Solid crystals are not perfect. These imperfections in the crystalline arrangement of the atoms in space occur as isolated points, along lines or as surfaces in the structure. This Chapter will discuss Point Defects in crystals. The most important role played by point defects in the behaviour of solids is in diffusion, i.e., the atom-by-atom transport of components through the crystal lattice.

1. School of Multidisciplinary Engineering
Center for Materials Engineering
Materials for Energy Conversion andStorage
(MatE 6003)
Water Splitting: From Electrode to Green Energy
System
Review Paper by: Xiao Li, et al.
Presentation by: Abenezer Mengistu
Presented to: Sintayehu Nibret (PHD)

2. Introduction
 Solid crystals are not perfect. These imperfections in the crystalline arrangement of the
atoms in space occur as isolated points, along lines or as surfaces in the structure.
This Chapter will discuss Point Defects in crystals.
 The most important role played by point defects in the behaviour of solids is in diffusion,
i.e., the atom-by-atom transport of components through the crystal lattice.
The elemental step in diffusion is the motion of an atom from a normal crystal site into an
adjacent point defect in the crystal.
Processes such as precipitation, phase changes, sintering, oxidation, solid state
bonding and some forms of creep depend upon the presence of point defects in the
system.

3. 1. Point Defects in Elemental Crystals
 The concept of a point defect presumes the existence of a periodic lattice of sites that
are normally occupied by atoms in a crystal.
Two primary classes of point defects are found to exist in elemental crystals: vacancies
and interstitials.
A vacancy exists in a crystal where a normal lattice site is unoccupied.
An interstitial defect occurs when an atom occupies a position in the crystal other than a
normal lattice site. Figure:1
Vacancy
Point Defect
Interstitial Point
Defect

4.  While each defect contributes an increase to the energy of the crystal, each defect also
increases the entropy of the crystal.
At equilibrium a crystal will contain some point defects: a crystal is not perfect at
equilibrium.
The concentration of point defects in an elemental crystal is normally very small: even where
it is highest, near the melting point, defects occur at only about one in 10,000 sites.
Concentrations of interstitials in elemental crystals are expected to be much smaller than
vacancies under the same conditions. Nonetheless, this small fraction of defect sites plays a
crucial role in materials science.

5.  To find the concentration of defects at equilibrium, consider a system composed of a
homogeneous crystalline phase (a) and its vapour (g).
The combined statement of the first and second laws must explicitly recognize that the crystal
may change its internal energy not only by changing entropy, volume and the number of moles
of each component, but also by changing the number of vacancies in the system.
……….. (1)
The change in internal energy for the crystal when taken through an arbitrary change in state is
…...(2)
1.1. Conditions for Equilibrium in a Crystal with Vacant Lattice Sites

6.  For the Vapor phase similarly:
….. (3)
 Rearrange and combine (2) and (3) then solve for the total change in entropy of the system
….. (4)
 Now take an isolated system: conditions for equilibrium are
……… (5)
………. (6)
…....... (7)

7.  Now put those conditions of equilibrium into (4)
……… (8)
In this equation nV is an independent variable: nV may vary independently in an isolated
system because lattice sites may be created or annihilated with no other changes in the
system.

8.  The maximum change in entropy that corresponds to the equilibrium state for the system is
found by setting all of the coefficients of the differentials of the independent variables in the
system equal to zero. This yields the familiar conditions for thermal, mechanical and chemical
equilibrium in this solid-vapour system.
 In addition, it yields the condition: …… (9)

9.  A mixture of vacant sites with sites occupied by normal atoms may be considered to be a
dilute solution of vacancies and normal atoms.
The chemical potential of any component, including vacancies, is identical with the partial molal
Gibbs free energy.
…… (10)
Where HV is the partial molal enthalpy of vacancies in the crystal, SXS
V is the partial molal
excess entropy and Sid
V is the ideal entropy term, - k In XV:
………. (11)
where Xα
V is the atom fraction of vacant sites in the crystal. Combining (11) & (9), at equilibrium:
…….…. (12)
1.2. The Concentration of Vacancies in an Elemental Crystal at Equilibrium

10.  Thus, the equilibrium fraction of vacant sites in a crystal is:
………. (13)
 In this equation Hv may be thought of as the enthalpy of formation of a vacancy from a
perfect crystal.
 is the excess entropy associated with this process, physically associated with changes
in the vibrational behaviour of atoms surrounding the vacant site.
Since the enthalpy of formation of a vacancy is positive, (13) demonstrates that the
concentration of vacancies increases with temperature.

