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Crystal Defects

By Dr.Srimala

The Structure of Matter

Liquid Crystal Image

Content

1.0 Perfect Crystal2.0 Processing, Microstructure and

Properties 3.0 Crystal defect

3.1 Vacancies and Interstitials3.2 Impurity Atoms3.3 Point Defects in Ionic Crystals3.4 Defect Complexes

4.0 Vacancies formation5.0 Divacancy6.0 Defects in the ionic compounds

6.1 Kroger-Vink notation6.2 Frenkel Defect6.3 Schottky defects

Crystal Defect

1.01.0 Perfect CrystalPerfect Crystal

A perfect crystal with every atom in the correct position does not exist.

Defect: imperfection or "mistake" in the regular periodic arrangement of atoms in a crystal

Defects, even in very small concentrations, can have a dramatic impact on the properties of a material.

Most materials properties are determined by the crystal defects present

“Crystals are like people: it is the defects in them which tend to make them interesting!” - Colin Humphreys

.

2.02.0 Processing, Microstructure and PropertiesProcessing, Microstructure and Properties

Processing is the manner in which the material is made. Processing is the manner in which the material is made. The processing determines the defects that are present. The processing determines the defects that are present.

Since materials properties are determined by crystal Since materials properties are determined by crystal defects there is a correlation between the processing of defects there is a correlation between the processing of the materials and the materials properties.the materials and the materials properties.

2

3.03.0 Crystal defectCrystal defect

Point defects

plane defects

bulk defects

Pure metal will contain numerous defects in its crystal structure such as

1. A foreign atom substitutionally placed2. A foreign atom interstitially placed3. Parent atoms interstitially placed4. Vacancies

5. Twins and stacking faults6. Grain boundaries

7. Dislocations8. Voids9. inclusions

3.13.1 Vacancies and InterstitialsVacancies and Interstitials

On the left are vacancies and on the right are interstitials

a. Schottky defecta. Schottky defect b. Interstitial (impurity) defectb. Interstitial (impurity) defect

3.23.2 Impurity AtomsImpurity Atoms

The colored atoms are impurity atoms. They are atoms of a different element.On the left the impurity atom sits between the black atoms and is called an interstitial impurity.On the right the impurity atom replaces or substitutes for a black atom and is called a substitutional impurity

Cotterill 1985

3.33.3 Point Defects in Ionic CrystalsPoint Defects in Ionic Crystals

•• Ionic crystals unlike other solids are Ionic crystals unlike other solids are made up of charged ions. made up of charged ions.

•• Point defects in ionic crystal are charged. Point defects in ionic crystal are charged. •• Charge neutrality must always be Charge neutrality must always be

maintained.maintained.

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3.43.4 Defect ComplexesDefect Complexes

FD FD -- Frenkel defect Frenkel defect -- cation cation hops from lattice site to hops from lattice site to interstitialinterstitial

SD SD -- Schottky defect Schottky defect -- anion anion and cation vacancies.and cation vacancies.

Cotterill 1985

To maintain charge neutrality several point defects are created

Line DefectsEdge dislocation

Migration aids ductile deformation

Fig 10-4 of Bloss, Crystallography and Crystal Chemistry.© MSA

Line DefectsScrew dislocation (aids mineral growth)

Fig 10-5 of Bloss, Crystallography and Crystal Chemistry. © MSA

Plane DefectsLineage structure or mosaic crystal

Boundary of slightly mis-oriented volumes within a single crystal

Lattices are close enough to provide continuity (so not separate crystals)

Fig 10-1 of Bloss, Crystallography and Crystal Chemistry. © MSA

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Plane DefectsDomain structure (antiphase domains)

Also has short-range but not long-range order

Fig 10-2 of Bloss, Crystallography and Crystal Chemistry. © MSA

Crystal DefectsPlane Defects

Stacking faultsCommon in clays and low-T disequilibriumA - B - C layers may be various clay types (illite, smectite,

etc.)

