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Material Science
Chapter I
Lattice Defects
CHAPTER 1
Lattice Defects
Prof. Dr. T. Fahmy
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
After completing this chapter, the students will able to:
 Define the unit cell.
 Compare between different types of unit cells.
 Mention different types of lattice defects.
 Explain the planar defects.
 Differentiate between Schottky and Frenkel defects.
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
Lattice Defects
Unit Cells
The unit cell is the smallest structure that repeats itself by translation
through the crystal. We construct these symmetrical units with the hard
spheres. The most common types of unit cells are the faced-centered cubic
(FCC), the body-centered cubic (FCC) and the hexagonal close-packed (HCP).
Other types exist, particularly among minerals. The simple cube (SC) is often
used for didactical purpose, no material has this structure.
Types of Unit Cell:
A unit cell is obtained by joining the lattice points. There are four types
of unit cell as follow:
 Primitive unit cell or simple cubic (SC):
In this type, the atoms are arranged only at the corners
of the unit cell.
1
8
The total lattice points = x 8  1
Non- Primitive unit cell
 Body centred cubic (bcc):
In this type, the atoms are arranged at the corners and at the center
of the unit cell.
1
8
The total lattice points = x 8  1  2
 Face centred cubic (fcc):
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
In this type, the atoms are arranged at the corners and at the centre of each face
of the unit cell.
1
8
1
2
The total lattice points = x 8  6 x  1  3  4
 Side centred:
In this type, the atoms are arranged at the corners and at the centre
of set of faces of the unit cell.
1
8
1
2
The total lattice points = x 8  2 x  1  1  2
The Energy of Crystalline Lattice:
The lattice energy of an ionic solid is a measure of the strength of bonds in that
ionic compound. In addition, it can be defined as the amount of energy that
holds a crystal together.
The calculation of crystalline lattice energy is
dependence on the types of bonds and electrostatic
forces between the atoms or the ions.
As already well known, each ion in the crystalline
lattice is under the effect of attraction and
repulsion forces depending on the type of
neighboring ions. Thus, the total energy of
crystalline lattice is a net of attraction and
repulsion forces.
Sodium Chloride Lattice (Na Cl)
Firstly:
 The electrostatic attraction energy (Eatr) between two ions is expressed
as follow:
Eatr  
Z 1 Z2 q2
4o r
(1.1)
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
Where:
Z1 is the charge of positive ion
Z2 is the charge of negative ion
q is the electronic charge
0 is the permittivity
Secondly:
 The electrostatic attraction energy (Eatr) of double ions is expressed as
follow:
Eatr  
A Z 1 Z2 q2
(1.2)
4o r
Where: A is the Madelung constant.
Thirdly:
The electrostatic repulsion energy according to Born can be expressed as
follow:
Erep 
B
rn
(1.3)
Hence, the total energy for every mole of ions can be written as follow:
Etotal  Eatr  Erep

A NZ 1 Z 2 q 2
4o r

(1.4)
NB
rn
E ( r)
Where: N is Avogadro’s constant.
repulsion
From this figure, we can conclude that, the
energy of repulsion is dominant at lower
B/rn
r=r0
r
distance, whereas, the energy of attraction
is dominant at higher distance.
-Z1Z2e2/4πε0r
attraction
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
The energy of repulsion equals to energy of attraction at distance r=r o. Hence,
the total energy will has a minimum value at r=r0.
So,
A NZ 1 Z 2 q 2
dE
N nB

 n 1
2
dr
4o r
r
but at r=ro , dE/dr =0
0 
A NZ 1 Z 2 q 2
4o r
B 
2

A Z 1 Z2 q2
4o n
N nB
r n 1
ron 1
(1.5)
By substituting from (1.5) in (1.4), we can get:
Etotal  
Etotal
A NZ 1 Z 2 q 2
4o ro

