NATIONAL OPEN UNIVERSITY OF NIGERIA

SCHOOL OF SCIENCE AND TECHNOLOGY

COURSE CODE: PHY 407

COURSE TITLE: SOLID STATE PHYSCICS II

1
PHY 407 SOLID STATE PHYSICS II

COURSE
GUIDE

PHY 407
SOLID STATE PHYSICS II

Course Team Dr. A. O. Adewale (Course Developer/Writer) -

UNILAG
Mr. M. A. Olopade (Course Editor) – UNILAG
Dr. S. O. Ajibola (Programme Leader) – NOUN
Mr. E. A.Ibanga (Course Coordinator) – NOUN

NATIONAL OPEN UNIVERSITY OF NIGERIA

2
PHY 407 MODULE 1

National Open University of Nigeria

Headquarters
14/16 Ahmadu Bello Way
Victoria Island, Lagos

Abuja Office
5 Dar es Salaam Street
Off Aminu Kano Crescent
Wuse II, Abuja

E-mail: centralinfo@nou.edu.ng
URL: www.nou.edu.ng

Published by
National Open University of Nigeria

Printed 2014

ISBN: 978-058-981-4

All Rights Reserved

3
PHY 407 SOLID STATE PHYSICS II

CONTENTS PAGE

Introduction ……………………………………………. iv
The Course …………………………………………….. iv
Course Aim ……………………………………………. iv
Course Objectives ……………………………………... v
Working through the Course ………………………….. v
Course Material ………………………………………... v
Study Units …………………………………………….. v
Textbooks ……………………………………………… vi
Assessment …………………………………………….. vi
Summary ………………………………………………. vi

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PHY 407 MODULE 1

INTRODUCTION

Solid state physics is a very important branch of physics that has to do

with the study of rigid matter, or solids, through methods such as quantum
mechanics, crystallography, electromagnetism, and metallurgy. It is known
to be the largest branch of condensed matter physics. In this course you
will learn how large-scale properties of solid materials result from their
atomic-scale properties.

Solid materials are formed from densely-packed atoms, which interact

intensely. These interactions produce the mechanical (e.g. hardness and
elasticity), thermal, electrical, magnetic and optical properties of solids.
Depending on the material involved and the conditions in which it was
formed, the atoms may be arranged in a regular, geometric pattern
(crystalline solids, which include metals and ordinary water ice) or
irregularly (an amorphous solid such as common window glass).

What is the importance of solid state physics? Solid state physics applies
everywhere around us. It involves how the macro-scale properties of
solid materials result from their micro-scale properties. Most of the
solids are arranged in such a way that their atoms are in an orderly
repetitive arrays so the solids we see around are the examples of solid
state physics e.g. sugar molecules, salt (Nacl), glass, etc. So, it forms the
theoretical basis of material science. Its applications are in the
technology of transistors and semiconductors. Quantum mechanics,
metallurgy, electromagnetism, crystallography, etc. are all part and
parcel of solid state physics.

THE COURSE

Solid State Physics II, is a 2-unit course made up of 1 module divided

into 3 units.

Unit 1 deals with dielectric properties of solids, unit 2 explains the

magnetic properties of solids and unit 3 elucidates the various
imperfections in solids.

COURSE AIMS

The aim of this course is to teach you properties of solids and

imperfections in solids.

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PHY 407 SOLID STATE PHYSICS II

COURSE OBJECTIVES

After studying this course, you should be able to:

x explain the dielectric properties of solid

x explain local electric field
x define dielectric constant, polarisability and susceptibility
x explain magnetisation of materials
x differentiate between diamagnetism and paramagnetism
x differentiate between ferromagnetism and antiferromagnetism
x explain magnetic resonance
x explain the various imperfections in solids.

WORKING THROUGH THE COURSE

Solid state physics is the theoretical foundation of material science. It

has applications in the technology of transistors and semiconductors,
which are important components in electronics. It is expected that you
must have pass the prerequisite which is PHY 307. It is hoped that
bearing this in mind, you would find enough motivation to thoroughly
work at this course.

THE COURSE MATERIAL

You will be provided with a course guide and study material containing
the study units.

At the end of the course, you will find a list of recommended textbooks
which are necessary as supplements to the course material. However,
note that it is not compulsory for you to acquire or indeed read all of
them.

STUDY UNITS

The following units are contained in this course:

Module 1

Unit 1 Dielectric Properties of Solids

Unit 2 Magnetism
Unit 3 Imperfection in Solids

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PHY 407 MODULE 1

TEXTBOOK

Charles, K. (2005). Introduction to Solid State Physics. USA: John

Wiley and Sons, Inc.

ASSESSMENT

There are two components of assessment for this course:

i. the self-assessment exercises, and

ii. the tutor-marked assignments.

The assessments are continuous assessment of each component of this

course.

SUMMARY

This course is designed to lay a foundation for you for further studies in
solid state physics. At the end of this course, you should be able to:

x explain polarisation and depolarisation fields

x discuss dielectrics
x define dipole moment
x define electric susceptibility and polarisability
x explain the relevance of Clausius-Mossotti equation to material
sciences
x explain all the symbols in Langevin expression and state their SI
units
x explain why the magnetic susceptibility of diamagnetic materials
will always be less than zero
x differentiate between diamagnetism and paramagnetism
x list five materials that are diamagnetic and paramagnetic
x differentiate between ferromagnetism and antiferromagnetism.

We wish you all the best as you work through the course.

MAIN
COURSE

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PHY 407 SOLID STATE PHYSICS II

CONTENTS PAGE

Module 1 ………………………………………….. 1

Unit 1 Dielectric Properties of Solids ………… 1

Unit 2 Magnetism ……………………………… 13
Unit 3 Imperfection in Solids …………………. 35

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PHY 407 MODULE 1

MODULE 1

Unit 1 Dielectric Properties of Solids

Unit 2 Magnetism
Unit 3 Imperfection in Solids

UNIT 1 DIELECTRIC PROPERTIES OF SOLIDS

CONTENTS

1.0 Introduction
2.0 Objectives
3.0 Main Content
3.1 Dielectric Properties
3.2 Local Electric Field
3.3 Dielectric Constant and Polarisability
3.4 Dipole Relaxation and Dielectric Losses
4.0 Conclusion
5.0 Summary
6.0 Tutor-Marked Assignment
7.0 Reference/Further Reading

1.0 INTRODUCTION

In the previous course on solid state physics, you learnt the crystal
structure of solids, thermal and elastic properties of the crystal lattice.
You will recall that a solid has been defined as a rigid substance in
which the average positions of the constituent atoms form a lattice
structure. In this course, our discussion is on the properties of solids. We
will distinguish between the properties of different materials. Glass
flows like liquids at high temperatures and copper exhibits
characteristics of a crystal structure.