ABCABCABCABABCABCAAAAAABAAAAAAAABABABABABCABABAB

Why do defects form?

The introduction of defects increases entropy ΔS and decreases free energy ΔG

A minimum value for ΔG is reached for an optimum concentration of defects

The structure with defects is more stable

)( STHG

4.04.0 Vacancies formationVacancies formation

• Will these vacancies remains in the lattice as stable defect?

• Or will they migrate to the surface restoring the perfection of the lattice?

Perfect crystal-all lattice sites are occupied-no vacancies are presentWe introduce vacancies- by removing atom from the perfect lattice.

5

• We analyze this problem by considering the effect of the vacancies upon Gibbs free energyof the lattice

• Let say we have N lattice sites, how does the free energy change as a function of the n, no of vacancies?

• If G decrease, then the vacancies thermodynamically stable

)( )( )(

cv SSnTHnGSnTHnperfectGGG

Let the free energy change due to vacancies be

H = energy per vacancy

SV = vibrational entropy per vacancy

Sc = configurational entropy of the entire crystal

There’s many different geometrical ways in which the vacancies may arrange on the lattice (mixing, randomizing of n vacancies in N aotms.

Ω

Ω

Therefore, we use symbol -is simply the number of distinguishably different ways that n vacancies may be arranged on N lattice sites.

To determine lets consider an analogous problem:

Given 100 lattice sites and the three atoms Fe, Co, and Ni, in how many ways can we put these atoms on the 100 lattice points?

Fe may go 100 different sitesFor each of these, Co may go 99 waysFor each of these, Ni may go 98 ways

Answers = 1000 x 99 x 98 = 100!/(100-3)!=970200

What is the answers to the same question if all three atoms are Fe atoms?

For the previous sets, we know that Fe may go 3 different waysFor each of these, Co may go 2 waysFor each of these, Ni may go 1 ways

Therefore, for the previous sets we have counted 3! permutation of three atoms. If the three atoms are the same these 3! permutation are indistinguishable and we must dividethe equation by 3!

Answers = 100!/[(100-3)!3!]=161700

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Vacancies are indistinguishable from one another, so by analogy we have

Where N is the total number of lattice sitesn is the number of vacancies

!)!(!

nnNNΩ

]ln)ln()(ln[)](ln)ln()(ln[

X largeor ln!ln

!)!(!

ln

c

c

c

nnnNnNNNkSnNnnnnNnNNNNkS

fXXXX

nnNNΩ

ΩkS

For the configurationally entropy

Since N and n are large apply Stirling’s approximation

After some algebraic manipulation we obtain

Vibrational entropy

'ln3v v

vkS

Is related to the ways in which the energy levels in the solid are occupied.

Where v’ is the final frequency of the atoms around the vacancy and v is their original frequency.

The vacancy tends to increase the vibrational amplitudeSo that v/v’ >1 and SV is positive. Therefore, the total entropy change associated with introducing vacancies must be positive cv SSn

• Now we can evaluate each term in equation below

• Figure 1 is a plot of the terms based on the equation

• A maximum decrease in free energy is obtained at this minimum, so this point represent the equilibrium condition and the value of n at this minimum is the equilibrium number of vacancies, ne

)( cv SSnTHnG H is the energy required to remove the atom to the surface or interface

H

cv SSnT

G

n

ne

Energy terms raises the free energy but the total free energy actually drops at first due to increase entropy upon introducing vacancies