A NZ 1 Z 2 q 2
4o n ro
A NZ 1 Z 2 q 2  1 

1  
4o ro  n 
This equation is called Lande-Born equation.
(1.6)
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
Lattice Defects
Lattice Defects
Introduction:
Crystalline solids exhibit 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 atom or molecules in most
crystalline materials is not perfect. The regular patterns are interrupted by
crystallographic defects. These defects are classified as follow:
Lattice defects are:
 Point Defects
 Schootky defects
 Frenkel defects
 Linear Defects
 Edge dislocation
 Screw dislocation
 Planar Defects
 Grain boundaries
 Antiphase boundaries
 Stacking faults
 Point Defects
 Voids
 Impurities
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
Point Defects:
Point defects can be divided into Schottky defects and Frenkel defects,
and these often occur in ionic crystals.
Point defects are defects that occur only at or around a single lattice point.
They are not extended in space in any dimension. Strict limits for how small a
point defect is, are generally not defined explicitly, but typically these defects
involve at most a few extra or missing atoms. Larger defects in an ordered
structure are usually considered dislocation loops. For historical reasons, many
point defects, especially in ionic crystals, are called centers: for example a
vacancy in many ionic solids is called a luminescence center, a color center, or
F-center. These dislocations permit ionic transport through crystals leading to
electrochemical reactions. These are frequently specified using Kröger–Vink
Notation.
Vacancy defects are lattice sites which would be occupied in a perfect
crystal, but are vacant. If a neighboring atom moves to occupy the vacant site,
the vacancy moves in the opposite direction to the site which used to be
occupied by the moving atom. The stability of the surrounding crystal structure
guarantees that the neighboring atoms will not simply collapse around the
vacancy. In some materials, neighboring atoms actually move away from a
vacancy, because they experience attraction from atoms in the surroundings. A
vacancy (or pair of vacancies in an ionic solid) is sometimes called a Schottky
defect.
Vacancy, in crystallography, absence of an atom or molecule from a point
that it would normally occupy in a crystal. Such an imperfection (crystal
defect) in the regular spacing of atoms changes the electrical and optical
properties of the crystal. Colour centres are vacancies that give colour to
many solids. Vacancies can be created by mechanical deformation of the
crystal, rapid cooling from high temperature, or the impact of radiation on
the crystal. In the so-called Schottky defect, an atom moves from the inside
of the crystal to its surface, leaving behind an isolated vacancy.
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
Schottky Defects:
A Schottky defect is a type of point defect in a crystal lattice. The defect
forms when oppositely charged ions leave their lattice sites, creating vacancies.
These vacancies are formed in stoichiometric units, to maintain an overall
neutral charge in the ionic solid. The vacancies are then free to move about as
their own entities. Normally these defects will lead to a decrease in the density
of the crystal.
Definition:
If an ionic crystal of type A+B-, an equal number of cations and anions are
missing from their lattice sites so that electrical neutrality as well as stoichiometry
is maintained, it is called Schottky Defect.
Let’s calculate the energy (Ev) of vacancy formation as follow:
If N = number of possible normal lattice sites and n = number of
Schottky defects. The total free energy of the crystal can be expressed as
follow:
F  E T S
(1.7)
Where:
F is the total free energy, E is the total internal energy, T is the temperature and
S is the entropy. The entropy (S) can be expressed as follow:
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
S  kB x ln P
(1.8)
Where:
kB is the Boltezmann’s constant and p is the probability of Schottky defect
formation.
The probability of Schottky defects
P=
N!
N  n ! n!
(1.9)
Then by substitution from equation (1.8 &1.9) in equation (1.7), we can get the
total free energy for n defects as


N!
F  n E  kBT x ln 

 N  n ! n !
(1.10)
Note that, using Sterling principle:
ln X !  X ln X  X
(1.11)
Then, the previous equation can be rewritten as follow:
F  n E  kBT x N ln N  N  N  nln N  n  N  n  n ln n  n
n can be determined by minimizing the free energy and at the thermal
equilibrium, we can obtain:
 F 

  0  EV  kB T  n N  n   n n
 n T



N n 
EV  kB T  n

n


N  n  
EV

  n
kB T 
n 
If n<<N, then N-n ~ N
(1.12)
EV
N n
k T
e B
n
Material Science
Chapter I
Lattice Defects
EV
N
k T
e B
n
n N e
Prof. Dr. T. Fahmy

EV
kB T
(1.13)
Then the number of defects and energy of vacancy formation can be obtained
using the previous equation. In addition, the concentration of vacancies can be
obtained using the following equation:
EV

n
k T
C e B
N
(1.14)
Examples
This type of defect is shown in compounds with:

highly ionic compounds

high co-ordination number

small difference in sizes of cations and anions
Examples: NaCl, KCl, CsCl, KBr, AgCl.
Experimental observations show that at room temperature in an NaCl crystal
there is one Schottky defect per 1016 ions.
Effect on Density
The total number of ions in a crystal with this defect is less than the theoretical
value of ions, thus, the density of the said crystal is less than normal.
Interstitial defects are atoms that occupy a site in the crystal structure at
which there is usually not an atom. They are generally high energy
configurations. Small atoms in some crystals can occupy interstices without
high energy, such as hydrogen in palladium.
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
Schematic illustration of some simple point defect types in a monatomic solid.