Solid state physics is the study of crystalline solids, through methods

such as quantum mechanics, crystallography, electromagnetism, and
metallurgy. It is the largest branch of condensed matter physics. Solid
state physics studies how the large-scale properties of solid materials
result from their atomic-scale properties. Thus, solid state physics forms
the theoretical basis of material science. It has direct applications, for
example in the technology of transistors and semiconductors.

Solid materials can be divided into two distinct groups on the basis of
their behaviour under the influence of an external electric field.

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PHY 407 SOLID STATE PHYSICS II

i. Conductors – those in which there are electrons which are free to

move in the presence of a field. Examples of these are metals,
carbon, etc.
ii. Insulators or dielectrics – those in which the electrons are
strongly bound to the atoms or molecules composing the material
and cannot be detached by the application of an electric field to
these materials. Examples of these are sulphur, porcelain, mica,
etc.

2.0 OBJECTIVES

At the end of this unit, you should be able to:

x explain the dielectric properties of solid

x explain local electric field
x define dielectric constant, polarisability and susceptibility.

3.0 MAIN CONTENT

3.1 Dielectric Properties

A dielectric is a non-conducting material, such as glass, rubber, or

waxed paper. The following are some of the properties of a dielectric:

i. when a dielectric material is inserted between the plates of a

capacitor, the capacitance increases by a factor K, called the
dielectric constant of the dielectric

C = KCo (1.1)

where Co is the capacitance in the absence of the dielectric

ii. the dielectric constant K has no unit and it is characteristic of a

given material

iii. f εo is the permittivity of free space and ε is the permittivity of the

dielectric material, thus

H
K (1.2)
Ho

iv. with the introduction of the dielectric materials, the energy

density becomes
1
u H E2 (1.3)
2

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PHY 407 MODULE 1

i. the potential difference V between the plates when a dielectric

material is inserted, provided that the charge remains unchanged,
is given as

Vo
V (1.4)
K
where Vo is the potential difference when the dielectric is not inserted.

Equation (1.4) states that the potential difference V between the plates
decreases by a factor of K.

ii. f the potential difference is kept unchanged when the dielectric

material is inserted, then we have,

Q = KQo (1.5)

thus, the charge on a capacitor with a fixed potential difference between

its plates is increased by factor K

iii. f the electric field between the parallel plate capacitor is Eo and it
is E after the material with dielectric constant K is inserted into
the space between the plates, then

Eo
E (1.6)
K

from equation (1.6), we see that the electric field within the dielectric is
reduced by a factor equal to the dielectric constant

iv. Net field E within the dielectric is given as

E = Eo - Eind (1.7)

where Eo is the original field and Eind is the electric field induced

If you use Equation (1.6), (1.7) becomes,

Eo
E o  Eind
K
§ 1·
E ind E o ¨1  ¸ (1.8)
© K¹

v. the induced electric field in the dielectric is related to the induced

charge density σind through the relationship Eind = σind/εo.

vi. from Equation (1.9), you have,

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PHY 407 SOLID STATE PHYSICS II

§ 1·
V ind V ¨1  ¸ (1.9)
© K¹
and
§ 1·
Qind Q ¨1  ¸ (1.10)
© K¹

since K is always greater than 1, these expressions [Equations (1.8),

(1.9), and (1.10)] show that Eind > Eo, σind > σ and Qind > Q.

3.2 Local Electric Field

It should be noted that:

x every material is made up of a very large number of

atoms/molecules
x an atom consists of a positively charged nucleus and negatively
charged electrons
x a system consisting of two equal and opposite charges q,
separated by a certain distance d, is an electric dipole
x when an atom or a molecule of a dielectric is placed in an electric
field, the positive and negative charges feel opposite forces and
are separated forming a dipole.

Suppose the charges -q and +q are placed, respectively, at d/2 and –d/2
from the origin, as shown in Figure 1. The magnitude of the potential
due to this system at a point X is given by

q §1 1·
)r  ¨  ¸ 1.11
4SH o ¨© r1 r2 ¸¹
where r1, r2 are the distances of X from + q and – q respectively.

r2 r1
r

θ
-q +q
d

Fig. 1.1: Dipole

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PHY 407 MODULE 1

If θ is the angle between the axis of the charges and the position vector
of X, we can write

1 1 d cosT
| 
r1 r 2r 2
and
1 1 d cosT
| 
r2 r 2r 2

Substituting in (11)

d cosT
)r 
q
1.12
4SH o r 2

The dipole moment, p, is the vector along the axis of the dipole pointing
in the direction – q to + q and having magnitude qd.

In terms of the dipole moment, you can see that Eq. 1.12 becomes

p cosT
)r 
p.r
1.13
4SH o r 2 4SH o r
3

Using ’§¨ ·¸  3 , the Eq. (1.13) can be written as

1 r
©r¹ r

§1·
)r 
p
 ˜ ’¨ ¸  p ˜ ’) o . 1.14
4SH o r 2
©r¹
where ) o is the potential of a unit charge.

The field of an electric dipole may also be expressed in the following

way,

§ p˜r ·
’¨ 3 ¸
1
E ’) 
4SH o ¨ r ¸
© ¹
1 ª3
«
 
p ˜r r  r pº
2

» 1.15a
4SH o ¬ r5 ¼

In Centimetre-Gram-Second (CGS) system of units, Eq. (1.15a)

becomes

E
 
3 p˜r r  r2 p
1.15b
r5

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PHY 407 SOLID STATE PHYSICS II

Considering a cubic crystal dielectric Figure 1.2 and assume an electric

field of magnitude Eo is applied parallel to one of the axes of the crystal.
The field will push the positively charged nucleus of the atoms of the
dielectric slightly in the direction of the field and negatively charged
electrons in the opposite direction, as shown in Figure 1.2. We say that
the atom is polarised under the influence of the applied external field
and the charges are called polarisation charges.

The polarisation P is defined as the dipole moment p per unit volume,

averaged over the volume of a cell. The total dipole moment is defined
as

P ¦q r
n n 1.16

Let us imagine that a small sphere is cut out of the specimen around the
reference point; then E1 is field of the polarisation charges on the inside
of the cavity left by the sphere, and E2 is the field of the atoms within
the cavity. The polarisation charges on the outer surface of the specimen
produce the depolarisation field, E3. The depolarisation field is opposite
to P, that is, the field E3 is called depolarisation field, because within the
body it tends to oppose the applied field Eo as in Figure 1.2. The
dielectric acquires a polarisation due to the applied electric field Eo. This
polarisation is the consequence of redistribution of charges inside the
dielectric.

The local electric field Eloc at any atom of the dielectric may be written
as

Eloc = Eo + E1 + E2 + E3 1.17

The contribution E1 + E2 + E3 is the total effect at one atom of the dipole

moments of all the other atoms in the system (in CGS units):

and in SI unit

E1  E 2  E3
1
«
 
ª3 p ˜ r r  r 2 p º
» 1.18b
4SH o ¬ r5 ¼

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PHY 407 MODULE 1

Fig. 1.2: The Depolarisation Field is Opposite to P

Lorentz showed that the field E1 due to the polarisation charges on the
surface formed a continuous distribution. The density of the surface
charge is equal to Pcosθ, where θ is the polar angle along the direction
of polarisation in Figure 3.

