2Crystal Physics 2013

8/12/2019 2Crystal Physics 2013

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KGiSL INSTITUTE OF TECHNOLOGY

ENGINEERING PHYSICS I

UNI T I - CRYSTAL PHYSICS

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DEPARTMENT OF SCIENCE & HUMANITIES

INTRODUCTION TO CRYSTALLOGRAPHY

A crystal is a solid composed of atoms or other microscopic particles arranged in an orderlyperiodic array.

The three general types of solids: amorphous, polycrystalline and single crystal aredistinguished by the same periodicity over entire regions within the materials. Order in

amorphous solids is limited to a few molecular distances. In polycrystalline materials, thesolid is made-up of grains which have same periodicity over smaller regions separated byboundaries from regions of other periodicity or same periodicity with different orientation.

Single crystals have long-range order. Many important properties of materials are found to

depend on the structure of crystals and on the electron states within the crystals. At thebeginning of the study of crystals it was their external form which was related to the

physical properties. In this way only a limited success was achieved. Towards the middle oflast century a deeper understanding developed regarding the correlation of the structure ofcrystals and{mechanical, thermal, electrical and magnetic} properties of solids. This isprimarily due to the advances in the band theory of electron states and in the theory of

bonding in solids. This knowledge has led to the development of newer and better materials

for electrical, electronic and structural engineering. The study of crystal physics aims tointerpret the macroscopic properties in terms of properties of the microscopic particles of

which the solid is composed. The study of the geometric form and other physical propertiesof crystalline solids by using x-rays, electron beams and neutron beams constitute thescience of Crystallography or Crystal Physics.

Single Pyri te Crystal and Amorphous soli ds:


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Di stincti on between Crystal li ne soli ds and Amorphous soli ds

A crystalis a three-dimensional periodic arrangements of atoms in space.

Crystal structure- The arrangement of atoms, ions, or molecules in a crystal. Crystals are

solids having, in all three dimensions of space, a regular repeating internal unit of structure.

Space Lattice- A regular periodic 3 dimensional arrangement of points in space.

Unit cell- The smallest group of atoms or molecules whose repetition at regular intervals inthree dimensions produces the lattices of a crystal

Primitive cellUnit cell of the crystal which has one lattice point per cell. Eg: Cubic cell -Each atom is shared by 8 cells. So per cell there are (1/8 x 8) = 1 lattice point/cell= 1

atom/cell in a SC.

S. No. Crystalline solids Amorphous solids1 The internal arrangement of

particles is regular so they possess definite

and regular geometry

The internal arrangement of particles is irregular. Thus

they do not have any definite geometry

2. They have sharp melting points They do not have sharp melting points

3. There is regularity in the external form

when crystals are formed

There is no regularity in the external form when

amorphous solids are formed

4. Crystalline solids give a regular cut

when cut with a sharp edged knife

Amorphous solids give irregular

cut.

5. They have characteristic heat of fusion They do not have characteristic heat of fusion

6. Crystalline solids are rigid and their shape

is not distorted by mild distorting forces

Amorphous solids are not rigid. They are distorted by

bending or compressive forces

7 Crystalline solids are regarded as true

solids

Amorphous solids are regarded as super cooled liquids or

pseudo solids

8 Crystalline solids are anisotropic Amorphous solids are isotropic


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Basis Group of atoms or molecules identical in composition.

Apart from the fundamental translation symmetry, a lattice may also exhibit othersymmetries:Rotations by 2/n where n=1,2,3,4,6. Reflection in a plane through a lattice point.

Inversion, i.e. a rotation of + a reflection in a plane orthogonal to the rotation axis. Lattice

Point groups collection of symmetry operations which when applied about a lattice point,leaves the lattice invariant. No. of different (independent) possible lattice types can be


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grouped, based on symmetry operations, or their combinations into Bravais lattices. There

are 5 types in two dimensions and 14 ones in three dimensions.

Lattice parameters of an unit cell

The parameters that define a unitcell are:a, b, c unit cell dimensions along x, y, z respectively , , angles between b,c ();a,c (); a,b ()

Crystal systems and Bravais lattices

The number of different Bravais lattices is determined by symmetry considerations

Two dimensional space five Bravais latticesThree dimensional space 14 Bravais lattices grouped into seven crystal systems

The classification of crystal lattice is based1) On equality (or inequality) of the lengths a,b,c of the edges of the conventional

cell,2) On the angles , , between each pair of them

3) and on possible occurrence of additional lattice points at the centre of opposite faces

or at the center of the cell.


