 11(a) Unit-cellrepresentation

>

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Figure 11(b) Schematics of arrangement of points inspace

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

112 Unit Cell

A unit cell is the smallest repeatable unit in a point lattice (Figure 12)

Choice of unit cell shape is not unique

Figure 12 Representation of a point lattice and anunit cell

Unit-cell parameters for a 3-D unit cells

axis lengths a b and c

anglesa szlig and γ

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

113 Motif and Crystal Structure

Crystal structure a combination of motif and point lattice

Motif is defined as a unit or a pattern For a crystal it can be an atom an ion or a group ofatoms or ions or a formula unit or formula units Often it is also called as Basis

When motif replaces points in a periodic point lattice it gives rise to what is called as acrystal with a defined structure

Figure 13 Formation of a periodic crystal structure

>

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

114 Types of Lattice

Lattice can further be classfied into two types

Primitive lattice having one formula unit or one lattice point or one unit of motif perunit cell and Non-primitive lattices having more than one lattice points or more than one unitof motif per unit cell

Figure 14 Primitive and Non-primitivelattices

>

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

115 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation

Symmetry elements underlying a point lattice (see the figure) Reflection reflection across a mirror plane

Rotation rotation around a crystallographic axis by an angle θ such as 360degθ isan integer of value 1 2 3 4 and 6 and is referred to as n -fold rotationInversion a point at xyz becomes its equivalent at (ndashx-y-z)Rotation-Inversion Rotation followed by inversion OR Rotation-ReflectionRotation followed by reflection

Figure 15 Basic symmetry operations in crystals

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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543 Poling of Piezoelectric Materials 544 Depolarization of Piezoelectrics 545 Common Piezoelectric Materials 546 Measurement of Piezoelectric Properties 547 Applications of Piezoelectric Ceramics

55 Pyroelectric Ceramics 551 Difference between and pyroelectric and ferroelectric material 552 Theory of Pyroelectric Materials 553 Measurement of Pyroelectric coefficient 554 Direct and Indirect effect 555 Common Pyroelectric Materials 556 Common Applications

56 Summary

6 Magnetic Ceramics

61 Magnetic Moments62 Macroscopic view of Magnetization 63 Classification of Magnetism 64 Diamagnetism65 Paramagnetism66 Ferromagnetism67 Antiferromagnetism 68 Ferrimagnetism69 A Comparison 610 Magnetic Losses and Frequency Dependence 611 Magnetic Ferrites 612 Summary

7 High temperature Superconductors

71 Background72 Meissner Effect73 The critical field Hc74 Theory of Superconductivity75 Discovery of high temperature superconductivity76 Mechanism of high temperature superconductivity77 Applications78 Summary

8 Multiferroic and Magnetoelectric Ceramics

81 Introduction82 Historical Perspective83 Requirements of a magnetoelectric and multiferroic material84 Magnetoelectric Coupling85 Type I Multiferroics86 Type II Multiferroics87 Two Phase Materials88 Summary

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9 Synthesis Methods

91 Bulk Preparation Methods92 Thin Film Preparation Methods93 Thin film deposition Issues94 Summary

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Electroceramics General Bibliography

General Bibliography

The following are the books which can be referred for general reading More references are providedin each module

Recommended Reading

1 Physical Ceramics Principles for Ceramic Science and Engineering Y-M Chiang D PBirnie and W D Kingery Wiley-VCH

2 Introduction to Ceramics 2nd Edition W D Kingery H K Bowen D R Uhlmann Wiley3 Principles of Electronic Ceramics by L L Hench and J K West Wiley4 Electroceramics Materials Properties Applications by A J Moulson and J M Herbert Wiley5 Nonstoichiometry Diffusion and Electrical Conductivity in Binary Metal Oxides (Science amp

Technology of Materials) PK Kofstad John Wiley and Sons Inc

Supplementary Reading

6 Introduction to Solid State Physics C Kittel Wiley7 Electrical Properties of Materials L Solymer and D Walsh Oxford University Press8 Introduction of Solid State Physics NW Ashcroft and ND Mermin Brooks Cole9 Solid State Physics AJ Dekker Prentice-Hall

10 Transition Metal Oxides An Introduction to Their Electronic Structure and Properties PA CoxOxford University Press

11 Basic Solid State Chemistry AR West Wiley12 Non-stoichiometric Oxides O Toft Soslashrensen Academic Press13 Dielectrics and Waves AR von Hippel John Wiley and Sons14 Feynman Lectures on Physics Volume 1-3 RP Feynman Addison Wesley Longman15 Materials Science and Engineering A first course V Raghavan Prentice Hall of India16 Materials Science And Engineering An Introduction WD Callister Wiley

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Module 1 Structure of Ceramic Materials Introduction

In this module we will first review the structure and bonding in the materials in general followed by abrief discussion on how atoms pack together in the solids and what are the types of intersticespresent in various structures Then we would briefly delve into the types of bonding with reference tothe nature of materials Together this information will form the basis for structures in ceramicmaterials which are typically bonded with a mix of ionic and covalent bonding Subsequently wewould discuss the structure of ceramic materials with purely covalent bonding followed by ratherdetailed description of ceramic materials with ionic bonding These are essentially based on packing ofanions closed packed forms where cations fill the interstices

The Module contains

Brief Review of Structure of Materials

A Brief Review of Bonding in Materials

Packing of Atoms in Metals

Interstices in Structures

Ionically Bonded Ceramic Structures

Compounds based on FCC Packing of ions

Other Cubic Structures

Orthogonal Structures

Structures based on HCP packing of ions

Summary

Suggested Reading

Materials Science and Engineering WD Callister Jr Wiley

Physical Ceramics Principles for Ceramic Science and Engineering Y-MChiang D P Birnie and W D Kingery Wiley-VCH

Introduction to Ceramics W D Kingery H K Bowen D R Uhlmann Wiley

Fundamentals of Ceramics Michael Barsoum McGraw Hill

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

11 Brief Review of Structure of Materials

In the following sections we will quickly look at the concepts of lattice unit-cell and crystalstructures which will be useful to understand the crystal structures of common ceramic compounds

111 Point Lattice

In a point lattice the following characteristics are obeyed

There is a periodic arrangement of points in space (Figure 11(a))

In addition each point must have identical neighbourhood(Figure 11(b))