³ R 2SR sin T P cosT cosT RdT 

2 4S P
E1 1.19
0
3

θ a

Fig. 3: Calculation of E1

The field E2 due to the dipoles within the spherical cavity is the only
term that depends on the crystal structure. It can be shown that for a
reference site with cubic surroundings in a sphere that E2 = 0 if all the
atoms may be replaced by point dipoles parallel to each other.

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PHY 407 SOLID STATE PHYSICS II

For specimens in the shape of ellipsoids oriented with one of the

principal axes parallel to the applied field, it can be shown that the
depolarisation field will be parallel to the applied field and

NP
E  1.20a
Ho
The constant N is known as the depolarisation factor.
If Px, Py, Pz are the components of the polarisation P referred to the
principal axes of an ellipsoid, then the components of the depolarisation
field are written

N x Px  N y Py  N z Pz
E3 x  ; E3 y ; E3 z ; 1.20b
Ho Ho Ho

SELF-ASSESSMENT EXERCISE

i. Explain polarisation and depolarisation fields.

ii. What are dielectrics?
iii. Define dipole moment.

3.3 Dielectric Constant and Polarisability

Let us now consider the dielectric behaviour of molecules which have a

permanent dipole moment. The dielectric constant ε has been defined for
an isotropic or cubic medium relative to vacuum as:

D P
H 1 1 F 1.21a
E HoE
where χ is the electric susceptibility.

The susceptibility (in SI unit) is related to the dielectric constant by

P
F H 1 1.21b
HoE

If the field is not too large, the strength of the induced dipole moment in
an atom i is proportional to the local electric field acting on the
dielectric. That is
pi D Eloc
pi D i Eloc
i
1.22

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PHY 407 MODULE 1

where αi is the polarisability and pi is the dipole moment of atom i.

Polarisability has the SI units of C·m2·V−1 = A2·s4·kg−1·m−1 but is more

often expressed as polarisability volume with units of cm3 or in Å3 =
10−24 cm3.

The polarisation of a crystal may be expressed approximately as the

product of the polarisabilities of the atoms times the local electric field:

P ¦N
i
i pi ¦E
i
i
loc N iD i 1.23

where Ni is the number per unit volume of atom i.

Electric susceptibility can now be defined as

P
¦E i
loc N iD i
F
E 4S
Eloc  P
3
Using Eqs. (1.23) and the Lorentz relation, you will have:

P
¦E i
i
loc N iD i ¦N D
i
i i
H 1
1.24
4S 4S 4S
¦E ¦N D
E i
E loc  i
loc N iD i 1 i i
3 i 3 i

Hence (in CGS)

Eq. (1.25), known as Clausius-Mossotti equation, expresses the relation

between the dielectric constant and the atomic polarisabilities.

You can calculate the dielectric constant of various materials from the
polarisability by use of the Clausius-Mossotti equation. This equation
holds fairly accurately for non-polar substances but it fails completely in
pure polar liquids or solids.

The total polarisability of an atom can be separated into three parts:

electronic, ionic, and orientational. The electronic contribution arises
from the deformation of the electron shell about a nucleus. The ionic or
atomic contribution comes from the displacement and deformation of a
charged ion with respect to other ions. The orientational or dipolar

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PHY 407 SOLID STATE PHYSICS II

polarisability arises when the substance is built up of molecules

possessing a permanent electric dipole moment which may be more or
less free to change orientation in an applied electric field.

In general, the total polarisability is given by

D D dipolar  D ionic  D electronic 1.26

which is the sum of the dipolar, ionic, and electronic polarisabilities,

respectively. The electronic contribution is present in any type of
substance, but the presence of the other two terms depends on the
material under consideration.

SELF-ASSESSMENT EXERCISE

i. Define electric susceptibility and polarisability.

ii. Explain the relevance of Clausius-Mossotti equation to material
sciences.

3.4 Dipole Relaxation and Dielectric Losses

Dipole relaxation time is the time interval characterising the restoration

of a disturbed system to its equilibrium configuration; the relaxation
frequency is defined as the reciprocal of the relaxation time. The
orientational contribution to the dielectric constant is a major cause to
the difference between the low frequency dielectric constant and the
high frequency dielectric constant. The orientational relaxation
frequencies are strongly dependent on the temperature and frequency as:

Do
D 1.27
1  iZ W

where τ is known as Debye relaxation time, αo is the static orientational

polarisability and ω is the angular frequency.
In liquids the relaxation time, for a molecule of radius a can be
expressed as:

4SK a 3
W 1.28
kT

4.0 CONCLUSION

In this unit we have reviewed the dielectric properties of a solid in a

capacitor. You have also seen the generation of local electric field,
polarisation and depolarisation fields. We also defined dielectric
constant, polarisability and susceptibility.
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PHY 407 MODULE 1

5.0 SUMMARY

x Solid materials can be divided into two distinct groups on the

basis of their behaviour under the influence of an external electric
field: Conductors and Insulators or dielectrics.
x The dipole moment, p, is the vector along the axis of the dipole
pointing in the direction – q to + q and having magnitude qd.
x The polarisation P is defined as the dipole moment p per unit
volume, averaged over the volume of a cell. The total dipole
moment is defined as P ¦ qn rn
x The polarisation charges on the outer surface of a specimen
produce the depolarisation field. The depolarisation field is
opposite to P.
x The local electric field Eloc at any atom of the dielectric may be
written as

Eloc = Eo +
 
3 p˜r r  r2 p
r5

and in SI unit

Eloc = Eo + 1 « 3 p ˜ r r5  r p »
ª 2
º
4SH o ¬ r ¼

x For specimens in the shape of ellipsoids oriented with one of the

principal axes parallel to the applied field, it can be shown that
the depolarisation field will be parallel to the applied field and
NP
E . The constant N is known as the depolarisation factor.
Ho
x The dielectric constant ε has been defined for an isotropic or
D P
cubic medium relative to vacuum as: H 1 1 F
E HoE
where χ is the electric susceptibility.
x The susceptibility (in SI unit) is related to the dielectric constant
by

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PHY 407 SOLID STATE PHYSICS II

x The polarisation of a crystal may be expressed approximately as

the product of the polarisabilities of the atoms times the local
electric field:
P ¦ N i pi ¦ Eloci
N iD i where Ni is the number per unit volume
i i

of atom i.
3 § H 1 ·
x In CGS, ¦N D
i
i i ¨ ¸ known
4S © H  2 ¹
as Clausius-Mossotti
equation, expresses the relation between the dielectric constant
and the atomic polarisabilities.