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S. No. Crystal System Lattice Parameters Bravais Lattice Example

1 Cubic a=b=c

===901. Simple2. Body centred3. Face Centred

PoloniumNa, Cu2O, Ag, Au,Cu

2 Tetragonal a=bc===90

1. Simple2. Body centred

TiO2, SnO2,KH2PO4

3 Orthorhombic abc== 90

1. Simple2. Body centred3. Face centred4. Base centred

C15H20O2, Sulphur,KNO3

4 Monoclinic abc===90

1. Simple2. Base centred

Gypsum, Na2SO4,FeSO4

5 Triclinic abc

90

1. Simple K2Cr2O7, CuSO4

6 Trigonal a=b=c

==901. Simple Calcite, Sb, Bi

7 Hexagonal a=bc== 90=120

1. Simple Quartz, Zn, Cd


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Crystal Planes

Within a crystal lattice it is possible to identify sets of equally spaced parallel planes. Theseare called lattice planes. In the figure density of lattice points on each plane of a set is thesame and all lattice points are contained on each set of planes.

The set of planes in 2D lattice

Miller Indices

Miller Indices are a symbolic vector representation for the orientation of an atomic planein a crystal lattice. It is defined as the reciprocals of the fractional intercepts which the plane

makes with the crystallographic axes. Miller indices are represented by a set of 3 integernumbers h,k and l given in brackets (hkl).

[ hkl ] represents direction perpendicular to the crystal plane

represented by (hkl).{ hkl } represents set of planes parallel to ( h k l ).

Procedure for finding Miller indices

Consider a crystal plane. Let us find its Miller indices as follows.

Step 1: Find intercepts of the plane along the coordinate axes X,Y, Z.

(The intercepts are measured as multiples of the fundamental vectors.)

Step 2: Take the reciprocal of these intercepts.Step 3: Reduce the reciprocals into whole numbers by multiplying each ofthem with their least common multiple. Let us represent these numbers as h, k and l.

Step 4: Write these integers within paratheses to get Miller indices.


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Example:

Step 1: Intercepts a, b, c are 1 1

Step 2: Reciprocals 1/1 1/1 1/

Step 3: Reduction 1 1 0

Step 4: (1 1 0)

Derivation of interplanar spacing of cubic crystal

The interplanar spacing or distance dhkl between adjacent planes of Miller indices (hkl) isdefined as the spacing between the first such plane and a parallel plane passing through the

origin.ais the lattice constant. Let h, k, and l are the Miller indices for a plane ABC. An

equation is to be derived for interplanar distanced of plane ABC from a plane parallel to it

and passing through o (origin) in terms ofh,k,land a. OD is the normal from O on plane

ABC. OD d = interplanar distance. The intercepts of the plane ABC on the three axis i.c.,X,

Y and Z axis are OA, OB and OC respectively.

Now, OA=a/h, OB=a/k,OC=a/l (i)

Also, cos =OD/OA= d/(a/h) =dh/a (ii)

cos = OD/OA= d/(a/k)= dk/a (iii)

cos =OD/OC = d/(a/l) = dl/a (iv)

Since, cos2+ cos2 + cos2 = 1. therefore

(dh/a)2+(dk/a)2+ (dl/a)2= 1 d2h2/a2 + d2k2/a2 + d2l2/a2= 1


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d2(h2+ k2 + l2) =a2

d2=a2 /(h2+ k2 + l2)

d= a2 /(h2+ k2 + l2)

This is the relation between interplanar spacing d, cube edge a and Miller indices (h k l )

Characteristics of an unit cell

Number of atoms per unit cell It is the number of atoms possessed by an unit cell.Nearest neighbour distance (2r)Distance between the centres

of nearest neighbouring atoms.

Atomic radius (r)Half the distance between the nearestneighbouring atoms.Coordination number (N)Number of equidistant nearest

neighbours

Atomic Packing Factor or Packing densityratio of the volumeOccupied by the atoms in an unit cell (v) to the total volume of

the unit cell (V).

Packing factor= v/ V

Crystal Structures

Simple Cubic (SC) Structur e

One atom at each of the eight Corners of the unit cell

Co-ordination number= 6

4-same layer1-top layer

1-bottom layer


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Nearest neighbour distance, 2r=a

Atomic radius, r= a/2

No. of atoms/cell=(1/8) x 8 = 1 (Primitive cell)

Packing factor= No. Of atoms/cell x volume of one atom

__________________________________

Total volume of the unit cell

=

= = 0.52 = 52%Example: Polonium

Body-centered Cubic (BCC) structure

Co-ordination number8 (Look at the atom at body centre)

No. of atoms/cell= [(1/8) x 8] + 1 = 2= corner atoms + body centered atom

Diagonal length= 4r3a = 4 r

Atomic radius r= (3 a/4)


