Structures and X-Ray Diffraction

Lectures 4-6

Last revised 4/13/10 5:13 am

Learning Objectives

• Know the definition of a crystal• Recognize lattice and motif in 2D and 3D

crystals• Classify structures by Bravais Lattice • Be able to draw, describe, and do

calculations with the cubic crystal structures.

Learning Objectives

• Calculate likely CNs of cations and anions, using r/R

• Calculate whether a compound is likely to form a glass, using Zachariasen’s rules

• Determine the likely unit cell of a material, given its x-ray powder diffraction pattern

• Determine stoichiometry, density, APFs, CNs of all ions, given a unit cell

Crystal = Lattice + Motif

• Lattice = mathematical framework of identical points, defined by the lattice parameters

• Motif (or Basis) = the pattern that is associated with each lattice point.– Will be one or more atoms, or ions, in a

crystal

a

b

Will these same lattice vectors work?

a

b

All lattice points are equivalent

a

b

All lattice points are equivalent

ab

All lattice points are equivalent

ab

All M.C. Escher works (c) 2006 The M.C. Escher Company - the Netherlands. All rights reserved.

All M.C. Escher works (c) 2006 The M.C. Escher Company - the Netherlands. All rights reserved.

Hard sphere model:

• Atoms touch, don’t overlap• Atoms have a size, defined by their radius• Bonds are non-directional

– bonding: what does it mean to form a bond?• What types of materials can we easily

think about with this model?• 1) metals• 2) ionic solids

Close Packed Planes

• Maximizes the number of NN

• More bonding = lower energy

• Two patterns for Layering– ABAB– ABCABC

Metal Structures

What determines the number of nearest neighbors in metals?

Metallic Bonding• Atoms can bond at any angle because

electrons are delocalized.• CN is not limited by orbitals, so metals will

maximize CN to minimize energy.– Close packed structures are common

• FCC and HCP– BCC is another common structure

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Hexagonal Close Packed crystal structure (HCP), Callister

FCC crystal structure, Callister

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Body Centered Cubic (BCC) unit cell

7 Crystal Systems, Callister

3 Cubic Bravais Lattices within the Cubic crystal system

• Simple Cubic• Body Centered Cubic (BCC)• Face Centered Cubic (FCC)

• a=b=c, alpha=beta=gamma=90 degrees

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Simple Cubic unit cell

Covalent Bonding

• What determines the number of nearest neighbors?

Covalent Bonding

• Maximize number of bonds to minimize energy

• Number of bonding orbitals limits coordination number

Ionic Bonding

• What determines the number of nearest neighbors?

Ionic Bonding

Maximize the number of bondsIons have different sizesOverall charge neutrality must be

maintained

Together, these place restrictions on the types of structures that can form.

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• For stability, the central ion must be in contact with the surrounding ions, otherwise their mutual charge repulsion will be too great.

• To reduce energy, the central ion will bond with the greatest number of surrounding ions possible.

Stability of ions in solids

Radius Ratio Rules Callister

Radius Ratio Rules

Callister Chapter 12 Shackelford Ch. 2

Purely geometrical

Applies to Non-Directional Bonding

Example: Predict the structure of NaCl

Atomic radius Ionic radiusNa 0.186 nm 0.098 nm

Cl 0.107 nm 0.181 nm

Radius Ratio Coordination Number< 0.155 2

0.155 – 0.225 30.225 – 0.414 40.414 – 0.732 6

0.732 – 1.0 81 12

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The NaCl Crystal Structure

Close Packed Planes

Interstices: The spaces between the atoms

– Octahedral

– Tetrahedral

In some ionic structures, the cations fit in interstices formed in close-packed anion planes

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NaCl has an FCC structure with 1 Na+ and 1 Cl- ion per lattice site.

Calculate the theoretical density of NaCl

r(Na+) = 0.102 nm

r(Cl-) = 0.181 nm

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Draw Axes &

Write down positionsOf the ions

Example: Predict the structure of GaAs

Radius Ratio Coordination Number< 0.155 2

0.155 – 0.225 30.225 – 0.414 40.414 – 0.732 6

0.732 – 1.0 81 12

Atomic radius Ionic radiusGa 0.135 nm 0.062 nm

As 0.125 nm 0.040 nm

Example: SiO2

• Assuming that this is an ionic structure, what is the coordination number of Si 4+ in SiO2 ?

Example: Predict the structure of SiO2

Radius Ratio Coordination Number< 0.155 2

0.155 – 0.225 30.225 – 0.414 40.414 – 0.732 6

0.732 – 1.0 81 12

Atomic radius Ionic radiusSi 0.117 nm 0.040 nm

O 0.060 nm 0.140 nm

Example: SiO2

What is the coordination number of Si 4+ in SiO2 ?