6.0 TUTOR-MARKED ASSIGNMENT

1. Two parallel plates have equal and opposite charges. When the
space between the plates is evacuated, the electric field is 1.2 x
105 V/m, the electric field within the dielectric is 2 x 105 V/m.
Calculate the induced charge density on the surface of the
dielectric.
2. A parallel plate air capacitor is made of 0.2 m square tin plates
and 1 cm apart. It is connected to a 50 V battery. What is the
charge on each plate?
3. The plates of a parallel-plate capacitor are 2 mm apart and 5 m2
in area. The plates are in vacuum. A potential difference of 2000
volts is applied across the capacitor. Calculate the magnitude of
the electric field between the plates.
4. Find the frequency dependence of the electronic polarisability of
an electron having the resonance frequency ωo, treating the
system as a simple harmonic oscillator.
5. Calculate the individual dipole moment p of a molecule of carbon
tetrachloride given the following data.

Relative permittivity εr = 2.24

Density = 1.60 g/cm3
Molecular weight = 156
Field = 107 volts/metre
Find also the average electron displacement.

7.0 REFERENCE/FURTHER READING

Charles, K. (2005). Introduction to Solid State Physics. USA: John

Wiley and Sons, Inc.

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PHY 407 MODULE 1

UNIT 2 MAGNETISM

CONTENTS

1.0 Introduction
2.0 Objectives
3.0 Main Content
3.1 Magnetisation
3.2 Diamagnetism
3.3 Paramagnetism
3.4 Ferromagnetism
3.4.1 Curie-Weiss Law
3.4.2 Spontaneous Magnetisation
3.4.3 Domain Theory of Ferromagnetism
3.4.4 The Bloch Wall
3.5 Domain Dimensions
3.6 Antiferromagnetism
3.7 Ferrites
3.8 Magnetic Resonance
3.8.1 Nuclear Magnetic Resonance
3.8.2 Ferromagnetic Resonance
3.8.3 Antiferromagnetic Resonance
4.0 Conclusion
5.0 Summary
6.0 Tutor-Marked Assignment
7.0 Reference/Further Reading

1.0 INTRODUCTION

You should understand that:

x a bar magnet has its magnetism concentrated mainly at the ends

or poles
x these poles are called north (N) and south (S)
x two bar magnets will either attract each other or repel each other
when their poles are placed close to each other
x the pole-force law or law of poles states that like magnetic poles
repel each other, and unlike magnetic poles attract each other
x magnetic poles always occur in pairs, never singly
x two opposite poles form a magnetic dipole, if a magnet is broken
in an attempt to separate the poles, one finds two new magnets
each having a north and a south pole
x the direction of a magnetic field B at any point is the direction
that the north pole of a compass at that location would point.

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PHY 407 SOLID STATE PHYSICS II

2.0 OBJECTIVES

At the end of this unit, you should be able to:

x explain magnetisation of materials

x differentiate between diamagnetism and paramagnetism
x differentiate between ferromagnetism and antiferromagnetism
x explain magnetic resonance.

3.0 MAIN CONTENT

3.1 Magnetisation

In this section we explain the magnetisation of materials and how it

relates to magnetic field intensity. A material substance acquires a
magnetic polarisation when placed in a magnetic field just as a dielectric
medium acquires an electric polarisation in an electric field. The
response of the materials to an applied magnetic field depends on the
properties of the individual atoms and molecules, and on their
interactions. The orbital motion of the electrons in atoms and molecules
provide currents which give rise to the magnetic dipoles. In many
materials the small electric currents associated with the orbital motion
and spin of the electrons average to zero. When such atoms are placed in
a magnetic field, minute electron currents are generated by induction in
the clouds of electrons, the direction of which is such that the magnetic
field associated with them opposes the inducing field B. These are
known as diamagnetic substances.

There are other materials in which the atoms do have an intrinsic

magnetic moment on account of the fact that the currents from the
orbital motions and spins of the electrons do not average to zero.
Although electron spins tend to pair and cancel one another, there are
atoms in which pairing is incomplete. When such materials are placed in
a magnetic field, the magnetic moments of the atoms and the induced
magnetism enhances the external field. This effect is more pronounced
at low temperatures. Such substances are called paramagnetic
substances.

Every atom or molecule may be regarded as a tiny magnetic dipole with

the magnetic moment

^
Qm l en IdS 2.1

where Qm is the magnetic pole strength, l is the pole separation, I is the

current and dS is the area of the loop. The effect of atomic magnets is

22
PHY 407 MODULE 1

described by a quantity called magnetisation M which is defined as the

magnetic dipole moment per unit volume.

In many materials it is found that the magnetisation M is linearly

proportional to magnetic field intensity H, that is

M Fm H 2.2

where χ

23
PHY 407 SOLID STATE PHYSICS II

The magnetic susceptibility per unit volume is defined as

eH
ZL  2.4
2mc

where M is the magnetic moment per unit volume, or magnetisation, and

H is the magnetic field intensity. Diamagnetism explains the tendency of
electrical charges to partially shield the interior of a body from applied
magnetic field. For an atom in a magnetic field the motion of the
electrons is the same as a possible motion in the absence of H except for
the superposition of a common precession of angular frequency
eH
ZL  2.4
2mc
The precession of the electron distribution is equivalent to diamagnetic
current

I
ZeeH 2mc
2.5
2S c

As the magnetic moment μ of a current loop is given by the product of

the current by the area of the loop, we have

P § Ze 2 · 2
¨¨ 2
¸¸ U 2.6
H © 4mc ¹

for Z electrons, where U 2 x 2  y 2 is the average of the square of the

perpendicular distance of the electron from the field axis. In terms of the
mean square distance r 2 x 2  y 2  z 2 from the nucleus, we have

3 2
r2 U 2.7
2
for a distribution of charge which on the average is spherically
symmetrical, so that x 2 y 2 z 2 . Then the diamagnetic susceptibility
per unit volume is, if N is the number of atoms per unit volume,

Ze 2 N 2
F  r 2.8
6m c 2

Equation (2.8) is the Langevin expression that be used to calculate

diamagnetic susceptibility.

24
PHY 407 MODULE 1

SELF-ASSESSMENT EXERCISE

i. Explain all the symbols in Langevin expression and state their SI

units.
ii. Explain why the magnetic susceptibility of diamagnetic materials
will always be less than zero.

3.3 Paramagnetism

We have discussed diamagnetism in the previous section. Now we will

discuss paramagnetism. Substances with a positive susceptibility are
called paramagnetic. Positive susceptibility can be found in:

x all atoms and molecules that has odd number of electrons.

Examples of such atoms and molecules include: sodium atoms;
nitrogen II oxide (NO)
x all free atoms and ions with a partly filled inner shell: transition
elements; ions isoelectronic with transition elements; rare earth
and actinide elements. Examples include Mn2+, Gd3+, U4+
x metals.