Lack of regular arrangementNo periodicityNon-identical neighbourhood

Regular arrangementPeriodicityIdentical neighbourhood of each point

Figure 11(a) Unit-cellrepresentation

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Figure 11(b) Schematics of arrangement of points inspace

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

112 Unit Cell

A unit cell is the smallest repeatable unit in a point lattice (Figure 12)

Choice of unit cell shape is not unique

Figure 12 Representation of a point lattice and anunit cell

Unit-cell parameters for a 3-D unit cells

axis lengths a b and c

anglesa szlig and γ

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

113 Motif and Crystal Structure

Crystal structure a combination of motif and point lattice

Motif is defined as a unit or a pattern For a crystal it can be an atom an ion or a group ofatoms or ions or a formula unit or formula units Often it is also called as Basis

When motif replaces points in a periodic point lattice it gives rise to what is called as acrystal with a defined structure

Figure 13 Formation of a periodic crystal structure

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

114 Types of Lattice

Lattice can further be classfied into two types

Primitive lattice having one formula unit or one lattice point or one unit of motif perunit cell and Non-primitive lattices having more than one lattice points or more than one unitof motif per unit cell

Figure 14 Primitive and Non-primitivelattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

115 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation

Symmetry elements underlying a point lattice (see the figure) Reflection reflection across a mirror plane

Rotation rotation around a crystallographic axis by an angle θ such as 360degθ isan integer of value 1 2 3 4 and 6 and is referred to as n -fold rotationInversion a point at xyz becomes its equivalent at (ndashx-y-z)Rotation-Inversion Rotation followed by inversion OR Rotation-ReflectionRotation followed by reflection

Figure 15 Basic symmetry operations in crystals

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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9 Synthesis Methods

91 Bulk Preparation Methods92 Thin Film Preparation Methods93 Thin film deposition Issues94 Summary

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Electroceramics General Bibliography

General Bibliography

The following are the books which can be referred for general reading More references are providedin each module

Recommended Reading

1 Physical Ceramics Principles for Ceramic Science and Engineering Y-M Chiang D PBirnie and W D Kingery Wiley-VCH

2 Introduction to Ceramics 2nd Edition W D Kingery H K Bowen D R Uhlmann Wiley3 Principles of Electronic Ceramics by L L Hench and J K West Wiley4 Electroceramics Materials Properties Applications by A J Moulson and J M Herbert Wiley5 Nonstoichiometry Diffusion and Electrical Conductivity in Binary Metal Oxides (Science amp

Technology of Materials) PK Kofstad John Wiley and Sons Inc

Supplementary Reading

6 Introduction to Solid State Physics C Kittel Wiley7 Electrical Properties of Materials L Solymer and D Walsh Oxford University Press8 Introduction of Solid State Physics NW Ashcroft and ND Mermin Brooks Cole9 Solid State Physics AJ Dekker Prentice-Hall

10 Transition Metal Oxides An Introduction to Their Electronic Structure and Properties PA CoxOxford University Press

11 Basic Solid State Chemistry AR West Wiley12 Non-stoichiometric Oxides O Toft Soslashrensen Academic Press13 Dielectrics and Waves AR von Hippel John Wiley and Sons14 Feynman Lectures on Physics Volume 1-3 RP Feynman Addison Wesley Longman15 Materials Science and Engineering A first course V Raghavan Prentice Hall of India16 Materials Science And Engineering An Introduction WD Callister Wiley

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Module 1 Structure of Ceramic Materials Introduction

In this module we will first review the structure and bonding in the materials in general followed by abrief discussion on how atoms pack together in the solids and what are the types of intersticespresent in various structures Then we would briefly delve into the types of bonding with reference tothe nature of materials Together this information will form the basis for structures in ceramicmaterials which are typically bonded with a mix of ionic and covalent bonding Subsequently wewould discuss the structure of ceramic materials with purely covalent bonding followed by ratherdetailed description of ceramic materials with ionic bonding These are essentially based on packing ofanions closed packed forms where cations fill the interstices

The Module contains

Brief Review of Structure of Materials

A Brief Review of Bonding in Materials

Packing of Atoms in Metals

Interstices in Structures

Ionically Bonded Ceramic Structures

Compounds based on FCC Packing of ions

Other Cubic Structures

Orthogonal Structures

Structures based on HCP packing of ions

Summary

Suggested Reading

Materials Science and Engineering WD Callister Jr Wiley

Physical Ceramics Principles for Ceramic Science and Engineering Y-MChiang D P Birnie and W D Kingery Wiley-VCH

Introduction to Ceramics W D Kingery H K Bowen D R Uhlmann Wiley

Fundamentals of Ceramics Michael Barsoum McGraw Hill

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

11 Brief Review of Structure of Materials

In the following sections we will quickly look at the concepts of lattice unit-cell and crystalstructures which will be useful to understand the crystal structures of common ceramic compounds

111 Point Lattice

In a point lattice the following characteristics are obeyed

There is a periodic arrangement of points in space (Figure 11(a))

In addition each point must have identical neighbourhood(Figure 11(b))

Lack of regular arrangementNo periodicityNon-identical neighbourhood

Regular arrangementPeriodicityIdentical neighbourhood of each point

Figure 11(a) Unit-cellrepresentation

>

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Figure 11(b) Schematics of arrangement of points inspace

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

112 Unit Cell

A unit cell is the smallest repeatable unit in a point lattice (Figure 12)

Choice of unit cell shape is not unique

Figure 12 Representation of a point lattice and anunit cell

Unit-cell parameters for a 3-D unit cells

axis lengths a b and c

anglesa szlig and γ

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

113 Motif and Crystal Structure

Crystal structure a combination of motif and point lattice

Motif is defined as a unit or a pattern For a crystal it can be an atom an ion or a group ofatoms or ions or a formula unit or formula units Often it is also called as Basis

When motif replaces points in a periodic point lattice it gives rise to what is called as acrystal with a defined structure

Figure 13 Formation of a periodic crystal structure

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

114 Types of Lattice

Lattice can further be classfied into two types

Primitive lattice having one formula unit or one lattice point or one unit of motif perunit cell and Non-primitive lattices having more than one lattice points or more than one unitof motif per unit cell

Figure 14 Primitive and Non-primitivelattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

