Crystal DefectsBy-

Saurav K. Rawat

(Rawat DA Greatt)1

CRYSTAL DEFECT

Crystalline solids exhibits a periodic crystal structure. The positions of atoms or molecules occur on repeating fixed distances , determined by the unit cell parameters. However ,the arrangement of atoms or molecules in most crystalline materials is not perfect. The regular patterns are interrupted by crystal defects. The equilibrium structure at any temperature other than 0k will have some disorder or imperfection .The presence of defects in a crystal is a thermodynamic requirement for stability.

A perfect arrangement gives the crystal the highest possible negative value of lattice energy and hence apparently such a structure should be most stable.

Stability, however, is determined by free energy G, which for a solid may be taken as A, the Helmholtz free energy expressed as

A = U – TS ……..Eq(1) Further, we can write lattice energy UL in place of internal energy

U. Acc. Eq.(1) A = UL will be minimum only when T = 0K, i.e., when the contribution of entropy towards free energy

is 0. Entropy of an disordered (imperfect) crystal is higher than that of

an ordered crystal. Thus, at any temperature other than 0K, two factors will contribute

to free energy

i) By making the internal energy UL more negative.

ii) The contribution from entropy ST will try to minimize the same by making the crystal more imperfect thus, increasing S.

PROPERTIES OF DEFECTS

STRUCTURAL PROPERTIES

ELECTRONIC PROPERTIES

CHEMICAL PROPERTIES

SCATTERING PROPERTIES

THERMODYNAMIC PROPERTIES

CLASSIFICATION OF DEFECTS

0 -DIMENSIONAL DEFECTS

POINT DEFECTS VACANCY INTERSTITIAL SUBSTITUTION

1-DIMENSIONAL DEFECT

LINE DEFECTS EDGE DISLOCATION SCREW DISLOCATION

2-DIMENSIONAL DEFECTS

PLANE DEFECTS GRAIN BOUNDARY STACKING FAULTS

POINT DEFECT

Point defects are defects that occur only at or around a single lattice point. Atoms in solid posses vibrational energy, some atoms have sufficient energy to break the bonds which hold them in equilibrium position. Hence, once the atoms are free they give rise to point defects.

POINT DEFECTS

VACANCY

Missing atom from an atomic site. Atoms around the vacancy displaced. Tensile stress field produced in the vicinity.

Vacancydistortion of planes

VACANCY DEFECTAt any finite temperature, some of the lattice points normally occupied by metals ions are vacant giving rise to vacancy defects.The no. of vacancies is determined by :- Temperature The total no. of metals atoms The average energy required to create a vacancy

Then,

n = f (N,Ev,T )The concentration of vacancies increases with :- Increasing temperature Decreasing activation energy The creation of a defect requires energy, i.e., the process is endothermic. In most cases Diffusion (mass transport by atomic motion can only occur because of vacancies.This defect results in decrease in density of the substance.

VACANCY CONCENTRATION IN A METAL

For low concentration of defects, a quantitative relation between n,N,Ev and T may be obtained through the use of Boltzmann statistics is justified since the lattice sites are distinguishable although the metal ions occupying them are not. Also, the interaction between defects is neglected.

The total no. of different ways (W) in which a metal ion can be removed is given by

W = N(N-1)(N-2)....(N-n+1)/n! = N!/(N-n)!n!

Further, the entropy change ( S) associated with the process is equal to kB ln W,

Where kB is the Boltzmann constant.

Therefore,

S= kB ln[N!/(N-n)!n!] ……..Eq.(1)

The logarithmic factorial terms of Eq.(1) can be simplified using the stirling’s approximation, which states that for large values of x,

ln x! = x ln x-x

Therefore ,

ln[N!/(N-n)!n!] =ln N! – ln(N-n)!-ln n! =N ln N-N-{(N-n) ln (N-n)-(N-n)}-n ln n + n =N ln N- (N-n) ln (N-n)-n ln n Now, the Helmholtz free energy change ( A) is given by A = E – T S = nEv – T S Where E represents the total internal energy change .At

equilibrium, (δ A/ δn)T =0 Now, δ/ δn[N ln N-(N-n ) ln (N-n)-n ln n] = - δ/ δn{(N-n) ln (N-n)+ n ln n} = -[δ{(N-n) ln (N-n)}/ δn + δ(n ln n)/ δn] = -{(N-n)/(N-n)(-1)+ ln (N-n) x (-1)+n/n+ln n}

= -{-1- ln (N-n) + 1 + ln n} = ln (N-n/n) Therefore, δ(T S)/ δn = kBT ln (N-n/n)

And, [δ( A)/ δn]T = 0 = Ev–kBT ln (N-n/n) Where the partial differentiation is carried out with respect to the

variable ‘n’, since the total no. of lattice sites/metal atoms is a constant.

