Crystal defects occur when the regular patterns of atoms in crystalline materials are interrupted. There are several types of crystal defects including point defects, line defects, and plane defects. Point defects are defects that occur at or around a single lattice point and include vacancies, interstitials, and substitutions. Vacancies occur when an atom is missing from its normal position in the crystal lattice. Interstitials occur when an atom occupies a position between normal lattice sites. Substitutions occur when a foreign atom replaces a host atom in the lattice. The presence of defects is necessary for crystals to have stability at any non-zero temperature due to the contribution of defects to entropy.

1. Crystal Defects
By-
Saurav K. Rawat
(Rawat DA Greatt)
1

2. 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.

3.  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.

5. PROPERTIES OF DEFECTS
 STRUCTURAL PROPERTIES
 ELECTRONIC PROPERTIES
 CHEMICAL PROPERTIES
 SCATTERING PROPERTIES
THERMODYNAMIC PROPERTIES

6. 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

7. 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.

8. POINT DEFECTS

9. VACANCY
 Missing atom from an atomic site.
 Atoms around the vacancy displaced.
 Tensile stress field produced in the vicinity.
Vacancy
distortion
of planes

10. VACANCY DEFECT
At 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.

11. 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

12.  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}

13.  = -{-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.

15. 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 .

16. 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!]

17. 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.

18. 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

19. 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.

20. 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

21. 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!]

22. 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)

23. 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

24. 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!]

25. The Stirlings formula gives,
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
At equilibrium,
[δ( A)/ δn]T = 0 = Ef – kBT ln {(N-n)(Ni-n)/n2}
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)
= ln (NNi) – 2 ln n
Therefore,
n = (NNi)1/2 exp ( - Ef/2kBT)
Usually, the no. of frenkel defect is small at low
temperature.

26. 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.

27. 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
.

28. 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

29. IMPURITY IN METALS

31. 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
Ca2+
Na+
Na+
Na + Cl -
Ca2+
cation
vacancy
without impurity Ca2+ impurity with impurity
• Substitutional anion impurity
O2-
Cl-anion
without impurity O2- impurity
vacancy
Cl-with
impurity

32. 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).

33. 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 = G0
O2 +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 - G0
O2 = RT ln pO2 ………Eq(2)

34. 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.

35. 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:
i) Cation vacancies
ii) Extra anions occupying interstitial sites .

36. 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.

37. COLOUR CENTRES
The 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.

38. 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.

39.  Zno heated in Zn vapour Zn
y
O (y>1)
 The excess cations occupy interstitial
voids.
 The e- s (2e- ) released stay associated
to the interstitial cation.

40. 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.

41. 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.

42. 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.

43. 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+

44. 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.

62. 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

66. HIGH AND LOW ANLE GRAIN BOUNDARIES

67. 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.

68. 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 .

69. 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

70. 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.

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

72. Rawat’s Creation-rwtdgreat@
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