11.  Experimental tests of these relations are usually indirect.
• Electrical resistivity is proportional to the vacancy concentration (Vacancies act as
scattering events for electrons).
• Measurement of the power required to heat two samples, which are identical except
for defect concentrations. (interstitial and vacancy annihilation processes during
annealing after deformation or neutron irradiation).
 In a few cases direct measurements of the equilibrium vacancy concentration as a function
of temperature have been carried out.

12. Figure 2. Comparison of molar volume of a crystal computed from dilatometric (linear expansion)
measurements and x-ray lattice parameter measurements. The difference between these curves at
any temperature is the volume of vacancies in the crystal. Source: Simmons, R. and Ballufh, R.,
Phys. Rev., 117, 52, 1960.

13. (13) suggests that a plot of the logarithm of the measured vacancy concentration vs. 1/T
should be linear with a slope equal to and an intercept (at 1/T = 0) equal to .
Figure 3: Arrhenius plot of atom fraction of
vacancies in a crystal vs. the reciprocal of the
temperature; the slope is proportional to the
enthalpy of formation of a vacancy and the
intercept gives the vibrational entropy of a
vacancy.

14. Table 1: Properties of Vacancies for Some Typical Metals

15.  The previous arguments easily extend to other defects in elemental crystals with analogous
results.
 At equilibrium, the chemical potential of an interstitial defect is zero, implying a relation between
defect concentration and temperature of the general form:
……… (14)
 where fD is the ratio of the number of interstitial sites to the number of normal lattice sites in the
crystal.
 Interstitial defects are not expected to participate in processes that occur near equilibrium due to
their insignificant concentration. However, they may play an important role in crystals that are far
from equilibrium (For example, in neutron irradiation).
1.3. Interstitial Defects and Divacancies

16.  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.
Analysis shows that such a defect would be significantly more mobile than a single vacancy.
However, the equilibrium concentration of such defects in a crystal will be much smaller than
that of single vacancies. Adapt the result in (13) to this case; at equilibrium:
…….. (15)
Where, the subscript ‘vv’ denotes properties of di vacancies.

17. It is useful to visualize the formation of a divacancy in two steps: (1)
1. The separate formation of two vacancies from a perfect crystal;
2. The formation of the divacancy configuration from two separated single vacancies.
 The enthalpy change associated with the first process is simple 2HV. Let , the interaction
enthalpy, be the change in enthalpy associated with the second process. Thus, the enthalpy
of formation of a divacancy may be written:
…………… (16)
The same argument may be used to write the excess entropy
…………. (17)

18. (15) may be written as:
………….. (18)
…………(19)

19.  from this we can conclude that equilibrium concentrations of divacancies probably do not
play a significant role in most processes.
However, under conditions, far from equilibrium, significant divacancy concentrations may be
developed and may play a role in enhancing diffusion rates and in electronic processes. Such
processes could be:
quenching from a high temperature where vacancy concentrations are high
neutron irradiation or
ion bombardment,

20. 2. Point Defects in Stoichiometric Compound Crystals
 Many intermediate phases and line compounds have crystal structures that have two (or
more) distinct classes of lattice sites called sub-lattices.
In ionic crystals one set of sites typically contains the cations (positively charged ions) and
the other the anions (negatively charged ions). In such crystals one set of sites is termed
the cation sites and the other the anion sites.
This characteristic holds even if the bonding in the crystal is not purely ionic; in this case
the more electronegative atom occupies the anion sites and the less electronegative atom
the cation sites.

21. Figure 4: Common crystal structures with two classes of lattice sites (sublattices):
(a) NaCl and (b) CsCl.

22. The anion element will be designated generically as ‘X’ , and the cation element will be
designated as ‘M’. A vacant lattice site will be designated with a ‘V’ .
These components (atoms or ions) may occupy cation (M) sites, anion (X) sites or interstitial
(i) sites. Each entity visualized has an associated electronic charge.
A broad variety of defects may be visualized to exist, even in a simple crystal with only two
types of normal lattice sites.
To facilitate the explicit description of this variety of entities a widely accepted notation was
devised by ‘Kroger and Vink’

23.  This notation exhibits three important elements in the identification of a particular defect:
a. The entity occupying the defect site (M, X, V or substitutional elements).
b. A subscript for the type of site occupied ( M, X or I ).
c. A superscript for the excess charge associated with the site, ( x ),( . ) or ( / ).
It is convenient to describe the charge in comparison with that which is normally associated
with a particular site, i.e., to describe the local excess charge.
If the entity occupying the site carries the charge of the specie normally occupying that site,
then a superscript ( x ) is used. Eg. Al2O3 : AlAl
X
and OO
X

24.  ( . ) Denotes excess positive charge, and ( / ) denotes excess negative charge.
Vacant lattice sites on either sub-lattice have an associated excess charge of equal
magnitude but opposite sign to the ion that normally occupies the site.