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Lattice constant a = 4r/3

Packing factor= No. Of atoms/cell x volume of one atom

__________________________________


=

=

= 0.68 = 68%

Example: Tungsten, Chromium, Molybdenum

Face centred cubic (FCC) structure

Co-ordination number4+4+4=12

No. of atoms/cell= [(1/8) x 8] + [(1/2)x6] = 4

= corner atoms + Face centered atoms

Face diagonal AC=2 a=4r


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Atomic radius r=(2/4)a= a/(22)

Lattice constant, a= 22r

Packing factor= No. Of atoms/cell x volume of one atom__________________________________


=

=

= 0.74 = 74%

Example: Copper, Aluminium, Nickel, Gold

Hexagonal close packed structur e (hcp)

Co-ordination number3+6+3=12

No. of atoms/cell= [(1/6) x 6] + [(1/6)x6] + [(1/2)x2] + 3 = 6= corner atoms + central atoms (upper & lower) + body centred atoms


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Atomic radius 2r = a

r = a/2

Relation between c and a

Refer the above fig.

AA = AB COS 30 = a

AX = AA

AC2 = AX2 + CX2

CX2 = AC2 - AX2

CX2= a2 -

From fig. CX =2, {|2}2 =


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{|}2 =

Therefore, {|}2 = 1.633

Calculation of packing factor

Area of the base = 6 x area of the triangle ABO

Area of the base = 6 x1xABx OO = 3 x a x

x a

Area of the base =

Volume of the unit cell V = Area of the base x height

=

x c

Packing factor = No. Of atoms / unit cell x volume of one atom

__________________________________


P.F. = 3

=

= 0.74 = 74%


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Diamond Structure

It is formed by carbon atoms. Each carbon atom is surrounded by four other carbon atomssituated at the corners of regular tetrahedral by the covalent linkages. The diamond cubic

structure is a combination of two interpenetrating FCC sub lattices displaced along the body

diagonal of the cubic cell by 1/4th length of that diagonal. Thus the origins of two FCC sublattices lie at (0, 0, 0) and (1/4, 1/4,1/4).

The points at 0 and 1/2 are on the FCC lattice, those at 1/4 and 3/4 are on a similar FCC

lattice displaced along the body diagonal by one-fourth of its length. In the diamond cubic

unit cell, there are eight corner atoms, six face centred atoms and four more atoms. No. of

atoms contributed by the corner atoms to an unit cell is 1/88 =1. No. of atoms

contributed by the face centred atoms to the unit cell is 1/2 6 = 3. There are four moreatoms inside the structure. No.of atoms present in a diamond cubic unit cell is 1 + 3 + 4 =

8. Since each carbon atom is surrounded by four more carbon atoms, the co-ordination

number is 4.

ATOMIC RADIUS(R)

From the figure,in the triangle WXY,

XY2= XW2+ WY2

= +

XY2=

In triangle XYZ,

XZ2= XY2+ YZ2


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Since XZ = 2r,

(2)2 =

+

r =

Packing factor = No. Of atoms / unit cell x volume of one atom

__________________________________Total volume of the unit cell

=8 x 4 x 33 x 3

51a =16 = 0.34 = 34 %

Graphite Structure

Graphite is also a crystalline form of carbon. In graphite each carbon atom is covalently

bonded to three carbon atoms to give trigonal geometry. Bond angle in graphite is 120oC.Each carbon atom in graphite is sp2 hybridized. Three out of four valence electrons of each

carbon atom are used in bond formation with three other carbon atoms while the fourthelectron is free to move in the structure of graphite. Basic trigonal units unite together to

give basic hexagonal ring. In graphite these rings form flat layers. These layers arearranged in parallel, one above the other. These layers are held together by weak

vanderwaals forces only.These layers can slide one over another.Thus it is very soft. Fourth

electron of each carbon atom which spreads uniformly over all carbon atoms. Due to thisreason graphite conducts electricity parallel to the of its plane.


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METHODS OF CRYSTAL GROWTH

Growth of crystal ranges from a small inexpensive technique to a complex sophisticated

expensive process and crystallization time ranges from minutes, hours, days and to months.

Single crystals may be produced by the transport of crystal constituents in the solid, liquidor vapour phase. On the basis of this, crystal growth may be classified into three categoriesas follows,

Solid Growth - Solid-to-Solid phase transformation

Liquid Growth - Liquid to Solid phase transformation

Vapour Growth - Vapour to Solid phase transformation

Based on the phase transformation process, crystal growth techniques are classified as solid

growth, vapour growth, melt growth and solution growth. Several techniques have been

developed for each method of growth to achieve better control over the growth and suit to

the requirements of newer materials.