Ev = kBT ln (N-n/n) Or (N-n/n)= exp(Ev/kBT) If the no. of vacancies is much smaller than the total no. of atoms,

i.e., if n<<N,then N-n=N So that n=N exp(-Ev/kBT) ……….eq.(2) Usually, the no. of vacancies (n) is a very small fraction of the no. of

metal ions/atoms, so that Eq.(2) can be used to calculate the number of fraction of vacancies so long as the temperature is far removed from the melting point.

SCHOTTKY DEFECT

A vacancy is created in a metal by the migration of a metal atom/ion to the surface; similarly, in a binary ionic crystal represented by A+B- ,a cation anion pair ( A+B-) will be missing from the respective lattice position ,the energy absorbed during the creation of SCHOTTKY DEFECT is more than compensated by the resultant disorder in the lattice/structure.

Schottky defect are dominant in closest-packed structures such as ROCK SALT (NaCl) structure .

SCHOTTKY DEFECT CONCENTRATION IN AN IONIC CRYSTAL

Let ‘n’ be the number of Schottky defects produced by removing nA+ cation and nB- anions, N the total no. of cation-anion pairs and Es the average energy required to create a Schottky defect. Then, the total no. of different ways in which ‘n’ Schottky defects can be produced will be given by

W = [N!/(N-n)!n!]2

The exponent 2 appears in the above relation because there are n cations as well as n anions so that the no. of different ways in which a cation or anion can be removed is given by

N!/(N-n)!n!

And the total no. of different ways by the compound probability, which is given by the product of the two. Therefore,

S = 2kB ln [N!/(N-n)!n!]

Using Stirling’s approximation, we get

ln[N!/(N-n)!n! = N ln N-(N-n) ln (N-n)-n ln n

Further, the Helmholtz free energy change

A = E – T S

=nEs –2kBT{N ln N-(N-n)ln (N-n)-n ln n}

At equilibrium,

[δ A/ δn]T = 0 = Es – 2kBT ln [(N-n)/n]

Or

Es = 2kBT ln [(N-n)/n]

Or

[(N-n)]= exp (Es/2kBT)

If n<<N, then N-n=N so that

n = N exp (-Es/2kBT) ……….Eq(1)

Eq(1) is an adequate expression for the determination of the no. of Schottky defects present in binary ionic crystal such as NaCl and MgO at ordinary temperatures, at which the interaction between defects could be negligible since their no. is relatively small.

The no. of Schottky defects depends on :

The total no. of ion-pairs, i.e., mass of the ionic crystal. The average energy required to produce a schottky defect. Temperature .

The fraction of Schottky defects increases exponentially with increasing temperature

INTERSTITIAL DEFECT

If an atom or ion is removed from its normal position in the lattice and is placed at an interstitial site, the point defect is known as an INTERSTITIAL . An example of interstitial impurity atoms is the carbon atoms that are added to iron to make steel. Carbon atoms, with a radius of 0.071nm, fit nicely in the open spaces between the larger (0.124nm) iron atoms.

SELF – INTERSTITIAL

• If the matrix atom occupies its own interstitial site, the defect is called self interstitial.• Self-interstitials in metals introduce large distortions in the surrounding lattice.

self-interstitial

distortion of planes

Let Ei be the average energy required to displace an atom from its regular lattice position to an interstitial,

Ni be the no. of interstitial sites and

N the total no. of atoms.

The no. of different ways in which ‘n’ interstitial defects can be formed is given by,

W = N!/(N-n)!n! Ni!/(Ni-n)!n!

Therefore , the associated changes in entropy and Helmholtz free energy will be,

S = kB ln [N!/(N-n)!n! Ni!/(Ni-n)!n!]

And

A = nEi – kBT ln [N!/(N-n)!n! Ni!/(Ni – n)!n!]

Using Stirling’s approximation, we get

ln [N!/(N-n)! n! Ni!/(Ni-n)! n!]

=ln N!/(N-n)! n! + ln Ni!/(Ni-n)! n!