A nearby pair of a vacancy and an interstitial is often called a Frenkel
defect or Frenkel pair. This is caused when an ion moves into an
interstitial site and creates a vacancy.
Example (1.1):
Calculate the equilibrium number of vacancies per cubic meter for copper at
1000 0C. The energy for vacancy formation is 0.9 eV/atom; the atomic weight
and density at 1000 0C for copper are 63.5 g/mol and 8.4 g/cm3, respectively.
Solution:
It is first necessary, however to determine the value of N, the number of
atomic sites per cubic meter for copper, from its atomic weight Acu, its density
, and Avogadro’s number NA, according to the following:
N
6.023x10
N
23
NA 
Acu


atoms / mol x 8.4 g / cm3 (10 6 cm3 / m 3 )
63.5 g / mol
N  8 x1028 atoms / m3
Thus, the number of vacancies at 1000 0C (1273 0K)is equal to:
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
 E 
n  NV  N exp   V 
 KB T 

0.9
n  8 x1028 exp  
5
 8.62 x10 x 1273



n  2.2 x1025 Vanacncies / m3
Frenkel Defects
The Frenkel defect is shown by ionic solids. The smaller ion (usually the
cation) is displaced from its lattice position to an interstitial site. It creates a
vacancy defect at its original site and an interstitial defect at its new location.
Definition:
A Frenkel defect is a type of point defect in a crystal lattice. The
defect forms when an atom or cation leaves its place in the lattice, creating
a vacancy, and becomes an interstitial by lodging in a nearby location not
usually occupied by an atom. Frenkel defects occur due to thermal
vibrations, and it is theorized that there will be no defects in a crystal at 0
K. The phenomenon is named after the Soviet physicist Yakov Frenkel,
who discovered it in 1926.
Effect on Density
This defect does not change the density of the solid as it involves only
the migration of the ions within the crystal, thus preserving both the volume as
well as mass.
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
Examples
It is shown in ionic solids with large size difference between the anion
and cation (with the cation usually smaller due to an increased effective nuclear
charge). Some solids which display this defect - ZnS, AgCl, AgBr, AgI (due to
the comparatively smaller size of Zn2+ and Ag+ ions)
To be noted : Ag Br shows both Frenkel as well as Schottky defects.
For example, consider a lattice formed by X and M ions. Suppose an M ion
leaves the M sublattice, leaving the X sublattice unchanged. The number of
interstitials formed will equal the number of vacancies formed.
One form of a Frenkel defect reaction in MgO with the oxygen ion
leaving the lattice and going into the interstitial site. This can be illustrated
with the example of the sodium chloride crystal structure. The diagrams below
are schematic two-dimensional representations.
The defect-free NaCl structure
Two Frenkel defects within the NaCl structure
If N = number of possible normal lattice sites, N/ = number of possible
interstitial sites. (N=N/ for most of the cases), and n = number of Frenkel defects.
Number of ways to arrange the interstitial atoms:
  
N /!
P N  /
N  n ! n!
/

Number of ways to arrange the vacancy defects:
(1.15)
Material Science
Chapter I
Pn  
Lattice Defects
Prof. Dr. T. Fahmy
N!
N  n ! n !
(1.16)
The total probability of interstitial atom and vacancy is:
 
Pt  P N / x P(n) 
N /!
N!
x
/
N  n ! n! N  n ! n!


(1.17)
But as known, the total energy of the system can be expressed as follow:
F  E T S
(1.18)
Where E is the activation of enthalpy (internal energy of the system) and S is
the activation of entropy and can be expressed as follow:

N /!
N ! 
S  k B ln P  k B ln  /
x

 N  n ! n! N  n ! n!


(1.19)
By substituting from equation (1.19) in equation (1.15), we obtain

N /!
N! 
F  E  kB T n  /
x

 N  n ! n! N  n! n!


If EI energy required to form one defect and by using Sterling's approximation:
ℓn(x!) =x ℓnx – x, we can get the following
F  n EI  kB T
n N ! n N
/
 
/

 n !  n( n!)  n N!  n N  n !  n( n!)