Paramagnetism is a form of magnetism whereby the paramagnetic

material is only attracted when in the presence of an externally applied
magnetic field. In contrast with this behaviour, diamagnetic materials
are repelled by magnetic fields. Paramagnetic materials have a relative
magnetic permeability greater or equal to unity (i.e., a positive magnetic
susceptibility) and hence, are attracted to magnetic fields. The magnetic
moment induced by the applied field is linear in the field strength and
rather weak. Paramagnetic materials have a small, positive susceptibility
to magnetic fields. These materials are slightly attracted by a magnetic
field and the material does not retain the magnetic properties when the
external field is removed. Paramagnetic properties are due to the
presence of some unpaired electrons, and from the realignment of the
electron paths caused by the external magnetic field. Paramagnetic
materials include magnesium, molybdenum, lithium, and tantalum.

Unlike ferromagnets, paramagnets do not retain any magnetisation in the

absence of an externally applied magnetic field, because thermal motion
randomises the spin orientations. Some paramagnetic materials retain
spin disorder at absolute zero, meaning they are paramagnetic in the
ground state. Thus the total magnetisation drops to zero when the
applied field is removed. Even in the presence of the field there is only a
small induced magnetisation because only a small fraction of the spins
will be oriented by the field. This fraction is proportional to the field
strength and this explains the linear dependency. The attraction
experienced by ferromagnetic materials is non-linear and much stronger,

25
PHY 407 SOLID STATE PHYSICS II

so that it is easily observed, for instance, by the attraction between a

refrigerator magnet and the iron of the refrigerator itself.

Consider a paramagnetic material containing N atoms per unit volume,

each having a magnetic moment μ. If a magnetic field H is applied to the
material magnetisation occurs from the orientation of the magnetic
moments and thermal disorder resists the tendency of the applied field to
orient the moments. The energy of interaction with the applied magnetic
field is

V P ˜ H 2.9

For thermal equilibrium, the magnetisation can be derived as

M NPLa  2.10

where a PH / kT , and the Langevin function L(a) is

La  ctnh a 
1
a

For a << 1, L(a) = a/3, and

M # NP 2 H / 3kT . 2.11

In the limit PH / kT  1 , the magnetic susceptibility is

F M H NP 2 / 3kT C T, 2.12

where C NP 2 / 3k is known as Curie constant. The 1/T temperature

dependence is known as the Curie law. Equation (2.12) is known as
Langevin equation.

SELF-ASSESSMENT EXERCISE

i. Differentiate between diamagnetism and paramagnetism.

ii. List five materials that are diamagnetic and paramagnetic.

3.4 Ferromagnetism

In the previous sections we differentiated between diamagnetism and

paramagnetism. Now, we will explain ferromagnetism and
antiferromagnetism. Ferromagnetism is the basic mechanism by which
certain materials (such as iron) form permanent magnets, or are attracted
to magnets. In physics, several different types of magnetism are

26
PHY 407 MODULE 1

distinguished. Ferromagnetism (including ferrimagnetism) is the

strongest type; it is the only type that creates forces strong enough to be
felt, and is responsible for the common phenomena of magnetism
encountered in everyday life. Other substances respond weakly to
magnetic fields with two other types of magnetism, paramagnetism and
diamagnetism, but the forces are so weak that they can only be detected
by sensitive instruments in a laboratory. An everyday example of
ferromagnetism is a refrigerator magnet used to hold notes on a
refrigerator door.

Permanent magnets (materials that can be magnetised by an external

magnetic field and remain magnetised after the external field is
removed) are either ferromagnetic or ferrimagnetic, as are other
materials that are noticeably attracted to them. Only a few substances
are ferromagnetic. The common ones are iron, nickel, cobalt and most of
their alloys, some compounds of rare earth metals, and a few naturally-
occurring minerals such as lodestone.

Ferromagnetism is very important in industry and modern technology,

and is the basis for many electrical and electromechanical devices such
as electromagnets, electric motors, generators, transformers, and
magnetic storage such as tape recorders, and hard disks.

We now derive the Curie-Weiss law. We discuss the physical origin and
properties of the saturation magnetisation in ferromagnetic. The
properties of ferromagnetic materials of interest in technical applications
are closely related to the domain structure. Iron, nickel and cobalt are
known as transition elements; they and their alloys react very strongly in
a magnetic field. They are ferromagnetic and have very high values of
magnetic susceptibility. In ferromagnetic materials, the mutual coupling
forces between neighboring molecular dipole moments are sufficiently
stronger than the randomising effect of thermal agitation and the dipoles
are all aligned parallel within a small region known as domain.

3.4.1 Curie-Weiss Law

Ferromagnetic materials are substance that possesses a magnetic

moment even in the absence of an applied magnetic field. The saturation
magnetisation Ms is defined as the spontaneous magnetic moment per
unit volume. The Curie point Tc is the temperature above which the
magnetic moment vanishes.

If the ionic and atomic magnetic moments of a paramagnetic substance

can be made to line up the same way by the addition of an interaction
then a ferromagnetic substance will be formed. This interaction is called
Weiss field or the molecular field or the exchange field. Weiss was the

27
PHY 407 SOLID STATE PHYSICS II

first to imagine such a field. The motion of thermal agitation of the

elementary particles opposes the orienting effect of the Weiss field.

Pierre Weiss assumes that the Weiss field BE is proportional to the

magnetisation:

BE OM 2.13

where λ is the Weiss field constant, independent of temperature. The

susceptibility above the Curie point can be deduced from the Curie law
(Eq. 2.12). That is:

M C
2.14
B E  OM T

Note that we have taken the magnetic field as the sum of the applied
field B and the Weiss field BE.

You can easily see that Eq. (2.14) can be rewritten as

M C
F 2.15
H T  CO

This gives a non-zero magnetisation for zero applied field at the Curies
point expressed by

Tc CO , 2.16
Therefore,

C
F . 2.17
T  Tc

Eq. (2.17) is known as the Curie-Weiss law. This law describes the
observed susceptibility variation in the paramagnetic region above the
Curie point.

Let us now determine the value of the mean field constant λ. An atom
with angular momentum quantum number J has 2J + 1 equally spaced
energy levels in a magnetic field and the magnetisation is given by

M NgJP B BJ x , 2.18

where x gJP B H kT , and the Brillouin function BJ is given by

28
PHY 407 MODULE 1

2J  1 § 2 J  1x · 1 § x ·
BJ ctnh¨ ¸ ctnh¨ ¸ 2.19
2J © 2J ¹ 2J © 2J ¹

For x  1 , the susceptibility is

F NJ J  1g 2 P B2 / 3kT Np 2 P B2 / 3kT , 2.20

where the effective number of Bohr magnetons is defined as

g>J J  1@ 2 .
1
p 2.21

The value of Weiss field constant can be determined as follows:

NS S  1g 2 P B2 / 3kTc ,
C
O1 2.22
Tc

where S is spin angular momentum quantum number.