25.  A Frenkel defect is formed on the
cation sub-lattice by removing an
M ion from a normal M site and
placing it in an interstitial site. It is
also possible to form a Frenkel
defect on the anion sub-lattice.
A Frenkel defect is called an
intrinsic defect because it may be
formed without any interaction with
the surroundings of the crystal.
2.1. Frenkel Defects
Figure 5: A Frenkel defect is formed on the cation sub-lattice by
removing an M atom from a normal cation lattice site and placing it
in an interstitial site.

26.  Consider a crystal MX in which the normal valance of M is +2 and that of X Is -2.
If this crystal contains Frenkel defects derived from cation sites, four distinct entities exist in
such a crystal: M𝑀
𝑋
, XX
X
, VM
′′
, Mi
′′
.
The number of each of these entities may be varied in the crystal; however, these variations
are not independent.
It is possible to define a chemical potential for each of these entities as the rate of change of
the Gibbs free energy of the crystal with the number of each particular entity at constant
temperature and pressure.

27. First, write an expression for the entropy of this homogeneous crystalline phase incorporating
changes in the numbers of each possible entity:
…… (20)
In an isolated system, d U′ = 0 and dV ′ = 0. Conservation of atoms of M and X requires
………… (21)
… (22)
 Since we are dealing with Frenkel defect:
……. (23)

28.  Insert (21), (22) and (23) into (20)
………… (24)
The quantity in brackets may be thought of as the affinity for the defect reaction:
[1]
The condition for equilibrium in the crystal may be obtained by setting the affinity equal to zero:
……. (25)
Each of the chemical potentials can be now written in the form of
….. (26)
Where ak represent activity of the ‘kth’ component.

29.  Substitute (27) into (25)
……. (28)
Can be written as:
….(29) where, ….. (30)
and ….. (30)
Since the solution is very dilute, Raoult’s law applies to the solute, MM
……. (31)

30. ….. (32)
Put this into (28)
…. (33)
Referring (23), therefore, where: Xfd is the atom fraction
of Frenkel defects in the structure. Thus
…….. (34)

31.  In an MX crystal a Schottky defect consists of a vacant cation site and a vacant anion site; in
this simple case the formation of Schottky defects does not disturb the electrical neutrality of
the crystal.
It is intrinsic since it may be formed in a perfect crystal without adding or subtracting atoms or
charges to the crystal.
2.2 Schottky Defects
Figure 6: A Schottky defect consists of
a vacant cation site and a vacant anion
site.

32.  The equilibrium concentration of Schottky defects in a crystal with formula MX begins with an
expression for the change in entropy allowing in this case for changes in the number of cation
and anion vacancies:
…. (35)
The Isolation constraints in this case are
…… (36)

33.  The condition that the ratio of anion to cation sites is constrained to be 1/1 requires that the
equal numbers of each type of vacancies be formed:
……… (37)
Put (37) into (35)
….. (38)
The coefficient of in this expression may be thought of as the affinity for the defect
reaction: ….. [2]

34.  which describes the formation of two vacancies in a region that is initially a perfect crystal;
the notation, “null”, in this context means the initially defect-free crystal.
The condition for equilibrium is obtained by setting the coefficient in (38) equal to zero.
… (39)
Following an argument that we use for Frenkel Defects, yields an expression for the
equilibrium concentration of Schottky defects at any temperature T:
…… (40)
Where is the excess entropy associated with the formation of the pair of vacancies from a
perfect crystal and is the corresponding enthalpy change.

35. A cation site is normally occupied by an M ion carrying a positive charge eZM, where ZM is the
valence of the cation and e is the magnitude of the charge carried by an electron.
If the site is vacant, i.e., if there is no M ion occupying it, then there exists an uncompensated
negative charge equal to (-eZM) associated with surrounding anions.
Similarly, a vacant anion site carries a charge of (-eZX) where ZX Is the normal valence of the
anion. Since ZX Is negative, this excess charge is positive.
These oppositely charged entities in a crystal may be expected to attract each other to form a
cation-anion vacancy pair.
2.3 Combined Defects in Binary Compounds

36. Figure 7: Vacancies on the two sub-lattices tend to
attract each other to form a complex defect, VMVX. The
net charge on this complex is e(zX - ZM)

37. If ZM and ZX and are equal, this vacancy complex has zero excess charge; for the more
general case for which they are not equal, the complex carries a net charge.
This relationship may be represented by the defect reaction:
…….. [4]
where the notation (ZX ZM) represents the complex defect consisting of the oppositely charged
pair of vacancies.