= N ln n+ Ni ln N-(N-n) –(Ni-n) ln (Ni-n) – 2n ln n

At equilibrium,

[δ( A)/ δn]T = 0 = Ei – kBT ln {(N-n)(Ni-n)/n2}

Since the quantities N and Ni are independent of n . Hence ,

(δN/ δn)T = 0 = (δNi/ δn)T

If N>>n and Ni>>n , then

Ei/kBT = ln (NNi/n2)

= ln (Nni) – 2 ln n

Therefore,

n = (NNi)1/2 exp ( - Ei/2kBT)

FRENKEL DEFECT

• When an atom leaves its regular site and occupy nearby interstitial site it give rise to FRENKEL DEFECT. • A crystal with FRENKEL DEFECT will have equal no. of vacancies and interstitials.• These defects are prevalent in open structures such as those of silver halides .•For eg. AgI , CaF2

FRENKEL DEFECT CONCENTRATION

Let Ef be the average energy required to displace a cation from its normal lattice position to an interstitial,

Ni be the no. of interstitial sites and

N the total no. of cation ( in a binary ionic crystal, A+ B-).

The no. of different ways in which ‘n’ Frenkel defects can be produced is then given by,

Therefore, the associated changes in entropy and Helmholtz free energy are given by,

And

W = N!/(N-n)!n! Ni!/(Ni-n)!n!

S = kB ln [N!/(N-n)!n! Ni!/(Ni-n)!n!]

A = nEf – kBT ln [N!/(N-n)!n! Ni!/(Ni – n)!n!]

The Stirlings formula gives,

At equilibrium,

Since the terms containing constant quantities like N and Ni disappear on differentiation.

If N>>n and Ni>>n, then

Ef / kBT = ln(NNi / n2)

Usually, the no. of frenkel defect is small at low temperature.

ln [N!/(N-n)! n! Ni!/(Ni-n)! n!] =ln N!/(N-n)! n! + ln Ni!/(Ni-n)! n! = N ln N+ Ni ln Ni -(N-n) ln (N-n)–(Ni-n) ln (Ni-n) – 2n ln n

[δ( A)/ δn]T = 0 = Ef – kBT ln {(N-n)(Ni-n)/n2}

= ln (NNi) – 2 ln nTherefore, n = (NNi)1/2 exp ( - Ef/2kBT)

FRENKEL & SCHOTTKY DEFECT

Many crystals show combination of defect i.e., one type of defect is associated with an equivalent number of defect of another type. Such combination defects are particularly important in ionic crystals for maintaining overall charge neutrality of the crystal. Two such defects of particular importance are the FRENKEL & SCHOTTKY DEFECTS.

SUBSTITUTIONAL DEFECT

Impurities occur because materials are never 100% pure. In the case of an impurity, the atom is often incorporated at a regular atomic site in a crystal structure.This is neither a vacant site nor is the atom on an interstitial site and it is called a SUBSTITUTIONAL defect.

Substitution of a lattice ion by an impurity ion of different charge will disturb the charge balance.

In order to maintain the charge neutrality of the lattice, such substitution may be accompanied either by the creation of a lattice vacancy or by the change of the oxidation state of an ion in the lattice.

Substitution of Ag+ by Cd2+ in Agcl leads to the creation of a lattice vacancy.

Substitution of Ni2+ by Li+ in NiO lattice gives rise to the change of the oxidation state of an ion in the lattice .

SUBSTITUTIONAL DEFECT

Substitution of Ag+ by Cd2+ in Agcl leads to the creation of a lattice vacancy.

Ag+ Cl- Ag+ Cl-

Cl- Cd2+ Cl- Ag+

Ag+ Cl- Cl-

Cl- Ag+ Cl- Ag+

vacancy produced by

Cd2+ substitution

Substitution of Ni2+ by Li+ in NiO lattice gives rise to the change of the oxidation state of an ion in the lattice .

Ni2+ O2- Ni2+ O2-

O2- Ni3+ O2- Ni2+

Ni2+ O2- Li+ O2-

O2- Ni2+ O2- Ni2+

valence defect produced

by Li+ substitution

IMPURITY IN METALS

IMPERFECTION IN CERAMICS

Since there are both anions and cations in ceramics, a substitutional impurity will replace the host ion most similar in terms of charge . Charge balance must be maintained when impurities are present. For eg. NaCl

• Substitutional cation impurity

without impurity Ca2+ impurity with impurity

Ca2+

Na+

Na+Ca2+

cation vacancy

Na+ Cl-

• Substitutional anion impurity

without impurity O2- impurity

O2-

Cl-

anion vacancy

Cl-

with impurity

NON-STOICHIOMETRIC DEFECT

Any deviation from stoichiometry in a predominantly ionic solid implies that one of the two elements present in it should be an excess of what is needed for stoichiometry.

For eg., in the oxide MO1-x , the excess M may be due to the presence of extra metal atoms in the interstitial positions.

It may be also due to the presence of oxygen vacancy in greater no. than metal vacancy.