3.4.2 Spontaneous Magnetisation

In this section you will see how we can determine the spontaneous024 54] TJETBT1 0 0 1 72.0

29
PHY 407 SOLID STATE PHYSICS II

temperatures well below the Curie point the electronic magnetic

moments of a ferromagnetic specimen are essentially all lined up.

Fig. 2.1: Weiss Domains Microstructure

Weiss explained this phenomenon by assuming that actual specimens

are divided into tiny magnetic domains (also known as Weiss domains).
Within each domain, the spins are aligned, but (if the bulk material is in
its lowest energy configuration, i.e. unmagnetised), the spins of separate
domains point in different directions and their magnetic fields cancel
out, so the object has no net large scale magnetic field. This implies that
within each domain the local magnetisation is saturated and the
directions of magnetisation of different domains need not necessarily be
parallel.

Ferromagnetic materials spontaneously divide into magnetic domains

because the exchange interaction keeps nearby spins aligned with each
other. This is a short-range force, so rotations can occur over hundreds
of times the distance between atoms. To keep the magnetostatic energy
low, this rotation is concentrated in the boundary between domains, or
domain wall. The direction that the magnetisation rotates within a
domain wall varies. In a Bloch wall, the magnetisation direction stays in
the plane of the wall; in a Néel wall, it rotates perpendicular to the plane.

Thus, an ordinary piece of iron generally has little or no net magnetic

moment. However, if it is placed in a strong enough external magnetic
field, the domains will re-orient in parallel with that field, and will
remain re-oriented when the field is turned off, thus creating a
"permanent" magnet. The domains do not go back to their original

30
PHY 407 MODULE 1

magnetisation changes in thousands of tiny discontinuous jumps as the

domain walls suddenly "snap" past defects.

This magnetisation as a function of the external field is described by a

hysteresis curve. Although this state of aligned domains is not a
minimal-energy configuration, it is extremely stable and has been
observed to persist for millions of years in seafloor magnetite aligned by
the Earth's magnetic field (whose poles can thereby be seen to flip at
long intervals).

Alloys used for the strongest permanent magnets are "hard" alloys made
with many defects in their crystal structure where the domain walls
"catch" and stabilise. The net magnetisation can be destroyed by heating
and then cooling (annealing) the material without an external field,
however. The thermal motion allows the domain boundaries to move,
releasing them from any defects, to return to their low-energy unaligned
state.

3.4.4 The Bloch Wall

The term Bloch wall denotes the transition layer which separates
adjacent domains magnetised in different directions. The essential idea
of the Bloch wall is that the entire change in spin direction between
domains magnetised in different directions occurs in a gradual way over
many atomic planes as shown in Figure 2.2.

This gradual change in direction is due to the fact that for a given total
change of spin direction the exchange energy is lower when the change
is distributed over many spins than when the change occurs suddenly.
Let us attempt to explain this behaviour. The exchange energy between
two spins making a small angle φ with each other is given as:

wex JS 2I 2 2.25

here J is the exchange integral and S is the spin quantum number.

Let the total desired change of angle be φo; if the change occurs in N
equal steps, the angle change between neighboring spins is φo/N, and the
exchange energy between each pair of neighboring atoms is

JS 2 Io N  .
2
wex 2.26

31
PHY 407 SOLID STATE PHYSICS II

Fig. 2.2: The Structure of the Bloch Wall Separating Domains

The total exchange energy of the line of N + 1 atoms is thus

Eex JS 2 Io2 N 2.27

Since the exchange energy of a wall is inversely proportional to the

thickness, the wall might spread out until it filled a sizable proportion of
the crystal, were it not for the restraining effect of the anisotropy energy,

32
PHY 407 MODULE 1

V anis | KNa 2.30

therefore

Vw S 2

J S 2 Na 2  KNa 2.31

which is a minimum with respect to N when

N S 2
J S 2 Ka 3 
12
. 2.32

The total wall energy per unit area is

V w 2S J KS 2 a 
12
2.33

3.5 Domain Dimensions

Let us now calculate the domain width for a flux-closure arrangement of

domains in Figure 2.3 in a uniaxial crystal.

D
Fig.2.3: Domain Width

The wall energy per unit area of the crystal surface is approximately

wwall Vw L D 2.34

The volume contained within the domains of closure is oriented in a

direction of hard magnetisation and involves an energy K per unit
volume, where K is the anisotropy constant. Per unit area of crystal
surface on one side, the volume in the domains of closure on both sides
is D/2, and so the anisotropy energy per unit area is

wanis KD 2 . 2.35

Hence, the total energy per unit area is

33
PHY 407 SOLID STATE PHYSICS II

w V w L D  KD 2 2.36

The condition for the minimum of w with respect to the domain width D
is

D 2V w L K 1 2 , 2.37

and the corresponding energy per unit area is

w 2V w LK 1 2 . 2.38
The energy per unit volume is

f domain 2V w K L1 2 2.39

3.6 Antiferromagnetism

The physical origin of the Weiss field is in the quantum-mechanical

exchange integral. It can be shown that the energy of interaction of
atoms i, j bearing spins Si, Sj contains a term

Eex 2 J S i ˜ S j 2.40

where J is the exchange integral and is related to the overlap of the

charge distributions i, j.

When the exchange integral J in Eq. (2.40) is positive, we have

ferromagnetism; when J is negative, we have antiferromagnetism.
In materials that exhibit antiferromagnetism, the magnetic moments of
atoms or molecules, usually related to the spins of electrons, align in a
regular pattern with neighboring spins pointing in opposite directions as
in Figure 2. 4.

Fig. 2.4: Antiferromagnetic Ordering

34
PHY 407 MODULE 1

SELF-ASSESSMENT EXERCISE

i. Differentiate between ferromagnetism and antiferromagnetism.

ii. Explain ferromagnetism in terms of the exchange integral J.

3.7 Ferrites

Ferrites of magnetic interest belong to the group of compounds

represented by the chemical formula MOFe2O3, where M is a divelent

35
PHY 407 SOLID STATE PHYSICS II

although nuclei from isotopes of many other elements (e.g. 2H, 6Li, 10B,
11
B

36
PHY 407 MODULE 1

U P ˜ B a 2.42
^
if B a Bo z, then
U P z Bo JBo I z 2.43

The allowed values of Iz are mI = I, I – 1, and U mI JBo .

In a magnetic field a nucleus with I = ½ has two energy levels
corresponding to mI = ±½. If ћωo denotes the energy difference between
the two levels, then ћωo = γћBo or

ωo = γBo 2.44

This is the fundamental condition for the magnetic resonance

absorption.