115 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation

Symmetry elements underlying a point lattice (see the figure) Reflection reflection across a mirror plane

Rotation rotation around a crystallographic axis by an angle θ such as 360degθ isan integer of value 1 2 3 4 and 6 and is referred to as n -fold rotationInversion a point at xyz becomes its equivalent at (ndashx-y-z)Rotation-Inversion Rotation followed by inversion OR Rotation-ReflectionRotation followed by reflection

Figure 15 Basic symmetry operations in crystals

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Electroceramics General Bibliography

General Bibliography

The following are the books which can be referred for general reading More references are providedin each module

Recommended Reading

1 Physical Ceramics Principles for Ceramic Science and Engineering Y-M Chiang D PBirnie and W D Kingery Wiley-VCH

2 Introduction to Ceramics 2nd Edition W D Kingery H K Bowen D R Uhlmann Wiley3 Principles of Electronic Ceramics by L L Hench and J K West Wiley4 Electroceramics Materials Properties Applications by A J Moulson and J M Herbert Wiley5 Nonstoichiometry Diffusion and Electrical Conductivity in Binary Metal Oxides (Science amp

Technology of Materials) PK Kofstad John Wiley and Sons Inc

Supplementary Reading

6 Introduction to Solid State Physics C Kittel Wiley7 Electrical Properties of Materials L Solymer and D Walsh Oxford University Press8 Introduction of Solid State Physics NW Ashcroft and ND Mermin Brooks Cole9 Solid State Physics AJ Dekker Prentice-Hall

10 Transition Metal Oxides An Introduction to Their Electronic Structure and Properties PA CoxOxford University Press

11 Basic Solid State Chemistry AR West Wiley12 Non-stoichiometric Oxides O Toft Soslashrensen Academic Press13 Dielectrics and Waves AR von Hippel John Wiley and Sons14 Feynman Lectures on Physics Volume 1-3 RP Feynman Addison Wesley Longman15 Materials Science and Engineering A first course V Raghavan Prentice Hall of India16 Materials Science And Engineering An Introduction WD Callister Wiley

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Module 1 Structure of Ceramic Materials Introduction

In this module we will first review the structure and bonding in the materials in general followed by abrief discussion on how atoms pack together in the solids and what are the types of intersticespresent in various structures Then we would briefly delve into the types of bonding with reference tothe nature of materials Together this information will form the basis for structures in ceramicmaterials which are typically bonded with a mix of ionic and covalent bonding Subsequently wewould discuss the structure of ceramic materials with purely covalent bonding followed by ratherdetailed description of ceramic materials with ionic bonding These are essentially based on packing ofanions closed packed forms where cations fill the interstices

The Module contains

Brief Review of Structure of Materials

A Brief Review of Bonding in Materials

Packing of Atoms in Metals

Interstices in Structures

Ionically Bonded Ceramic Structures

Compounds based on FCC Packing of ions

Other Cubic Structures

Orthogonal Structures

Structures based on HCP packing of ions

Summary

Suggested Reading

Materials Science and Engineering WD Callister Jr Wiley

Physical Ceramics Principles for Ceramic Science and Engineering Y-MChiang D P Birnie and W D Kingery Wiley-VCH

Introduction to Ceramics W D Kingery H K Bowen D R Uhlmann Wiley

Fundamentals of Ceramics Michael Barsoum McGraw Hill

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

11 Brief Review of Structure of Materials

In the following sections we will quickly look at the concepts of lattice unit-cell and crystalstructures which will be useful to understand the crystal structures of common ceramic compounds

111 Point Lattice

In a point lattice the following characteristics are obeyed

There is a periodic arrangement of points in space (Figure 11(a))

In addition each point must have identical neighbourhood(Figure 11(b))

Lack of regular arrangementNo periodicityNon-identical neighbourhood

Regular arrangementPeriodicityIdentical neighbourhood of each point

Figure 11(a) Unit-cellrepresentation

>

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Figure 11(b) Schematics of arrangement of points inspace

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

112 Unit Cell

A unit cell is the smallest repeatable unit in a point lattice (Figure 12)

Choice of unit cell shape is not unique

Figure 12 Representation of a point lattice and anunit cell

Unit-cell parameters for a 3-D unit cells

axis lengths a b and c

anglesa szlig and γ

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

113 Motif and Crystal Structure

Crystal structure a combination of motif and point lattice

Motif is defined as a unit or a pattern For a crystal it can be an atom an ion or a group ofatoms or ions or a formula unit or formula units Often it is also called as Basis

When motif replaces points in a periodic point lattice it gives rise to what is called as acrystal with a defined structure

Figure 13 Formation of a periodic crystal structure

>

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

114 Types of Lattice

Lattice can further be classfied into two types

Primitive lattice having one formula unit or one lattice point or one unit of motif perunit cell and Non-primitive lattices having more than one lattice points or more than one unitof motif per unit cell

Figure 14 Primitive and Non-primitivelattices

>

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

115 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation

Symmetry elements underlying a point lattice (see the figure) Reflection reflection across a mirror plane

Rotation rotation around a crystallographic axis by an angle θ such as 360degθ isan integer of value 1 2 3 4 and 6 and is referred to as n -fold rotationInversion a point at xyz becomes its equivalent at (ndashx-y-z)Rotation-Inversion Rotation followed by inversion OR Rotation-ReflectionRotation followed by reflection

Figure 15 Basic symmetry operations in crystals

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramic Materials Introduction

In this module we will first review the structure and bonding in the materials in general followed by abrief discussion on how atoms pack together in the solids and what are the types of intersticespresent in various structures Then we would briefly delve into the types of bonding with reference tothe nature of materials Together this information will form the basis for structures in ceramicmaterials which are typically bonded with a mix of ionic and covalent bonding Subsequently wewould discuss the structure of ceramic materials with purely covalent bonding followed by ratherdetailed description of ceramic materials with ionic bonding These are essentially based on packing ofanions closed packed forms where cations fill the interstices

The Module contains

Brief Review of Structure of Materials

A Brief Review of Bonding in Materials

Packing of Atoms in Metals

Interstices in Structures

Ionically Bonded Ceramic Structures

Compounds based on FCC Packing of ions

Other Cubic Structures

Orthogonal Structures

Structures based on HCP packing of ions

Summary

Suggested Reading

Materials Science and Engineering WD Callister Jr Wiley

Physical Ceramics Principles for Ceramic Science and Engineering Y-MChiang D P Birnie and W D Kingery Wiley-VCH