 
 


 N / n N /  N /  N /  n n N /  n  N /  n  n n( n)  n 
F  n EI  k B T 

 N n N   N  N  n  n N  n   N  n   n n( n)  n 

  
 

F  n EI  k B T N / n N /  N /  n n N /  n  N n N   N  n n N  n  2n n( n)
n can be determined by minimizing the free energy and at the thermal
equilibrium, we can obtain:

Material Science
Chapter I
Lattice Defects
 
 
 

Prof. Dr. T. Fahmy
 F 
/

  0  EI  kB T n N  n 1  n N  n 1  2  2 n( n)
 n T
EI  k B T n N /  n  n N  n  2 n( n)




N  n  
N/  n
EI  kB T  n
 n

( n)
( n) 




N /  n N  n  
EI  k B T  n

n2


If N/ and N ˃˃ n



EI
N/N 
  n
kB T 
n 2 
n2  N / N e
n
/
N N e

EI
2 kB T

EI
kB T
(1.20)

Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
Linear Defects:
Linear defects can be described by gauge theories.
 Dislocations are linear defects around which some of the atoms of the
crystal lattice are misaligned.
There are two basic types of dislocations, the edge dislocation and the screw
dislocation. "Mixed" dislocations, combining aspects of both types, are also
common.
An edge dislocation is shown. The dislocation line is presented in blue, the Burgers
vector b in black.
Edge dislocation
Edge dislocation is caused by the
termination of a plane of atoms in
the middle of a crystal. In such a
case, the adjacent planes are not
straight, but instead bend around
the edge of the terminating plane
so that the crystal structure is
perfectly ordered on either side.
The analogy with a stack of paper
is apt: if a half a piece of paper is
inserted in a stack of paper, the
defect in the stack is only
noticeable at the edge of the half
sheet.
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
Edge dislocation is a linear defect that centers around the line that is
defined along the extra portion of plane of atoms (half plane)
The screw dislocation
The screw dislocation is more difficult to visualise, but basically
comprises a structure in which a helical path is traced around the linear defect
(dislocation line) by the atomic planes of atoms in the crystal lattice.
Screw dislocation can be regarded as formed by applying shear stress to
produce distortion.
The presence of dislocation results in lattice strain (distortion). The direction
and magnitude of such distortion is expressed in terms of a Burgers vector (b).
For an edge type, b is perpendicular to the dislocation line, whereas in the cases
of the screw type it is parallel. In metallic materials, b is aligned with closepacked crytallographic directions and its magnitude is equivalent to one
interatomic spacing.
Dislocations can move if the atoms from one of the surrounding planes
break their bonds and rebond with the atoms at the terminating edge. It is the
presence of dislocations and their ability to readily move (and interact) under
the influence of stresses induced by external loads that leads to the
characteristic malleability of metallic materials.
Dislocations can be observed using transmission electron microscopy,
field ion microscopy and atom probe techniques. Deep level transient
spectroscopy has been used for studying the electrical activity of dislocations in
semiconductors, mainly silicon.
Material Science

Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
Disclinations are line defects corresponding to "adding" or "subtracting"
an angle around a line. Basically, this means that if you track the crystal
orientation around the line defect, you get a rotation. Usually they play a
role only in liquid crystals.
Finally:
A dislocation is a line defect within a crystal which arise during crystal
growth or as a results of mechanical deformation of a crystal.
Planar defects

Grain boundaries occur where the crystallographic direction of the
lattice abruptly changes. This usually occurs when two crystals begin
growing separately and then meet.
The internal interfaces that separate neighboring misoriented single
crystals in a polycrystalline solid. Most solids such as metals, ceramics, and
semiconductors have a crystalline structure, which means that they are made of
atoms which are arranged in a three-dimensional periodic manner within the
constituent crystals. Most engineering materials are polycrystalline in nature in
that they are made of many small single crystals which are misoriented with
respect to each other and meet at internal interfaces called grain boundaries.
These interfaces, which are frequently planar, have a two-dimensionally
periodic atomic structure. A polycrystalline cube 1 cm on edge, with grains
0.0001 cm in diameter, would contain 1012 crystals with a grain boundary area
of several square meters. Thus, grain boundaries play an important role in
controlling the electrical and mechanical properties of the polycrystalline solid.
It is believed that the properties are influenced by the detailed atomic structure
of the grain boundaries, as well as by the defects that are present, such as
Material Science
Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
dislocations and ledges. Grain boundaries generally have very different atomic
configurations and local atomic densities than those of the perfect crystal, and
so they act as sinks for impurity atoms which tend to segregate to interfaces
Grain boundaries

Antiphase boundaries occur in ordered alloys: in this case, the
crystallographic direction remains the same, but each side of the boundary
has an opposite phase: For example, if the ordering is usually
ABABABAB, an antiphase boundary takes the form of ABABBABA.