3.8.2 Ferromagnetic Resonance

Let us summarise the unusual characteristics of ferromagnetic

resonance:

x The transverse susceptibility components χ‫ ׳‬and χ‫ ״‬are very large

because the magnetisation of a ferromagnet in a given static field
is very much larger than the magnetisation of electronic or
nuclear paramagnets in the same field.
x The shape of the specimen plays an important role. Because the
magnetisation is large, the demagnetisation field is large.
x The strong exchange coupling between the ferromagnetic
electrons tends to suppress the dipolar contribution to the line
width, so that the ferromagnetic resonance lines can be quite
sharp (<1 G) under favourable conditions.
x Saturation effects occur at low rf power levels.

Consider a specimen of a cubic ferromagnetic insulator in the form of an

ellipsoid with principal axes parallel to x, y, z axes of a Cartesian
coordinate system. The components of the internal magnetic field Bi, in
the ellipsoid are related to the applied field by

The Lorentz field and the exchange field do not contribute to the torque.
The components of the spin equation of motion M = γ(M x Bi) for an
^
applied static field Bo z,

37
PHY 407 SOLID STATE PHYSICS II

J M y Bzi  M z B yi  J >Bo  N y  N z M @M y
dM x
dt
dM y
J >M  N x M x   M x Bo  N z M @ J >Bo  N x  N z M @M x . 2.45
dt

To first order we may set dMz/dt = 0 and Mz = M.

Solving (44) with time dependence exp(-iωt) and assuming

iZ J >Bo  N y  N z M @
0,
 J >Bo  N x  N z M @ iZ

we have that the ferromagnetic resonance frequency in applied field Bo

is

Zo2 J 2 >Bo  N y  N z P o M @>Bo  N x  N z P o M @ 2.46

The frequency ωo is called the frequency of the uniform mode.

For a sphere Nx = Ny = Nz so that ωo = γBo. For a flat plate with Bo
perpendicular to the plate Nx = Ny = 0; Nz = 4π, whence the
ferromagnetic resonance frequency is

Zo J Bo  P o M  2.47

If Bo is parallel to the plane of the plate, the xz plane, then Nx = Nz = 0;

Ny = 4π, and

J >Bo Bo  P o M @
1
Zo 2 2.48

The explanation above showed the effects of specimen shape on the

resonance frequency.

3.8.3 Antiferromagnetic Resonance

Let us consider a uniaxial antiferromagnet with spins on two sublattices,

1 and 2. Assume that the magnetisation M1 on sublattice 1 is directed
^
along the +z direction by an anisotropy field B A z . Anisotropy energy is
the energy in a ferromagnetic crystal which directs the magnetisation
along certain crystallographic axes called directions of easy
magnetisation. The anisotropy field is a consequence of anisotropy
energy density. The magnetisation M2 is directed along the –z direction
by an anisotropy field If one sublattice is directed along +z, the
other will be directed along –z.

38
PHY 407 SOLID STATE PHYSICS II

We have set M 1z M ; M 2z M .
If we can define M 1 M 1x  iM 1y ; M 2 M 2x  iM 2y , then (2.50) and (2.51)
become

 iZM 1 >
iJ M 1 B A  OM   M 2 OM  ; @
 iZM 2 >
iJ M 2 B A  OM   M 1 OM  .@
These equations have a solution if, with exchange field BE ≡ λM,

J B A  B E   Z JBE
0.
JBE J B A  B E   Z

Thus, the antiferromagnetic resonance frequency is given by

Zo2 J 2 BA BA  2BE . 2.52

SELF-ASSESSMENT EXERCISE

Explain nuclear magnetic resonance.

4.0 CONCLUSION

You have seen the basic idea of magnetisation of materials in this unit.
The differences between diamagnetism and paramagnetism have been
highlighted to you and you should be able to differentiate between
ferromagnetism and antiferromagnetism. We concluded this unit with
explanation on magnetic resonance.

5.0 SUMMARY

x A material substance acquires a magnetic polarisation when

placed in a magnetic field just as a dielectric medium acquires an
electric polarisation in an electric field. The response of the
materials to an applied magnetic field depends on the properties
of the individual atoms and molecules, and on their interactions.
The orbital motion of the electrons in atoms and molecules
provide currents which give rise to the magnetic dipoles. In many
materials the small electric currents associated with the orbital
motion and spin of the electrons average to zero. When such
atoms are placed in a magnetic field, minute electron currents are
generated by induction in the clouds of electrons, the direction of
which is such that the magnetic field associated with them

40
PHY 407 MODULE 1

opposes the inducing field B. These are known as diamagnetic

substances.
x There are other materials in which the atoms do have an intrinsic
magnetic moment on account of the fact that the currents from
the orbital motions and spins of the electrons do not average to
zero. Although electron spins tend to pair and cancel one another,
there are atoms in which pairing is incomplete. When such
materials are placed in a magnetic field, the magnetic moments of
the atoms and the induced magnetism enhances the external field.
This effect is more pronounced at low temperatures. Such
substances are called paramagnetic substances.
C
x F is known as the Curie-Weiss law. This law describes
T  Tc
the observed susceptibility variation in the paramagnetic region
above the Curie point.
x The value of Weiss field constant can be determined as follows:
NS S  1g 2 P B2 / 3kTc , where S is spin angular momentum
C
O1
Tc
quantum number.
x The physical origin of the Weiss field is in the quantum-
mechanical exchange integral. It can be shown that the energy of
interaction of atoms i, j bearing spins Si, Sj contains a term
Eex 2 J S i ˜ S j , where J is the exchange integral and is related to
the overlap of the charge distributions i, j. When the exchange
integral J is positive, we have ferromagnetism; when J is
negative, we have antiferromagnetism.
x For a sphere Nx = Ny = Nz so that ωo = γBo. For a flat plate with
Bo perpendicular to the plate Nx = Ny = 0; Nz = 4π, whence the
ferromagnetic resonance frequency is
Zo J Bo  P o M  . If Bo is parallel to the plane of the plate, the xz
0; Ny = 4π, and Zo J >Bo Bo  P o M @ 2
1
plane, then Nx = Nz =
. The explanation above showed the effects of specimen shape
on the resonance frequency.
x Antiferromagnetic resonance frequency is given by
Zo2 J 2 BA BA  2BE 

6.0 TUTOR-MARKED ASSIGNMENT

1. Describe different methods of measurement of susceptibilities.

2. Explain Langevin theory of paramagnetism.
3. The molecular weight of a compound is 400, its density is 2 x
103kg/m3 and its magnetic susceptibility at 293 K is 2.56 x 10-4.
Calculate the permanent magnetic dipole moment associated with
each molecule.

41
PHY 407 SOLID STATE PHYSICS II

7.0 REFERENCE/FURTHER READING

Charles, K. (2005). Introduction to Solid State Physics. USA: John

Wiley and Sons, Inc.