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0dn

GdTherefore at equilibrium,

]ln)ln()1()()([

]}ln)ln()(ln{ln[

0)(

nnnnN

nNnNkTSTH

nnnNnNNNdndkTSTH

dndSTSTH

TSSTnHndnd

dnGd

v

v

cv

cv

X

][ ].[

][ ].[

][

ln

v

nNn

RSeksp

RTHeksp

nNn

kSeksp

kTHeksp

nNn

kS

kTHeksp

nNn

nNnkTSTH

v

v

v

v

No of vacancy

No of atomsk=1.38x10-23J/K

Vacancy fraction

= Vacancy concentration

For 1 mole vacancies

R=8.314

ln XV

- 1/T

Slope = ΔHV / k

0

ΔSV / k

Arrhenius plot of vacancies fraction versus temperature

Intercept gives the vibrational entropy of vacancies

slope is proportional to the enthalpy of formation of vacancy

The value of typically range from 80 to160kJ/mole and

from 1.0 to 2.0.

Example Cu: = 1.5, = 113kJ/mole, the number of atoms per volume from the density is 5x1022 sites/cm3. Therefore the number of vacancies per cubic centimeter is then found to be

Temperature 1000oC, vacancies/cm3 =5x1018

200oC, = 2x103

H

kSv

kSv

H

8

This results illustrates two interesting things

1. A perfect metal is not thermodynamically stable. The free energy is lowered when vacancies are added and the equilibrium number of vacancies quiet large although it is still a small fraction of the total number of lattice sites

2. The number of vacancies is temperature dependent.

Exercise 1.1

Take the energy to form a mole of vacancies in Cu as 20,000 calories and the vibrational entropy as 1.0k per vacancy, where k is Boltzman’s constant. Compute the number of vacancies per cubic centimeter of Cu at 20oC and at its melting point (1083oC). Take the density of Cu as 8.94 g/cm3. JMR for Cu= 64

320

38

151522

1522

3

22

o

3

22

322

23

v

m vacancy/c1038.1 1356K,For

m vacancy/c1084.2

1038.31038.3104122.8

1038.3104122.8

]0.1[ ].)293(314.8

106.83[104122.8

29320TFor

]0.1[ ].314.8

106.83[104122.8

][

atom/cm 104122.8N

1002.664

8.94N Cu,For

vacancy/0.1Se83.6kJ/molJ/mole x4.1842000

cal/mole 2000

xn

xn

nxxxn

xnx

n

ekspxekspnx

n

KC

kkeksp

Txeksp

nxn

kS

kTHeksp

nNn

x

xx

kHH

v

v

v

Answer 1 .1Answer 1 .1

5.05.0 DivacancyDivacancy

Defects may occur in combinations in an elemental crystal.The most common of these is the divacancy, which is a pair

of adjacent vacant lattice sites

by two steps1. The separate formation of two vacancies from a perfect

crystal2. The formation of the divacancy configuration from two

separated single vacancies

How to visualize the formation of a

divacancy?

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The enthalpy change associated with the first process is simply 2HvThe enthalpy change associated with second process is Hint (interaction enthalpy)

Thus the enthalpy of formation of a divacancy can be written

The same argument can be used to express excess entropy

Therefore for divacancy

intΔΔ2 HHH vvv

T

kHvveksp

kSekspX vv

vv

intΔΔ2 SSS vvv

]/Δ[exp ]/Δexp[}{ ]/Δ[exp ]/Δexp[.

]}Δ][/1[exp]Δ][/1exp[{ ]ΔΔ2][/1[exp.

]ΔΔ2][/1exp[

intintv

intint

int

int

kTHkSXXkTHkS

HkTSkHHkT

SSkX

vv

vvXvv

v

vvv

2

2

Divacancy (cont….)