Introduction to Ceramics W D Kingery H K Bowen D R Uhlmann Wiley

Fundamentals of Ceramics Michael Barsoum McGraw Hill

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

11 Brief Review of Structure of Materials

In the following sections we will quickly look at the concepts of lattice unit-cell and crystalstructures which will be useful to understand the crystal structures of common ceramic compounds

111 Point Lattice

In a point lattice the following characteristics are obeyed

There is a periodic arrangement of points in space (Figure 11(a))

In addition each point must have identical neighbourhood(Figure 11(b))

Lack of regular arrangementNo periodicityNon-identical neighbourhood

Regular arrangementPeriodicityIdentical neighbourhood of each point

Figure 11(a) Unit-cellrepresentation

>

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Figure 11(b) Schematics of arrangement of points inspace

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

112 Unit Cell

A unit cell is the smallest repeatable unit in a point lattice (Figure 12)

Choice of unit cell shape is not unique

Figure 12 Representation of a point lattice and anunit cell

Unit-cell parameters for a 3-D unit cells

axis lengths a b and c

anglesa szlig and γ

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

113 Motif and Crystal Structure

Crystal structure a combination of motif and point lattice

Motif is defined as a unit or a pattern For a crystal it can be an atom an ion or a group ofatoms or ions or a formula unit or formula units Often it is also called as Basis

When motif replaces points in a periodic point lattice it gives rise to what is called as acrystal with a defined structure

Figure 13 Formation of a periodic crystal structure

>

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

114 Types of Lattice

Lattice can further be classfied into two types

Primitive lattice having one formula unit or one lattice point or one unit of motif perunit cell and Non-primitive lattices having more than one lattice points or more than one unitof motif per unit cell

Figure 14 Primitive and Non-primitivelattices

>

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

115 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation

Symmetry elements underlying a point lattice (see the figure) Reflection reflection across a mirror plane

Rotation rotation around a crystallographic axis by an angle θ such as 360degθ isan integer of value 1 2 3 4 and 6 and is referred to as n -fold rotationInversion a point at xyz becomes its equivalent at (ndashx-y-z)Rotation-Inversion Rotation followed by inversion OR Rotation-ReflectionRotation followed by reflection

Figure 15 Basic symmetry operations in crystals

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

11 Brief Review of Structure of Materials

In the following sections we will quickly look at the concepts of lattice unit-cell and crystalstructures which will be useful to understand the crystal structures of common ceramic compounds

111 Point Lattice

In a point lattice the following characteristics are obeyed

There is a periodic arrangement of points in space (Figure 11(a))

In addition each point must have identical neighbourhood(Figure 11(b))

Lack of regular arrangementNo periodicityNon-identical neighbourhood

Regular arrangementPeriodicityIdentical neighbourhood of each point

Figure 11(a) Unit-cellrepresentation

>

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Figure 11(b) Schematics of arrangement of points inspace

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

112 Unit Cell

A unit cell is the smallest repeatable unit in a point lattice (Figure 12)

Choice of unit cell shape is not unique

Figure 12 Representation of a point lattice and anunit cell

Unit-cell parameters for a 3-D unit cells

axis lengths a b and c

anglesa szlig and γ

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

113 Motif and Crystal Structure

Crystal structure a combination of motif and point lattice

Motif is defined as a unit or a pattern For a crystal it can be an atom an ion or a group ofatoms or ions or a formula unit or formula units Often it is also called as Basis

When motif replaces points in a periodic point lattice it gives rise to what is called as acrystal with a defined structure

Figure 13 Formation of a periodic crystal structure

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

114 Types of Lattice

Lattice can further be classfied into two types

Primitive lattice having one formula unit or one lattice point or one unit of motif perunit cell and Non-primitive lattices having more than one lattice points or more than one unitof motif per unit cell

Figure 14 Primitive and Non-primitivelattices

>

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

115 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation

Symmetry elements underlying a point lattice (see the figure) Reflection reflection across a mirror plane

Rotation rotation around a crystallographic axis by an angle θ such as 360degθ isan integer of value 1 2 3 4 and 6 and is referred to as n -fold rotationInversion a point at xyz becomes its equivalent at (ndashx-y-z)Rotation-Inversion Rotation followed by inversion OR Rotation-ReflectionRotation followed by reflection

Figure 15 Basic symmetry operations in crystals

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Figure 11(b) Schematics of arrangement of points inspace

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

112 Unit Cell

A unit cell is the smallest repeatable unit in a point lattice (Figure 12)

Choice of unit cell shape is not unique

Figure 12 Representation of a point lattice and anunit cell

Unit-cell parameters for a 3-D unit cells

axis lengths a b and c

anglesa szlig and γ

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

113 Motif and Crystal Structure

Crystal structure a combination of motif and point lattice

Motif is defined as a unit or a pattern For a crystal it can be an atom an ion or a group ofatoms or ions or a formula unit or formula units Often it is also called as Basis

When motif replaces points in a periodic point lattice it gives rise to what is called as acrystal with a defined structure

Figure 13 Formation of a periodic crystal structure

>

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

114 Types of Lattice

Lattice can further be classfied into two types

Primitive lattice having one formula unit or one lattice point or one unit of motif perunit cell and Non-primitive lattices having more than one lattice points or more than one unitof motif per unit cell

Figure 14 Primitive and Non-primitivelattices

>

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

115 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation

Symmetry elements underlying a point lattice (see the figure) Reflection reflection across a mirror plane

Rotation rotation around a crystallographic axis by an angle θ such as 360degθ isan integer of value 1 2 3 4 and 6 and is referred to as n -fold rotationInversion a point at xyz becomes its equivalent at (ndashx-y-z)Rotation-Inversion Rotation followed by inversion OR Rotation-ReflectionRotation followed by reflection

Figure 15 Basic symmetry operations in crystals

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

>

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

112 Unit Cell

A unit cell is the smallest repeatable unit in a point lattice (Figure 12)

Choice of unit cell shape is not unique

Figure 12 Representation of a point lattice and anunit cell

Unit-cell parameters for a 3-D unit cells

axis lengths a b and c

anglesa szlig and γ

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

113 Motif and Crystal Structure

Crystal structure a combination of motif and point lattice

Motif is defined as a unit or a pattern For a crystal it can be an atom an ion or a group ofatoms or ions or a formula unit or formula units Often it is also called as Basis