Stacking faults occur in a number of crystal 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:
both structures have close packed atomic planes with sixfold symmetry—
the atoms form equilateral triangles. When stacking one of these layers on
top of another, the atoms are not directly on top of one another—the first
two layers are identical for hcp and fcc, and labelled AB. If the third layer
is placed so that its atoms are directly above those of the first layer, the
stacking will be ABA—this is the hcp structure, and it continues
ABABABAB. However, there is another possible location for the third
layer, such that its atoms are not above the first layer. Instead, it is the
atoms in the fourth layer that are directly above the first layer. This
produces the stacking ABCABCABC, and is actually a cubic arrangement
of the atoms. A stacking fault is a one or two layer interruption in the
Material Science
Chapter I
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Prof. Dr. T. Fahmy
stacking sequence, for example, if the sequence ABCABABCAB were
found in an fcc structure.
Bulk defects

Voids are small regions where there are no atoms, and can be thought of
as clusters of vacancies.
Impurities can cluster together to form small regions of a different phase. These
are often called precipitates.

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 the
crystal structure. This is neither a vacant site nor is the atom on an
interstitial site and it is called a substitutional defect. The atom is not
supposed to be anywhere in the crystal, and is thus an impurity. There are
two different types of substitutional defects. Isovalent substitution and
aliovalent substitution. Isovalent substitution is where the ion that is
substituting the original ion is of the same oxidation state as the ion it is
replacing. Aliovalent substitution is where the ion that is substituting the
original ion is of a different oxidation state as the ion it is replacing.
Aliovalent substitutions change the overall charge within the ionic
compound, but the ionic compound must be neutral. Therefore a charge
compensation mechanism is required. Hence either one of the metals is
partially or fully oxidised or reduced, or ion vacancies are created.

Antisite defects occur in an ordered alloy or compound when atoms of
different type exchange positions. For example, some alloys have a regular
structure in which every other atom is a different species; for illustration
assume that type A atoms sit on the corners of a cubic lattice, and type B
Material Science
Chapter I
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Prof. Dr. T. Fahmy
atoms sit in the center of the cubes. If one cube has an A atom at its center,
the atom is on a site usually occupied by a B atom, and is thus an antisite
defect. This is neither a vacancy nor an interstitial, nor an impurity.

Topological defects are regions in a crystal where the normal chemical
bonding environment is topologically different from the surroundings. For
instance, in a perfect sheet of graphite (graphene) all atoms are in rings
containing six atoms. If the sheet contains regions where the number of
atoms in a ring is different from six, while the total number of atoms
remains the same, a topological defect has formed. An example is the Stone
Wales defect in nanotubes, which consists of two adjacent 5-membered and
two 7-membered atom rings.
Schematic illustration of defects in a compound solid, using GaAs as an
example.

Also amorphous solids may contain defects. These are naturally
somewhat hard to define, but sometimes their nature can be quite easily
understood. For instance, in ideally bonded amorphous silica all Si atoms
have 4 bonds to O atoms and all O atoms have 2 bonds to Si atom. Thus
e.g. an O atom with only one Si bond (a dangling bond) can be considered a
defect in silica.
Note that:
a dangling bond is an unsatisfied valence on an immobilised atom. In
order to gain enough electrons to fill their valence shells (see also octet rule),
many atoms will form covalent bonds with other atoms. In the simplest case,
that of a single bond, two atoms each contribute one unpaired electron, and
the resulting pair of electrons is shared between both atoms. Atoms which
possess too few bonding partners to satisfy their valences and which possess
unpaired electrons are termed free radicals; so, often, are molecules
containing such atoms. When a free radical exists in an immobilized
environment, for example, a solid, it is referred to as an "immobilized free
radical" or a "dangling bond".
Material Science

Chapter I
Lattice Defects
Prof. Dr. T. Fahmy
Complexes can form between different kinds of point defects. For
example, if a vacancy encounters an impurity, the two may bind together if
the impurity is too large for the lattice. Interstitials can form 'split
interstitial' or 'dumbbell' structures where two atoms effectively share an
atomic site, resulting in neither atom actually occupying the site.