Exercise 1.2

Suppose the interaction parameters for divacancies are about 10 percent of the values of corresponding single defect parameters;

and

Calculate the equilibrium concentration of divacancies at 600K and at 1300K

vHH Δ1.0Δ int

vS SΔ1.0Δ int

moleJHv /95800Δ KmoleJSv ./85.8Δ

Answer 1 .2Answer 1 .2

kJ 02.182kJ 58.91.058.92Δ1.0Δ2

ΔΔ2 int

vv

vvvv

vvv

HHHH

HHH

J/mole.K 815.1685.81.085.82

Δ1.0Δ2ΔΔ2 int

vv

vvvv

vvv

SSSS

SSS

7

15

10666.3819.14

1300K,At

100744.1467.34

489.36 022.2 600.3148

182020314.8815.16

600K,At

T

xekspX

xekspX

ekspekspX

ekspekspX

kHvveksp

kSekspX

vv

vv

vv

vv

vvvv

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6.06.0 Defects in the ionic compoundsDefects in the ionic compounds

The anion (more electronegative) component in a ionic compound is typically a nonmetallic element and designated as X

The cation (more electropositive) component is metallic element and designated as M

A vacant lattice site is designated as V

Atoms or ions can occupy cation (M) sites, anion (X) 0r interstitial (i) sites

To facilitate the description, Kroger-Vinknotation are widely accepted

Kroger-Vink notation

X=the entity occupying the defect site (M,X,V or substituitional elements)

Y= type of site occupied (M,X,i)Z=the excess charge associated with the site

(• = positive, ’ = negative, X = neutral)

ZYX

6.16.1 KrogerKroger--Vink notationVink notation

Kroger-Vink notation

symbol Excess charge

defects

VM’’ -2 vacancy on M sublattice

VX ̇ ˙ +2 vacancy on X sublattice

Mi˙˙ +2 M atom in interstitial site

Xi’’ -2 X atom in interstitial site

Mx˙˙˙˙ +4 M atom on X site

Defect of a compound with composition MX and normal valance of M as +2, X as -2

Kroger-Vink notation

symbol Excess charge

defects

XM’’’’ -4 X atom on M site

VMVX 0 Divacancy on M and X sites

LM ̇+1 Solute cation L with +3 charge on M site

YX˙ +1 Solute cation Y with -1 charge on M site

e’ -1 Free electron

h ˙ +1 Electron holes

Cont….

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Important clues

Al2O3-VAl’’’ MgO – VMg’’

1. Superscript (x) is used if entity occupying the site carries

Example: Al2O3 normal sites would be AlAlx Oo

x

Cromium ion on a cation site in alumina CrAlx

because it carries same charge

2. Cation vacancies- leaves excess negative chargeAnion vacancies-leaves excess positive charge

Example: Cation vacancy in KCl – Vk’

6.2 Frenkel Defect

A frenkel defect is formed on the cation sublattice by removing an M ion from a normal M site and placing it in an interstitial site.

A frenkel defect is called an intrinsic defect because it can formed without any interaction with the surroundings of the crystal.

VM’’

Mi˙˙

Consider a crystal MX in which the normal valance of Mis +2 and X is -2. If this crystal contains frenkel defectsderived from cation sites, four distinct entites exist in suchcrystal

Kroger-Vink notation for frenkel defect can be written asMM=Mi˙˙ +VM’’since all X atoms remain on anion sites while M atomsare distributed over cation and interstitial sites.Furthermore each frenkel defect consists of one vacancyand one interstitial atom.

Mi˙˙, VM’’, MMx , XX

x

2 fd

2

fdfd

.

expexp .

XVXMXVXMXVX

kTH

kS

MXVX

MiM

iM

iM

]2

exp[]2

exp[ fffd kT

HkSX

The condition at equilibrium

Since the number of vacancies same as number of interstitial,

Therefore, equilibrium concentration of frenkel defects is

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6.36.3 Schottky defectsSchottky defects

In an MX crystal, a schottky defect consists of a vacant cation site and a vacant anion site.

For a crystal with formulaMXNull=Vx˙+VM’which describe the formation of two vacancies in a region that is initially a prefect crystal. In this context, null means the initially defect free crystal

The formation of Schottky defect does not disturb the electrical neutrality of the crystal-intrinsic defect

Equilibrium concentration of schottky defects is

]2

sdexp[]2

exp[ sdsd kTH

kS

X