When motif replaces points in a periodic point lattice it gives rise to what is called as acrystal with a defined structure

Figure 13 Formation of a periodic crystal structure

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

114 Types of Lattice

Lattice can further be classfied into two types

Primitive lattice having one formula unit or one lattice point or one unit of motif perunit cell and Non-primitive lattices having more than one lattice points or more than one unitof motif per unit cell

Figure 14 Primitive and Non-primitivelattices

>

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

115 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation

Symmetry elements underlying a point lattice (see the figure) Reflection reflection across a mirror plane

Rotation rotation around a crystallographic axis by an angle θ such as 360degθ isan integer of value 1 2 3 4 and 6 and is referred to as n -fold rotationInversion a point at xyz becomes its equivalent at (ndashx-y-z)Rotation-Inversion Rotation followed by inversion OR Rotation-ReflectionRotation followed by reflection

Figure 15 Basic symmetry operations in crystals

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

113 Motif and Crystal Structure

Crystal structure a combination of motif and point lattice

Motif is defined as a unit or a pattern For a crystal it can be an atom an ion or a group ofatoms or ions or a formula unit or formula units Often it is also called as Basis

When motif replaces points in a periodic point lattice it gives rise to what is called as acrystal with a defined structure

Figure 13 Formation of a periodic crystal structure

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

114 Types of Lattice

Lattice can further be classfied into two types

Primitive lattice having one formula unit or one lattice point or one unit of motif perunit cell and Non-primitive lattices having more than one lattice points or more than one unitof motif per unit cell

Figure 14 Primitive and Non-primitivelattices

>

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

115 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation

Symmetry elements underlying a point lattice (see the figure) Reflection reflection across a mirror plane

Rotation rotation around a crystallographic axis by an angle θ such as 360degθ isan integer of value 1 2 3 4 and 6 and is referred to as n -fold rotationInversion a point at xyz becomes its equivalent at (ndashx-y-z)Rotation-Inversion Rotation followed by inversion OR Rotation-ReflectionRotation followed by reflection

Figure 15 Basic symmetry operations in crystals

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

>

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

114 Types of Lattice

Lattice can further be classfied into two types

Primitive lattice having one formula unit or one lattice point or one unit of motif perunit cell and Non-primitive lattices having more than one lattice points or more than one unitof motif per unit cell

Figure 14 Primitive and Non-primitivelattices

>

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

115 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation

Symmetry elements underlying a point lattice (see the figure) Reflection reflection across a mirror plane

Rotation rotation around a crystallographic axis by an angle θ such as 360degθ isan integer of value 1 2 3 4 and 6 and is referred to as n -fold rotationInversion a point at xyz becomes its equivalent at (ndashx-y-z)Rotation-Inversion Rotation followed by inversion OR Rotation-ReflectionRotation followed by reflection

Figure 15 Basic symmetry operations in crystals

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

115 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation

Symmetry elements underlying a point lattice (see the figure) Reflection reflection across a mirror plane

Rotation rotation around a crystallographic axis by an angle θ such as 360degθ isan integer of value 1 2 3 4 and 6 and is referred to as n -fold rotationInversion a point at xyz becomes its equivalent at (ndashx-y-z)Rotation-Inversion Rotation followed by inversion OR Rotation-ReflectionRotation followed by reflection

Figure 15 Basic symmetry operations in crystals

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

>

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

116 Crystal Systems

As you can see now the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point

However as one starts defining various shapes we come up with seven categories called ascrystal systems in which all possible unit cells shapes would fit provided space filling criteriais fulfilled

Seven crystal systems are shown below

Crystal system and latticeparameters

Minimumsymmetryelements

Cubic

a = b =c

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

>

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 16 Seven crystal systems

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

>

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

117 Bravais Lattices

Taking seven crystal systems and symmetry elements into account Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below

Cubica=b=c

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 17 Fourteen Bravais lattices

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

>

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Brief Review of Structure in Materials

118 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W H Miller ) ascribed to determine various planes and directions While planes are determined little empirically directions are nothing but vectors

1181 Crystallographic Planes

Identification of various faces seen on the crystal

( hkl ) for a plane or hkl for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(11)

ha kb and cl are the intercepts of the plane on x y and z axes

a b c are the unit cell lengths

h k l are integers called Miller indices and the plane is represented as (h k l)

Any negative indices in Miller indices of a plane is written with a bar on top such as

1182 Directions

These are basically atomic directions in the crystal

Miller indices are [ u v w ] for a direction or lt u v w gt for identical set of directionswhere u v w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

>

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Figure 18(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction

Example [231] direction would be

13 intercept on cell a-length

12 intercept on cell b-length and

16 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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of directions in the form lt uvw gt

Figure 18(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal If you are not sure then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics

A Brief Review of Bonding in Materials

12 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials For basics of bonding you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms ie how todetermine the equilibrium distance bond energy and fundamental properties like youngrsquos modulusBonding in materials can be divided in two categories

Primary bondingSecondary bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

121 Primary Bonding There are three types of primary bonding mechanisms metallic covalent and ionic bonding

Figure 19 Metallic Bonding

1211 Metallic bonding

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds good malleability high electrical and thermal conductivity Mostmetals such as Ni Fe Cu Au Ag etc exhibit this kind of bonding

1212 Covalent Bonding

In this bonding atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si C Ge and SiC and gases like methane

Figure 110 Schematic of covalent bonding

1213 Ionic Bonding

This bonding occurs due to large differences in the electronegativities of two elements forexample in NaCl MgO etc

This type of bonding typically leads to high bond energies high bond strength high modulusbrittle nature generally low thermal and electrical conductivities making them excellentinsulators

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Figure 111 Schematic of ionic bonding

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

122 Secondary Bonding

It arises from the interaction between charge dipoles

1221 Fluctuating Dipoles

Observed in gases like hydrogen

Figure 112 Secondary bonding due to fluctuating dipoles

1222 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 113 Secondary bonding due to permanentdipoles

Examples are materials like polymers

Figure 114 Secondary bonding inpolymers

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics A Brief Review of Bonding in Materials