What is the CN of O 2-?

First, check to see whether SiO2 is ionic or not• O has an electronegativity of 3.5• Si has an electronegativity of 1.8• What is the difference in electronegativity? 1.7• This means that Si-O bonds are about 50%

ionic, and 50% covalent.

Are we justified in using r/R?

• O has an electronegativity of 3.5• Si has an electronegativity of 1.8

The difference in electronegativity is 1.7This means that Si-O bonds are 55% ionic,

and 45% covalent.

What if SiO2 had covalent bonding?

• If the bonds were completely covalent, what type of bond would they be? (what would be their shape and configuration?)

• What orbitals would be used to bond?

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CN of the larger species

• Charge Neutrality: number of positive charges = number of negative charges

• CN of Si4+ = 4• This means that there are 4 O2- around

each Si4+

• How many Si4+ are there around each O2-?• (What is the CN of O2-?)

iClicker: How many Si4+ are there around each O2-?

A. 1B. 2C. 3D. 4E. 6

Problem-Solving approach

• How did you approach this problem?

Draw a pictureWrite down an equationGuess and check

Guess and Check Method

• What if the CN(O2-) were 4?

• There would be 4 Si4+ around each O2-.• Check charge neutrality:

4(4+) =? 2(2-)This will not work!

• Should the next guess be higher or lower?

• CN(O2-) = 2• How can the SiO44- tetrahedra be

attached?

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Silicate CeramicsMost common elements on earth are Si & O

SiO2 (silica) crystal structures are quartz, crystobalite, & tridymite

The strong Si-O bond leads to a high melting material (1710ºC)

Si4+

O2-

Adapted from Figs. 12.9-10, Callister 7e.

crystobalite

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Silica Glass

• The angle between tetrahedra is variable

• Rings and chains can organize themselves in 3 dimensions in many non- regular ways

• This ability is what makes SiO2 so good at forming a glass network.

Amorphous Structures

SiO2 can crystallize into an ordered structure if given enough time

SiO2 can also form a glass if cooled quickly

Glass = amorphous

SRO and LRO

Glasses have Short Range Order (SRO)Can predict the number of NN

Glasses do not have Long Range Order (LRO)No lattice vectorsNo long range predictability of positions of individual

ions or atoms

Zachariasen’s Rules for Glass Formation1. Cation coordination number is 3 or 4 (small)

(triangles and tetrahedra)

2. Oxygen coordination number is 1 or 2 BRIDGING or NON-BRIDGING oxygen

3. Oxygen polyhedra share corners, not edges or faces

(Some faces can be shared as long as other corners are shared.)

4. For 3D networks, at least 3 corners must be shared–

There will be some non-bridging oxygen as well.

Structure Determination

How can we experimentally determine whether a material is crystalline or amorphous?

What is the fundamental difference between crystalline and amorphous structures?

X-Ray Diffraction

Crystal Structure

• Crystal structure = Lattice + Motif

Crystal Planes

Crystal Planes

x

y

(010)d010

Crystal Planes

x

y

(100)d100

Crystal Planes

x

y

(110)d110

Crystal Planes

x

y

(120)d120

222 lkhadhkl

++=

d-spacing of (hkl) planes in a cubic system

X-Ray Diffraction Reflection Rules for (hkl) planes

Example XRD Pattern

(111)

(200)

(220)

(331)(222)

Indices for points, directions and planes

• Points = 3 fractional coordinates within a unit cell

• Directions = [u v w] square brackets!– No fractions

• Planes = (h k l) Parentheses– No fractions

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Density

Mass per volume

The unit cell can be repeated to form the entire structure, so the density of the unit cell is the same as the density of the entire solid.

Vm

=ρ

Atomic Density

• Number of atoms per volume– Calculate atomic density for the three cubic

structures

Atomic Packing Factor

• Volume of atoms per total volume– Calculate APF for the three cubic structures

Calculate the Density of Cr

• r(Cr) = 0.125 nm• Cr = 52.00 amu

• What else do we need to know?

• Actual density is 7.19 g/cm3

Density of Ni

Ni has an FCC structure. Calculate the theoretical density of Ni.

What additional information do we need?

Density of Ni

r(Ni)=0.125 nmNi = 58.67 amu

8.90 g/cc

More Unit Cell Practice

• Identify the Bravais Lattice• How many of each atom per unit cell• What is the chemical formula

(stoichiometry)• How many nearest neighbors does each

have?• What is the volume of the unit cell?

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NOTE: This was an animation showing how the motif is madeOf 2 carbon atoms related by a ¼ ¼ ¼ translation. All of the atomsAre carbon, the blue color was just to distinguish the two positions.

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Graphite from Callister