42
PHY 407 MODULE 1

UNIT 3 IMPERFECTION IN SOLIDS

CONTENTS

1.0 Introduction
2.0 Objective
3.0 Main Content
3.1 Crystalline Defects
3.1.1 Point Defects
3.1.2 Linear Defects (Dislocations)
3.1.3 Planar (Interfacial) Defects
3.1.4 Volume (Bulk) Defects
4.0 Conclusion
5.0 Summary
6.0 Tutor-Marked Assignment
7.0 References/Further Reading

1.0 INTRODUCTION

The crystalline structures that we have looked at all have imperfections.

We will quantify these imperfections here. The common point
imperfections in crystals are chemical impurities, vacant lattice sites,
and extra atoms not in regular lattice positions. Linear imperfections are
treated under dislocations. The crystal surface is a planar imperfection,
with electron, phonon, and magnon surface states.

2.0 OBJECTIVE

At the end of this unit, you should be able to:

x explain the various imperfections in solids.

3.0 MAIN CONTENT

3.1 Crystalline Defects

A crystalline defect is a lattice irregularity having one or more of its

dimensions on the order of an atomic dimension. There are five major
categories of crystalline defects:

x Zero dimensional: Point defects

x One dimensional: Linear defects (dislocations)
x Two dimensional: Planar (surface) defects
x Three dimensional: Volume (bulk) defects
x Vibrations.

43
PHY 407 SOLID STATE PHYSICS II

SELF-ASSESSMENT EXERCISE

i. Explain the crystalline defects.

ii. List five major categories of crystalline defects.

3.1.1 Point Defects

There are two main types of geometrical defects in a crystal. There are
those which are very localised and are of atomic dimensions. These are
called point defects. An example of this is an impurity atom which can
be either a substitutional or an interstitial impurity.

(a) When some atoms are not exactly in their right place, the lattice
is said to contain imperfections or defects.
(b) Many properties of solids e.g. the electrical resistance and
mechanical strength are governed by the presence of certain types
of defects in the lattice.

Point defects allow for diffusion to occur in the solid state.

There are different categories of point defects:

x Vacancy

These are produced by thermal vibrations of the crystal lattice

and/or from need to maintain charge neutrality.

x Self-interstitial or interstitialcy

A position on the crystal lattice that is not normally occupied is

called an interstitial site. If it is occupied by the same atomic
species it is a self-interstitial or self-interstitialcy. It creates
large distortions in the crystal lattice and so the concentrations
are small.

x Impurities

x solid solutions–homogeneous single phase materials that are

completely analogous to a liquid solution (alcohol and water)
x solute/solvent/solubility – miscibility
x random solid solution vs. ordered solid solution
x substitutional solid solution (Cu/Ni) vs. interstitial solid solution
(Fe/C -
D Iron has 0.025% C and J iron has 2.08% C)
x nonstoichiometric compounds (Fe1-xO)

44
PHY 407 MODULE 1

x second phases – completely analogous to a liquid mixture (oil

and water)
x Hume-Rothery Rules – used to predict solubility (miscibility):

1. less than 15% difference in atomic radii

2. same crystal structure
3. similar electronegativities
4. same valence.

x Schottky defect - a pair of oppositely charged ion vacancies.

x Frenkel defect - a vacancy-interstitialcy combination.

3.1.2 Linear Defects (Dislocations)

Unlike point defects, these are types of disorder which extend beyond
the volume of one or two atoms. It is a line defect which can extend
right through a crystal or it can form closed loops.

A line defect is a lattice distortion created about a line formed by the

solidification process, plastic deformation, vacancy condensation or
atomic mismatch in solid solutions.

Dislocations explain the observation of plastic deformation at lower

stress than would be required in a perfect lattice. They also explain the
phenomenon of work hardening. There are two basic types of
dislocations, edge dislocation and screw dislocation.

x Edge dislocation

These are 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. See
figure 1.

x Screw dislocation

This 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. See Figure 3.1.

45
PHY 407 SOLID STATE PHYSICS II

3.1.3 Planar (Interfacial) Defects

x External surfaces

Atoms at any surface are not in their perfect crystal positions

because they will have decrease number neighbors than the atoms
inside the volume of the material. Hence, these atoms will be at
higher energies. During solidification, materials try to minimise
this.

x Grain boundaries

Grains are formed during the solidification process. A grain

boundary is the area of mismatch between volumes of material
that have a common orientation of the crystallographic axes. The
atoms at grain boundaries are not in their perfect crystal positions
and hence, the grain boundary is less dense. These atoms are at
higher energies than the atoms inside the volume of a grain.
The thickness is of the order of 2-5 atoms wide.

There are different categories of grain boundaries:

x low angle tilt boundary – a few isolated edge dislocations

x high angle tilt boundary – more complex.

46
PHY 407 MODULE 1

x average grain diameter

x grain density.

The size and shape of grains are determined by a number of factors

during solidification:

x lots of nucleation sites Ÿ fine grains

x fewer nucleation sites Ÿ coarse grains
x equal growth in all directions Ÿ equiaxed grains
x thermal gradients Ÿ elongated or columnar grains
x slow growth Ÿ coarse grains
x rapid growth Ÿ fine grains.

Most materials are polycrystalline (or polygranular). There are some

applications where the expense and time to produce single crystal
materials (and hence no grain boundaries) is justified:

x turbine blades
x silicon wafer chips.
x Twin boundaries – a special type of grain boundary across
which there exists a mirror image of the crystal lattice. It is
produced by mechanical shear stresses and/or annealing some
materials.
x Stacking faults – the interruption of the stacking sequence of
close packed planes.
x Phase boundaries - the surface area between the grains in
multiphase materials.
x Ferromagnetic domain walls – the boundary between regions
that have a different orientation of magnetic dipoles.

3.1.4 Volume (Bulk) Defects

These are introduced during processing and fabrication.

x Pores
x Cracks
x Foreign inclusions.

4.0 CONCLUSION

In this unit, we have examined the properties of dielectric materials. In

addition, we differentiated between paramagnetism and diamagnetism;
ferromagnetism and anti-ferromagnetism. We also explained magnetic
resonance. We ended the unit with a brief explanation on the various
imperfections in solids.

47
PHY 407 SOLID STATE PHYSICS II

5.0 SUMMARY

x A crystalline defect is a lattice irregularity having one or more of

its dimensions on the order of an atomic dimension. There are
five major categories of crystalline defects:
x Zero dimensional: Point defects
x One dimensional: Linear defects (dislocations)
x Two dimensional: Planar (surface) defects
x Three dimensional: Volume (bulk) defects
x Vibrations.

6.0 TUTOR-MARKED ASSIGNMENT

1. Distinguish between point defects and dislocations.

2. Explain the meaning of the followings: electronic, orientational
and ionic polarization
3. List the different categories of defects.

7.0 REFERENCES/FURTHER READING

Charles, K. (2005). Introduction to Solid State Physics. USA: John

Wiley and Sons, Inc.

http://fog.ccsf.cc.ca.us

48