123 Bonding Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm large E and small a

)

Example MgO ndash 1000 kJ

mol Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example Si - 450 kJmol Tm-

1410oC

Directional

(Typically SemiconductorsPolymers and some Ceramics)

Metallic

Variable

Large Tungsten (W)

Small Mercury (Hg)

Moderate Al 68 kJmol Tm~

670oC

Nonndashdirectional (Metals)

Secondary

Smallest

Characterized by low Tm low

E and large a

Directional

Interndashchain (Polymer)

Interndashmolecular

Symbols Tm Melting point a Coefficient of thermal expansion E Elastic modulus

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Packing of Atoms in Metals

13 Packing of Atoms in Metals

In solids we consider atoms as hard incompressible spheres which can be packed in variousforms First we will see how atoms pack in metals

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures Some metals are a little loosely packed inthe form of body-centered cubic structure Very rarely atoms pack in metals in the form ofsimple cubic structure

131 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure

Structure contains only one atom per unit-cell

Figure 115 Simple cubic structure

132 Body Centered Cubic or BCC Structure

Many metals like W Fe (room temperature form) possess BCC structure

Contains 2 atoms per unit-cell

Figure 116 BCC Structure

One of the important parameters of interest is packing factor determining how loosly or densely astructure is packed by atoms

Packing Factor Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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If r is the atomic radii in these structures then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

133 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other

Figure 117 Closed packing of atoms in FCCHCPmetals

ABC ABC ABC stacking leads to the formation of cubic closed packed (CCP) or face centered

>

>

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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cubic (FCC) structure which has higher symmetry than other structures The closed packed A B Cplanes are (111) planes in the structure

AB AB AB stacking leads to hexagonal closed packed (HCP) structure The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure

Figure 118 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 074

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Interstices in Structures 14 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100 there are empty spaces inside which are called as interstices

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger

141 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 119 Interstices in a FCCstructure

So by simple geometry you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them

rtet = 0225 rrtet = 0225 r

142 Interstices in BCC Structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Figure 120 Interstices in a BCCstructure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

15 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials In most ionicallybonded materials there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors mostceramics such as NaCl MgO BaTiO3 Fe3O4 etc are predominantly ionically bonded Covalent

bonding as we saw in preceding sections arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure In contrastionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as PaulingrsquosRules which we will see later in this moduleIn this section we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

151 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond) Si Ge andSiC

For example in diamond the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms Thus the unit cell of diamond contains a total of 8 atoms

The structure is typically called as diamond cubic structure

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality

One unit-cell consists of two FCC motifs one at (0 0 0) and another at (frac14 frac14 frac14)

What it means is that there are two FCC unit-cells of C intermingled into each otherwith origin of one at (000) and another at (frac14frac14frac14)

In case of compounds FCC lattice can be formed by one type of atom and remaining atomsusually from the same group occupy half of the tetrahedral sites

Figure 121 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms

Figure 121 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell This is because the tetrahedral site size in a normal FCC unit cell is 0225r while inthis structure the size of the atom sitting at the site is much larger ie same size as thebase lattice atom(Self Evaluation)

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Structure of Covalent Ceramics

152 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different

Graphite has a layered structure where in each layer carbon atoms are sp2 hybridized andthey make a hexagonal pattern However the bonding between individual layers is Van derWalls type of bonding That is why Graphite is a soft material and is used as a lubricant

Figure 122 Structure ofGraphite

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

16 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities Hence when these ions are brought together they form a very strong ionicbond

Typically since anions are bigger in size than cations anions tend to form the base latticeand cations fill in the intersticesHowever it is not so simple As there is an involvement of two different types of ions to forma crystal structure there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure These rules are called Paulingrsquos rules

Based on these rules typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them Fortunately most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions As a result we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Ionically Bonded Ceramic Structures

161 Paulingrsquos Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) ie (rcra) Also another point to note is that the ionic radius of each ion is also

dependent on its co-ordination

Ligancy orCoordination

number

Range ofRadius Ratio

(rcra)

Configuration

2 00-0155 Linear

3 0155-0225 Triangular

4 0225-0424 Tetrahedral

6 0414-0732 Octahedral

8 0732-10 Cubic

12 10 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule)

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations This is especially true for solids with smaller cations and cations

with bigger charges eg Ti4+ and Zr4+ For example in SiO2 due to +4 charge on Si atoms

corner linking of tetrahedrons is preferred

In a crystal containing different cations those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3

The number of essentially different kinds of constituents in a crystal tends to be small Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment There may be two or three types of polyhedra such as tetrahedra oroctahedra but there will not be many different types (Rule of parsimony)

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics

Ionically Bonded Ceramic Structures

162 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot at least whether the charge is neutral and stoichiomteric or not Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination ie

In a stoichiometric and charge neutral solid the bond strengths of cations must be equal tothose of anions Alternatively you can work out bond strength of one ion and from this youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases the common valence state

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics

Compounds based on FCC Packing of Ions

17 Compounds based on FCC Packing of ions

In these structures typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases although in some cases some other co-ordination may be preferred Here wewill discuss the following structures

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

171 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure

Cations (M) fill the octahedral sites

100 occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion

Radius ratio rcra is typically between 0414 - 0732 with some exceptions

Examples of ceramic materials with such structure as NaCl MgO NiO FeO etc

Figure 123 Schematic of structure of Rocksalt structuredcompounds

Lattice type FCC and motif will be M at 0 0 0 and X at frac12 0 0

Four formula units per unit cell

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow

Compound rc (nm) ra (nm) rcra

NaCl 0102 0181 0564

MgO 0072 0140 0514

SrO 0118 0140 0842

NiO 0069 0140 0492

FeO 0078 0140 0557

MnO 0053 0140 0378

PbO 0119 0140 085

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations ie Na+ =

Valence of anions

which is the valence of chlorine and hence proposed packing is appropriate

The octahedrons shares at the edges If there was corner sharing of polyhedra co-ordinationnumber will be 2 for anions which wonrsquot maintain the stoichiomentry as you can verify usingbond-strength relationship

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction

Figure 124 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled this would bring cationsquite close to each other leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion

The complete structure can be viewed as two FCC lattices one of Na and another of Clinterpenetrating into each other