38. The condition for equilibrium with respect to this interaction is determined by setting the
corresponding affinity equal to zero:
…... (41) …… (42)
The entropy difference is arguably small. The enthalpy difference is negative since the
oppositely charged vacancies are spontaneously attracted.
Thus the concentration of complexes is larger than the product of the two single vacancy
concentrations by an amount that depends upon the energy of the attraction.

39.  This section introduces the strategy for determining defect concentrations in a crystal of the
compound MuXV that is isolated from its surroundings.
The entities that may exist in this crystal when it comes to equilibrium include MM, Mi, VM, XX,
Xi and VX. Defect complexes may also exist but are neglected.
the expression for the change in entropy will contain a term of the form 𝜇𝑘dnk for each of the
six entities listed above.
……. (43)
2.4. Multivariate Equilibrium among Defects in a Stoichiometric
Compound Crystal

40. In addition to the usual isolation constraints on changes in internal energy and volume, three
constraining equations operate on the changes in the numbers of each entity in this isolated
crystal:
a. Conservation of M atoms: ……. (44)
b. Conservation of X atoms: ……. (45)
c. Conservation of the ratio of sites in the two sub-lattices: …...(46)
where ns denotes the number of each kind of site in the lattice. Since sites are either occupied
or vacant,

41.  Thus three of the six dnk’s are dependent upon the other three. Choose dnMi, dnXi and dnVX
as independent variables. The third equation permits expression of dnVM in terms of these
variables:
………. (48)
Substitute these constraints into Equation (43) and collect terms:
……(49)

42. To find the conditions for equilibrium, set the coefficients equal to zero. These three
coefficients are the affinities for the three defect chemistry equations:
…… [5]
…….[6]
…….[7]
The first two equations correspond to the formation of Frenkel defects on the cation and
anion sublattices, respectively; the last equation is the Schottky reaction for an MuXV crystal.

43. It is customary to describe the concentrations of each of the entities as a fraction of the sites
in the corresponding sub-lattice.
These measures of composition are designated with brackets.
Thus [VM] is the ratio of the number of vacant cation sites to the total number of cation sites:
nVM / nSM.
Similarly, [Xi] is the ratio of the number of interstitial X ions to the number of normal anion
sites in the crystal: nXi / nSX.

44. Focus upon 1 mole of the compound MMXV; more explicitly, focus upon a quantity of the
crystal that contains (u + v)N0 lattice sites, where N0 is Avagadro’s number.
The number of M sites is u(N0); and the number of X sites is v(N0).
Therefore the total number of cation vacancies in 1 mole of compound may be written as :
nVM = uN0[VM], the number of cation interstitials is nMi = uN0[Mi], , the number of anion
vacancies is nVX = vN0[VX] and the number of anion interstitials may be written nXi = vN0[Xi].
The conditions for equilibrium for [5]:

45. Since the defect concentrations are generally very small, [MM] may be taken to be 1. A similar
approximation is also valid for [XX].
The conditions for equilibrium corresponding to the defect [5] to [7] are:
…… (50)
…… (51)
……. (52)
The K0
r coefficients contain the entropy factors in these equations.

46. Each of the four defect entities has an excess charge associated with it. If z is the normal
charge on an occupied cation site, then -(u/v)z is the charge on a normal anion site.
For example, for the crystal Al2O3, z = +3 and the charge on the anion site is - (2/3)(+3) = - 2.
The excess charge associated with each entity is thus:
a. Cation vacancy, VM: -z.
b. Cation interstitial, Mi: +z.
c. Anion vacancy, VX: +(u/v)z.
d. Anion interstitial, Xi : -(u/v)z.

47. Charge neutrality requires that the total excess charge sum to zero.
Mathematically:
Substitute the expressions for the defect concentrations and simplify:
……. (53)
If the entropy and energy factors are known, then (50) to (53) provide four equations among
four variables and the concentrations of all four defects may be computed for the crystal at
any given temperature.

48. Thank you for your
attention!!