This would also require that the metal is present in more than one oxidation state as the charge neutrality of the solid cannot be maintained otherwise .

Charge neutrality may also be maintained by electronic defects (electrons or holes trapped at the defects).

THERMODYNAMIC CONSIDERATION

If one of the two components of the solid is more volatile than the other, the solid exists in equilibrium with the vapour of the more volatile component. This would mean that the stoichiometric composition MO of the solid is only one of the many possible compositions that would depend on the vapour pressure of the more volatile component.

The thermodynamic quantity to be considered is the chemical potential (molar Gibbs function) of the volatile component. We have

GO2 = G0O2 +RT ln pO2 ……..Eq(1)

Where G and G0 are the Gibbs function and standard Gibbs function respectively. It is adequate to use the relative partial molar Gibbs function GO2 where

GO2 = GO2 - G0O2 = RT ln pO2 ………

Eq(2)

The quantity GO2 depends not only on T but also on the composition.

If the system has two components and a single phase (such as MO), the phase rule tells us that it should have two degrees of freedom. One of these two is the temperature T and other is the composition.

It is thus possible to think of such a system as a single phase over a range of composition. If two phases are simultaneously present in the system, it will have a single degree of freedom which is the temperature.

Non- stoichiometric defects depends upon whether the positive ions are in excess or negative ions are in excess .

Metal Excess Defects

In these defects , the positive ions are in excess.

These may arise due to :

i) Anionic vacancies

ii) Presence of extra cations in interstitial sites.

Metal Deficient Defects

These contain less no. of positive ions than negative ions.

These arise due to:

iii) Cation vacancies

iv) Extra anions occupying interstitial sites .

ANION VACANCIES

In this case, negative ions may be missing from their lattice sites leaving holes in which the electrons remain entrapped to maintain the electrical neutrality. There is an excess of positive ion although, the crystal as whole is electrically neutral. In alkali metal halides, anion vacancies are produced when alkali metal halide crystals are heated in the atmosphere of the alkali metal vapours.

COLOUR CENTRESThe defect that very much influences the electronic properties of the solids.These are the electronic defects.Solid has an anion vacancy. Hence this site has a strong positive potential and can trap electrons. It will need some energy to free the trapped electron.When sodium chloride is heated in sodium vapour, it acquires a yellow colour. The heating in excess sodium creates new lattice sites for sodium.To maintain the structure, an equal no. of chloride sites are also created, but the latter are vacant.The electron of the sodium atom that is ionized in the Nacl matrix is trapped by the chloride ion vacancy. This trapped electron can be freed into the crystal by absorbing visible light and hence the colour .A trapped electron at an anion vacancy is an electronic defect and it is known as F-centre(from German farbe meaning colour).Many of the point defects may have a charge different from the normal charge of the original site due to trapped electrons or holes. These electrons and holes can be released to the conduction band or to the valence band of the solid and this will modify its electronic properties.

EXCESS CATIONS OCCUPYING INTERSTITIAL SITES

•In this case, there are extra positive ions occupying interstitial sites and the electrons in another interstitial sites to maintain the electrical neutrality. •For eg., Zinc Oxide•Zinc Oxide (ZnO) is white in colour at room temperature . On heating it loses oxygen at high temperatures and turns yellow in colour.

ZnO Heat Zn2+ + 1/2 O2 + 2e-

•The excess of Zn2+ ions are trapped in interstitial sites and equal no. of electrons are trapped in the neighbourhood to balance the electrical charge .•These electrons give rise to enhanced electrical conductivity.

Zno heated in Zn vapour ZnyO (y>1) The excess cations occupy interstitial

voids. The e- s (2e- ) released stay associated to

the interstitial cation.

COSEQUENCE OF METAL EXCESS DEFECTS

• The crystals with metal excess defects conduct electricity due to free electrons. • However, the conductivity is low because of the no. of defects and therefore, the no. of free electrons is very small.• Because of low conductivity as compared to conductivity of the metals, these are called semi-conductors.• These compounds are also called n-type semi conductors. Since the current is carried by the electrons in the normal way.• The crystal with metal defects are generally coloured. •For eg., non-stoichiometric sodium chloride is yellow, potassium chloride is violet and lithium chloride is pink.

CATION VACANCIES

In some cases, the positive ions may be missing from their lattice sites.

The extra negative charge may be balanced by some nearby metal ion acquiring two positive charges instead of one.

This type of defect is possible in metals which show variable oxidation state.

For eg., ferrous oxide, ferrous sulphide, nickel oxide etc.