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

172 Antifluorite (A2X) and Fluorite (AX2) Structures

1721 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination Anions 8 Cations 4

Chemical formula M2X

Example Li2O Na2O K2O

Radius ratio (rcra) 0225-0414

Examples r(Li+) 0059 nm r(Na+) 0099 nm r(O-) 014 nm

Figure 125 Antifluorite structure

Lattice type FCC

Motif ndash X 0 0 0 M -

Four formula units per unit cell

In this structure in many cases although rcra ratio predict an octahedral co-ordination

tetrahedral coordination is preferred to fulfill the stoichiometry requirements In turn anionsare cubic coordinated by cations (CN 8)

The structure shows corner sharing of tetrahedra

1722 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

ExampleUO2 ZrO2 CaF2 CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites

Figure 126 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion

Figure 127 A more appropriate representation of fluoritestructure

Co-ordination number Cations - 8 Anions - 4

Lattice FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14 frac34 frac34 frac34

Examples of ionic radii of a few ions

U4+ 01 nm Zr4+ 0084 nm Ce4+ 0097 nm O2- 014 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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The structure as you can also see has a large void in the center of unit cell made by cations

These empty spaces make such oxides good ionic conductors which is useful in applicationssuch as energy storage eg batteries

For having some fun with the structure we can also draw as projection of this material on(110) plane Here you can see the row of empty octahedral sites along [110]-direction

Figure 128 View of (110) plane of fluorite structure

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

173 Zinc Blende (MX) Structure

MX type compounds also called as sphalerite structured compounds based on a mineralname of sphalerite

Mostly oxides and sulphides follow this structure Examples are ZnO ZnS BeO etc

Some covalently bonded materials and compounds have similar structure such as GaAs SiCBN You can also visualize diamond also having similar structure with both anion and cationbeing of same type

Typically compounds with tetrahedral co-ordination assume this structure

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices

Due to stoichiometry half of the tetrahedral sites are filled

Compounds with radius ratio 0225-0414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio especiallycovalently bonded compounds

ExamplesZn2+ - 006 nmBe2+ - 0027 nmO2- - 014 nm S2-- 0184 nm

Figure 129 Zinc Blende or Sphaleritestructure

Coordination numbers M - 4 X - 4

Lattice type FCC

Motif M ndash 0 0 0 X ndash frac14 frac14 frac14

4 formula units per unit cell

Tetrahedra are shared at corners

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Compounds based on FCC Packing of Ions

174 Spinel Structure

Formulae ndash (A2+)(B3+)2O4 or AB2O4 or AOB2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites ie18th of tetrahedral and frac12 of theoctahedral sites are occupied

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2times2times2 so it is a big structure consisting of 32 oxygen atoms 8 A atoms and 16 B atoms

Depending on how cations occupy different interstices spinel structure can be Normal orInverse

1741 Normal Spinel

Chemical formula (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4 FeAl2O4 CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4

In this structure all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites

Apply bond strength rule to verify the stoichiometry

Cations - A2+ - 24 B3+ - 36

Oxygen valence = (24x1)+ (36x3) = 2

Figure 130 Schematic of spinel structure

1742 Inverse Spinel B(AB)O4

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Chemical formula (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4

Most ferrite follow this structure such as Fe3O4 (or FeOFe2O3) NiFe2O4 CoFe2O4 etc

In this structure frac12 of the B3+ ions occupy the tetrahedral sites and remaining frac12 B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line)

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Module 1 Structure of Ceramics Other Cubic Structures

18 Other Cubic Structures

There are a few structures which appear as if they are based on cubic closed packing ofanions However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions Here we discuss the perovskite structure based onABO3 structure CsCl structure and ReO3 structure

181 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3 SrTiO3 PbTiO3 etc which happen to be

technologically very useful compounds as we will see in later modules

In ABO3 structured compounds A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms

Bond strength check

Cation Ba 212 = 16 and Ti 46 = 23

Oxygen valence = 16 x Coordination number by Ba + 23 x coordination number by Ti

Figure 131 Perovskite structure

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

Objectives_template

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

Objectives_template

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

Objectives_template

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Figure 139 Layer by layer filling in Wurtzite

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

Objectives_template

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

Objectives_template

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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Figure 132 Polyhedra model of perovskitestructure

Lattice type Primitive Cubic (NOT FCC)

Motif A ion - 0 0 0 B ion ndash frac12 frac12 frac12 O ion - frac12 frac12 0 0 frac12 frac12 frac12 0 frac12

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra

An important parameters about perovskites is the their ldquoTolerance Factor (t)rdquo which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence the face diagonal of the unit cell would be times the unit-celllength as result t = 1 for a perfect cubic perovskite

However due to variations in ionic radii of various ions many perovskites show deviationsfrom t = 1 and may not even have a cubic structure Deviations from t = 1 signify the level oflattice distortion

For example BaTiO3 has cubic structure only above ~120degC while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT

Perovskites can also have various combinations of ionic valence such as

eg A2+B4+O4 BaTiO3 PbTiO3 CaTiO3 SrTiO3 etc

eg A3+B3+O4 LaAlO3 LaGaO3 BiFeO3 etc

Mixed Perovskites

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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A2+(B2+13B5+

23)O3 eg Pb(Mg13Nb23)O3

A2+(B3+12B5+

12)O3 eg Pb(Sc12Ta12)O3

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

Objectives_template

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Figure 139 Layer by layer filling in Wurtzite

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

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Figure 145 Structure on 10-10 plane in LiNbO3

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

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Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

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Module 1 Structure of Ceramics Other Cubic Structures

182 ReO3 Structure

Stoichiometry MX3

Lattice type Primitive cubic

Atomic Positions M- 0 0 0 X - frac12 0 0 0 frac12 0 0 0 frac12

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3 UO3 WO3

Used for gas sensing and electrochromic applications

Figure 133 ReO3 structure and polyhedramodel

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Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

Objectives_template

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Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

Objectives_template

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Figure 139 Layer by layer filling in Wurtzite

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

Objectives_template

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Figure 141 View of 1010 plane ofCorundum

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

Objectives_template

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

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Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture44_4htm[5252012 32456 PM]