GROWTH FROM SOLUTION

Materials, which have high solubility and have variation in solubility with temperature can begrown easily by solution method. There are two methods in solution growth depending on

the solvents and the solubility of the solute. They are

1. High temperature solution growth

2. Low temperature solution growth

In high-temperature solutions, the constituents of the material to be crystallized are

dissolved in a suitable solvent and crystallization occurs as the solution becomes critically

supersaturated. The supersaturated may bepromoted by evaporation of the solvent, bycooling the solution or by a transport process in which the solute is made to flow from ahotter to a cooler region. The high temperature crystal growth can be divided into two

major categories:

1. Growth from single component system.2. Growth from multi component system.

This method is widely used for the growth of oxide crystals. The procedure is to heat the

container having flux and the solute to a temperature so that all the solute materials


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dissolve. This temperature is maintained for a soak period of several hours and then the

temperature is lowered very slowly.

Basic apparatus for solution growth

Hydrothermal growth

Hydrothermal implies conditions of high pressure as well as high temperature. Substances

like calcite, quartz is considered to be insoluble in water but at high temperature and

pressure, these substances are soluble. This method of crystal growth at high temperatureand pressure is known as hydrothermal method. Temperatures are typically in the range of

400 C to 600 C and the pressure involved is large (hundreds or thousands of

atmospheres).Growth is usually carried out in steel autoclaves with gold or silver linings. Depending on the

pressure the autoclaves are grouped into low, medium and high-pressure autoclaves. The

concentration gradient required to produce growth is provided by a temperature difference

between the nutrient and growth areas. The requirement of high pressure presents practical

difficulties and there are only a few crystals of good quality and large dimensions are grown

by this technique. Quartz is the outstanding example of industrial hydrothermal

crystallization. One serious disadvantage of this technique is the frequent incorporation of

OH

-

ions into the crystal, which makes them unsuitable for many applications.

GEL GROWTH

It is an alternative technique to solution growth with controlled diffusion and the growth

process is free from convection. Gel is a two-component system of a semisolid rich in liquid

and inert in nature. The material, which decomposes before melting, can be grown in this


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medium by counter diffusing two suitable reactants. Crystals with dimensions of several mm

can be grown in a period of 3 to 4 weeks. The crystals grown by this technique have highdegree of perfection and fewer defects since the growth takes place at room temperature.

GROWTH FROM MELT

All materials can be grown in single crystal form from the melt provided they melt

congruently without decomposition at the melting point and do not undergo any phase

transformation between the melting point and room temperature. Depending on the thermal

characteristics, the following techniques are employed.

1. Bridgman technique

2. Czhochralski technique

Bridgman technique

Bridgman method is based on crystal growth from a melt but a temperature gradient

furnace is gradually lowered and crystallization begins at the cooler end, fixed crystal and

changing temperature gradient.

Bridgman method involve controlled solidification of a stoichiometric melt of the material

to be crystallized in a temperature gradient.

Enables oriented solidification

Melt passes through a temperature gradient.

Crystallization occurs at a cooler end.


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This method benefit from seed crystals, predetermined orientation and controlled

atmospheres.

Advantages:

1. Relatively lower cost

2. Melt composition can be controlled during the growth.

Dis-advantages:

1. Growth rate is very low.

2. This technique cannot be used for materials which decompose before melting.

CZOCHRALSKI METHOD

This technique originates from pioneering work by Czochralski in 1917 who pulled single

crystals of metals. Since crystal pulling was first developed as a technique for growing single

crystals, it has been used to grow germanium and silicon and extended to grow a wide

range of compound semiconductors, oxides, metals, and halides. It is the dominant

technique for the commercial production of most of these materials.

Single crystal growth from the melt precursors

Crystal seed of material to be grown placed in contact with the surface of the melt.

Temperature of melt held just above melting point, highest viscosity, lowest vapour

pressure favors the crystal growth.

Seed gradually pulled out of the melt.

Melt solidifies on surface of the seed


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Melt and seed usually rotated counter clockwise with respect to each other to maintain

constant temperature and to facilitate uniformity of the melt during crystal growth,produces higher quality crystals and less defects.

Inert atomosphere often under pressure around growing & melt during crystal & melt to

prevent any materials loss and undesirable reactions like oxidation nitration etc.

The Czochralski crystal pulling technique is invaluable for growing many large single

crystals as a rod to be cut into wafers and polished for various applications like Si, Ge,

lithium niobate.

Advantages:

1. Relatively lower cost

2. Ability to produce larger wafer sizes as in float zone process.

Dis-advantages:

1. Melt is accumulated with dopants during the process.

2. Impurities or metals can dissolve from the crucible and built into the crystal.

Crystal growth from vapour phase

In this technique the material to be grown is supplied in the form of vapour. The powders

of the desired crystalline material are atomized by exposing it to an electric arc or hot


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flame. A seed crystal is brought somewhere near the melting point so that the arriving

atoms or molecules will have sufficiently high mobility on the growing surface.

At very low temperature many crystals in a very high state of purity has been grown from

the vapour phase.

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2Crystal Physics 2013