In case of iron pyrites (FeS), two out of three ferrous ions in a lattice may be converted into Fe3+ state and the third Fe2+ ion may be missing from its lattice site.

Therefore, the crystal contains Fe2+ and Fe3+ ions.

This gives rise to exchange of electrons from one Fe2+ ion to Fe3+ ion in which Fe2+ changes to Fe3+ changes to Fe2+ ion.

As a result crystal has metallic lustre .

Because of the natural colour of iron pyrites and metallic lustre some samples of minerals shine like gold.

Similarly, FeO is mostly found with a composition of Fe0.95 O it may actually range from Fe0.93O to Fe0.96O.

In crystal of FeO, some Fe2+ ions are missing and the loss of positive charge is made up by presence of required no. of Fe3+ ions. Moreover, since there is exchange of electrons, the substances become conductors.

FeO heated in oxygen atmosphere FexO (x<1)Vacant cation sites are permanent.Charge is compensated by conversion of ferrous to ferric ion. Fe2+ Fe3+ + e-For every vacancy (of Fe cation) two ferrous ions are converted to ferric ions provide the 2 e- s required by excess oxygen.

EXTRA ANIONS OCCUPYING INTERSTITIAL SITES

In this case, the extra anion may be occupying interstitial positions. The extra negative charge is balanced by the extra charges (oxidation of equal no. of cations to higher oxidation states) on the adjacent metal ions.

Such type of defect is not common because the negative ions are usually very large and they cannot easily fit into the interstitial sites.

A+ B- A+ B-

B-

B- A+ B- A+

A+ B- A2+ B-

B- A+ B- A+

CONSEQUENCE OF METAL DEFICIENT DEFECT

Crystals with metal deficient defects are semi-conductors.

The conductivity is due to the movement of electrons from one ion to another.

For eg., When an electron moves from ion A+ , it changes to A2+ . It is also called movement of positive hole and the substances are called p-type semi-conductors.

PLANAR OR SURFACE DEFECTS

It is a two dimensional defect. Planar defects arise due to atomic planes during mechanical and thermal treatments. The change may be of the orientation or of the stacking sequence of the planes.

Planar defects are of the following types:

Grain boundaries

Tilt boundaries

Twin boundaries

Stacking faults

HIGH AND LOW ANLE GRAIN BOUNDARIES

STACKING FAULTS

Stacking faults occur in a number of crystals structures, but the common example is in close-packed structures. Face-centered cubic (fcc) structures differ from hexagonal close packed (hcp) structures only in stacking order.

FCC and HCP packings are represented by ABCABCABC…… and ABABAB…..arrangements, respectively.

A deviation from the sequence in either of the stacking gives rise to a stacking faults,

e.g., ABCABABCAB…. in FCC and ABABBABA…… in HCP .

Stacking faults occur either due to dissociation of a dislocation or during the process of crystal growth.

However, the orientations on both sides of the stacking fault are the same except for translation with respect to one another by a fraction of a lattice vector.

DIFFUSION IN SOLIDS

At any finite temperature, atoms,ions or molecules occupying the lattice sites in crystalline materials vibrate about their equilibrium positions.

The presence of point defect such as vacancy (in metals) and Schottky defects (in ionic crystals) permits atoms or ions to wander over distances much larger than the interatomic or interionic distances.

Such a motion of atoms or ions to far away lattice sites is called Diffusion and it is responsible for several important processes like solid state reactions, etc.

Diffusion is a process in which transport of matter occurs due to thermally activated transfer of atoms or ions .

Fick suggested two differential equations describing transport of matter under the influence of a concentration gradient.

These are Fick’s first and second laws of diffusion

J = D(δ c/ δx) ……….Eq(1)

J = D(δc/ δx2) ……….Eq(2)

Eq(1) is an expression of the Fick’s first law and Eq(2) is the expression of Fick’s second law.

Where D = diffusion coefficient or diffusivity.

J = flux density

(δ c/ δx) = concentration gradient

The diffusion constant is strongly temperature dependent and is usually expressed by an Arrhenius-type relationship

D= D0 exp (- E/kBT)

Or

D = D0 exp (- E/RT)

Where E is the energy of activation for diffusion expressed appropriately

DEFECTS DOES NOT NECESSARILY IMPLY A BAD THING :-

Addition of C to Fe to make Steel.Addition of Cu to Ni to make thermocouple wires.Addition of Ge to Si to make thermoelectrical materials.Addition of Cr to Fe for corrosion resistance.Introduction of grain boundaries to strengthen materials.

REFERENCE

Principles of the Solid State by H.V. KEER Solid State Chemistry by D.K. Chakrabarty Internet

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