Module 1 Structure of Ceramics Other Cubic Structures

183 CsCl Structure

MX type compounds parent compound being CsCl

Examples Halides such as CSCl AgI AgBr etc

Radius ratio governs cubic co-ordination of both cations and anions

Lattice type Primitive cubic lattice

Motif Anions (X) 0 0 0 Cations (M) frac12 frac12 frac12

One formula unit per unit cell

Figure 134 (a) CsCl structure (b) Ball-stickmodel

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Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

Objectives_template

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Figure 136 Atomic coordination inYBCO

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_2htm[5252012 32456 PM]

Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Figure 139 Layer by layer filling in Wurtzite

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Figure 141 View of 1010 plane ofCorundum

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

Objectives_template

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One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

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Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

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Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture44_5htm[5252012 32456 PM]

Module 1 Structure of Ceramics Orthogonal Structures

19 Orthogonal Structures

Many superconductors follow the structures which are perovskite based ie the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most casesThe examples are superconductors such as YBa2Cu3O7 ferroelectrics such as Bi4Ti3O12

etc In some other compounds such as La-Sr-Cu-O the structure is composed of alternatingperovskite and rocksalt structure units Such a representation makes it easy to understandthem

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly

191 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig 135) which also contains perovskite units

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure This gives a transition temperature of~92 K below which the compound has zero electrical resistance ie is a superconductor

Y3Cu3+3O9rarrYBa2Cu3+

3O8rarr YBa2Cu2+2Cu3+O7-x

Figure 135 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit (DM Smyth PP1-10 in ceramic superconductors IIResearch Update 1988 MFYan Ed The American Ceramic Society1988)

Here Cu coordination is of interest

Cu2+ atoms have four-fold coordination along Cu-O chains

Cu3+ atoms have five-fold coordination in the Cu-O planes

Objectives_template

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Figure 136 Atomic coordination inYBCO

Objectives_template

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_2htm[5252012 32456 PM]

Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Figure 139 Layer by layer filling in Wurtzite

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

Objectives_template

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Figure 141 View of 1010 plane ofCorundum

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

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Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

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Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture44_5htm[5252012 32456 PM]

Figure 136 Atomic coordination inYBCO

Objectives_template

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Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_2htm[5252012 32456 PM]

Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Figure 139 Layer by layer filling in Wurtzite

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Figure 141 View of 1010 plane ofCorundum

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture44_6htm[5252012 32456 PM]

Module 1 Structure of Ceramics Orthogonal Structures

192 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound LaO

and one perovskite structured compound LaCuO3 and can also be written as LaOLaCuO3

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K

Figure 137 (a) Origin of La2-xSrxCuO4 structure shown in(b) as two perovskite and cells

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_2htm[5252012 32456 PM]

Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Figure 139 Layer by layer filling in Wurtzite

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Figure 141 View of 1010 plane ofCorundum

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_2htm[5252012 32456 PM]

Module 1 Structure of Ceramics

Structures based on HCP Packing of Ions

110 Structures based on HCP packing of ions

Similar to FCC packing of anions many ceramic structures are also based on another type of closed packing of anions ie hexagonal closed packed (HCP) In this category we will look at the following structures

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Figure 139 Layer by layer filling in Wurtzite

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Figure 141 View of 1010 plane ofCorundum

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1101 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO SiC

Co-ordination of both anions and cations is 4 as in Sphalerite structured compounds

Anions form an HCP lattice with frac12 of the tetrahedral sites occupied by cations

The only difference to Sphalerite structure is that here anions pack in the form ofABCABChellip stacking

Figure 138 Wurtzite structure and polyhedralmodel

As you can notice all the tetrahedrons point in one direction ie along the c-axis of the unit-cell and they share the corners

Lattice type Primitive HCP

Motif M 0 0 0 and X and

The filling of structure can be seen below

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Figure 139 Layer by layer filling in Wurtzite

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Figure 141 View of 1010 plane ofCorundum

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_3htm[5252012 32457 PM]

Figure 139 Layer by layer filling in Wurtzite

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Figure 141 View of 1010 plane ofCorundum

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1102 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound

Other examples are compounds like Cr2O3 Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry

Coordination numbers M 6 X 4

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices

Figure 140 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 23 rd filling of octahedral sites

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Figure 141 View of 1010 plane ofCorundum

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_4htm[5252012 32457 PM]

Figure 141 View of 1010 plane ofCorundum

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions 1103 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3

Other compounds which follow this structure are CdTiO3 CoTiO3 CrRhO3 FeRhO3 FeVO3 LiNbO3

MgGeO3 MgTiO3

This structure is very similar to Corundum or a - Al2O3

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer ie half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell

Hence the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

Coordination numbers both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations ie 2 Fe and 2 Ti

Bond strength rule gives correct oxygen valence

+ =2=Oxygen

valence

Figure 142 Layered filling of Ilmenite

>

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_5htm[5252012 32457 PM]

One unit-cell consists of six layers of oxygen ions

A side view of the structure on 10-10 plane as shown below shows the columns of cations along the c-

axis with 23rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site

Figure 143 Ilmenite structure on 10-10plane

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1104 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3

The parent compound LiNbO3 is ferroelectric in nature and hence is technologically

important

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field

Such materials are used in electro-optic devices

Figure 144 Atomic arrangement of a layer inLiNbO3 structure

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_6htm[5252012 32457 PM]

Figure 145 Structure on 10-10 plane in LiNbO3

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Module 1 Structure of Ceramics Structures based on HCP Packing of Ions

1105 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite

It is formed by quasi-HCP packing of anions

Half of the octahedral sites are filled by cations

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice

Anisotropic diffusion properties of cations are found in TiO2

Materials shows large and anisotropic refractive index and high birefringence

TiO2 is often used as pigments and is non-toxic

Figure 146 Structure of a layer of oxygen and Titanium inRutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_7htm[5252012 32457 PM]

Figure 147 Unit-cell of Rutile

Figure 148 Polyhedral model of Rutile

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

Objectives_template

fileC|Documents20and20Settingsiitkrana1Desktopnew_electroceramics_14may2012lecture55_8htm[5252012 32457 PM]

Module 1 Structure of Ceramics Summary

Summary

In ionic solids anions typically form the base lattice

Interstices can be completely or partially filled depending on the size of cations andstoichiometry

Paulingrsquos rules play an important role in structure determination and deviations lead tostructural distortions

Most ceramic compounds follow three types of common structures based on packing ofanions ie

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures