Unit 4 Dielectric Properties of Materials (12 hrs)

Polarization- Local Electric Field at an Atom- Depolarization Field- Electric Susceptibility-

Polarizability- Clausius Mosotti Equation- Classical Theory of Electric Polarizability- Normal
and Anomalous Dispersion- Cauchy and Sellmeir relations- Langevin-Debye equation-
Complex Dielectric Constant- Optical Phenomena- Application: Plasma Oscillations- Plasma
Frequency- Plasmons
Polarization
When a dielectric slab is placed in an electric field, the positive and negative charges
are displaced from their equilibrium position by vey small distances. This results in the
formation of a large number of dipoles each having some dipole moment in the direction of
the field. Now the dielectric is said to be polarised.
The dipole moment per unit volume of a dielectric material is called Polarization (P).
The polarization charges induce an electric field opposite to the applied field within the
material. Thus the effect of polarization is to reduce the magnitude of the external field E0.
The magnitude of the resultant field is
E = E0  EP
For ordinary electric fields, the polarization P is proportional to the macroscopic field E
P E
P = 0  E
where 0 is the permittivity of free space and  is the electric susceptibility.
Local Electric Field at an Atom

In dielectric solids, the atoms or molecules experience not only the external applied
electric field but also the electric field produced by the dipoles. The resultant electric field
acting on the atoms or molecules of dielectric substance is called the local field or an
internal field.

To find an expression for local electric field on a dielectric molecule or an atom,

consider a dielectric placed between the plates of a capacitor. Consider an imaginary
spherical cavity around an atom A inside the dielectric

The internal field at the atom site A can be made up of four components E1, E2, E3
and E4

Eloc = E1 + E2 + E3 + E4

Field E1

This field is due to the charge density on the plates

D
E =
ε
where D = 0E + P , called the dielectric displacement
ε E+P
E =
ε
P
E =E+
ε

Field E2

This is the field intensity at A due to the polarizing charges lying on the external
surface of the dielectric. This field is opposite to E0. It is also called depolarization field.
−P
E =
ε

Field E3

It is the field at the centre A due to the polarization charges on surface of the cavity
(Lorentz cavity). This field is called Lorentz field.

The value of this field depends on the shape of the specimen. For symmetrical
specimens in the form of sphere, cylinder, disc etc the medium is uniformly polarised. In
such cases
P
E =
3ε

Field E4

This field is due to the dipoles contained in the cavity. This field depends on the
crystal structure of the solid under consideration. For a spherically symmetric case like
cubic structure its value is zero.
P −P P
E =E+ + + +0
ε ε 3ε
P
= E+
3ε

This is known as Lorentz relation

Depolarization Field
When a dielectric is placed in a uniform electric field E0, then the positive and
negative charges are in the dielectric are displaced from their equilibrium position. So the
molecules of the dielectric gets polarised. This polarization charges inside the dielectric cancel
each other. But the charges at the end faces set up an electric field in a direction opposite to
the direction of the applied field. This electric field is called depolarization field. The value of
this field depends on the geometrical shape of the external surface. This field is given by the
expression , where P is the polarization

Electric Susceptibility (Dielectric susceptibility)

When a dielectric slab is placed in an electric field, it gets polarised. The dipole
moment per unit volume of a dielectric material is called Polarization (P). For ordinary electric
fields, the polarization P is proportional to the macroscopic field E
P = 0  E
where 0 is the permittivity of free space and  is the electric susceptibility.
P
=
ε E
But P = ε (ε − 1)E
ε (ε − 1)E
=
ε E
=ε −1
Polarizability-
When a dielectric is placed in a uniform electric field E, the molecules get polarised.
The dipole moment p is proportional to the field
p E
p = E
where  is called the polarisability of the molecule.
Polarizability is a measure of the ability of a material to become polarized in the
presence of an applied electric field.
Sources of Polarisability
The net polarisability of a dielectric material results mainly from the following four types of
processes
i) Electronic polarisation
ii) Ionic polarisation
iii) Dipolar or Orientation polarisation
iv) Space charge polarisation
Electronic Polarization:
This effect occurs in all atoms under the application of an electric field. The nucleus
of the atom and the center of its electron cloud shift away from each other, creating a tiny
dipole with very small polarization effect. This leads to the electronic polarisability e
Ionic Polarization:
Ionic polarization is a mechanism that contributes to the relative permittivity of a
material. This type of polarization typically occurs in ionic crystal elements like NaCl, KCl,
LiBr, ceramic materials etc.
 In ionic solids, the ions are symmetrically arranged in a crystal lattice with a net zero
polarization. Once an electric field is applied, the cations and anions are attracted to
opposite directions. This creates a relatively large ionic displacement (compared to
electronic displacement), which can give rise to high dielectric constants. Hence ceramics
are popularly used in capacitors. This is the source of ionic polarisability i
Dipole (or Orientation) Polarization:
Certain solids have permanent molecular dipoles. Under an electric field, these
dipoles rotate in the direction of the applied field. This creates a net average dipole
moment per molecule. Dipole orientation is more common in polymers since their atomic
structure permits reorientation. This leads to the dipolar polarisability d
Space Charge (or Interfacial) Polarization:
Interfacial or space charge polarization occurs when there is an accumulation of
charge at an interface between two materials or between two regions within a material
because of an external field. In ceramics, this phenomenon arises from extraneous charges
that come from contaminants or irregular geometry in the interfaces of the polycrystalline
solids. These charges are partly mobile and migrate under an applied field, causing this
extrinsic type of polarization. This is the source of interfacial polarisability c
The space charge polarizability is given by
αc =α−αe−αo
where α, αe and αo refer to the total, electronic, and orientation polarizations respectively.
The total amount of dielectric polarization in a material is the sum of the electronic,
orientational, and interfacial polarizabilities.
α =αe+αo+αi
Clausius Mosotti Equation
It is the relation between the polarizability α of a molecule and the dielectric
constant ε of a dielectric substance made up of molecules with this polarizability.
Consider a molecule of a dielectric medium having dipole moment p. This dipole
moment is proportional to the electric field. Then
p E
p =  E ……..(1)
where  is called the molecular polarizability
If there are N molecules per unit volume, the polarization P is given by
P = Np
= N E ………(2)
If  is the density, NA is Avogadro's number and M is the molecular weight, then
N
N=
M
N
P = E
M
 In dielectric solids, the molecules experience the external applied electric field Eo and
the electric field produced by the dipoles. The resultant electric field acting on the atoms or
molecules of dielectric substance is called the local field or an internal field it is given by
P
E=E +
3ε

We have, the electric displacement

D= ε E +P

P= D−ε E

=ε ε E −ε E

= ε E (ε − 1) … … … . . (3)

where r is the relative permittivity of the medium

ε E (ε − 1)
E=E +
3ε
E (ε − 1)
=E +
3
E ε −E
=E +
3
2E + E ε
=
3
E
= (ε + 2)
3
Sub in eqn (2)
E
P = N (ε + 2) … … … (4 )
3
From eqns (3) and (4)
E
ε E (ε − 1) = N (ε + 2)
3
N
ε (ε − 1) = (ε + 2)
3
(ε − 1) N
=
(ε + 2) 3ε
This is known as the Clausius –Mossotti equation
Classical Theory of Electric Polarizability
Consider an atom consisting of a heavy, immobile and undeformable nucleus of
charge Ze surrounded by an electronic shell. Under the influence of an external field, let the
electron be displaced through a small distance x. An electron in an atom polarised by this
field can be treated as a harmonic oscillator experiencing a restoring force that originates
from the Coulomb attraction between the nucleus and the electron.
Now the force acting on the electron is
d x
m = −kx
dt
Where k is the force constant b/w the positive heavy nucleus and the electron
But 𝑘 = 𝑚𝜔 where o is the natural frequency of oscillation of atom
d x
m = −mω x
dt
d x
m + mω x = 0
dt

When an alternating field Eo sin t is applied, the force acting on the electron will modify
the above equation as
d x
m + mω x = −eE sin t
dt
Under the influence of the alternating electric field the displacement is modified as
x = x sin t
So
dx
= x  cos t
dt
d x
= − ω x sin t
dt
Sun in the above eqn
− mω x sin t + mω x sin t = −eE sin t
− mω x + mω x = −eE
m(ω − ω )x = −eE
−eE
x =
m(ω − ω )
The amplitude of the corresponding dipole moment is
𝑝 = −𝑒𝑥
𝑒 𝐸
=
𝑚(𝜔 − 𝜔 )
Now the electronic polarisability is given by
𝑝
𝛼 =
𝐸
 𝑒
=
𝑚(𝜔 −𝜔 )
This relation indicates the frequency dependence of electronic polarisability
If   o
𝑒
𝛼 =
𝑚𝜔
This equation is independent of the frequency of the electric field.
Dielectric Dispersion
Dielectric dispersion is the dependence of the permittivity of a dielectric material on
the frequency of an applied electric field. Because there is a lag between changes in
polarization and changes in the electric field, the permittivity of the dielectric is a
complicated function of frequency of the electric field.
Normal and Anomalous Dispersion
Maxwell’s theory of electromagnetic field provides a relation between the dielectric
constant and the refractive index. The relation between refractive index ‘n’ and frequency
‘’ is
𝑐
 =
𝑛
Where c is the velocity of the e.m wave.
But n = √ε μ
c
 =
√ε μ
For transparent media r = 1
So 𝜀 =𝑛
It is also observed that in frequency regions where the dielectric absorbs energy
from the electromagnetic field, the refractive index is frequency dependent function. Such
a phenomenon is called dispersion.
For visible light frequencies, if the refractive index increases with increase in
frequency, the dispersion is called normal dispersion.
If the refractive index decreases with increase in frequency at frequencies near to a
characteristic frequency of maximum absorption, then it is called anomalous dispersion
Cauchy and Sellmeir relations
According to classical theory of electronic polarization in an alternating field, the
refractive index is given by
𝐴
𝑛 =1+ … … . . (1)
𝜔 −𝜔

Where A =
 But ω = 2π f
c
= 2π

And ω = 2π 

A
n =1+
4π c 4π c
−
 
A 1
=1+
4π c 1 1
−
 
A 1 1
=1+ −
4π c  

A 
= 1+ 1−
4π c 

A  
=1+ 1+ + + ⋯..
4π c  

Taking the square root

b d
n=a+ ++ … ….
 
Where a, b, d,……. Are constants
This is the Cauchy’s dispersion relation. This formula is valid for gases.
In the infinite wavelength limit
A
n = 1 +
ω

A
=1+
4π c

4π c 4π c
ω −ω = −
 
1 1
= 4π c −
 
 −
= 4𝜋 𝑐
 
 Eqn (1) 
A  
n = 1 +
4π c  −

B
=1+
 −


Where B = ∑

This is called Sellmeyer’s dispersion formula

Langevin-Debye equation
Consider an electric dipole of moment p placed in an electric field E. then the torque acting
on the dipole is
τ = pE sin θ
Where  is the angle between the moment and the field direction
The potential energy of the dipole is
V = −pE cos θ
The polarization is
P = Np 〈cos θ〉
Where N is the number of molecules per unit volume and 〈cos 𝜃〉 is the average over a
distribution in thermal equilibrium.
According to Boltzmann distribution law the relative probability of finding a molecule
in an element of solid angle d is proportional to eV/kT and
∫e / cos θ dω
〈cos θ〉 = /
∫e dω
But 𝑑𝜔 = 2𝜋 sin 𝜃 𝑑𝜃

∫e / cos θ 2π sin θ dθ
〈cos θ〉 = /
∫e 2π sin θ dθ
/
∫e cos θ sin θ dθ
= /
∫e sin θ dθ
The integration is to be carried over all solid angles. So that
/
∫ e cos θ sin θ dθ
〈cos θ〉 =
∫ e / sin θ dθ
pE
Let =x
kT
And cosθ = y
 −𝑠𝑖𝑛𝜃 𝑑𝜃 = 𝑑𝑦
∫ e y dy
〈cos θ〉 =
∫ e dy
d
= log e dy
dx
d e
= log
dx x
d d
= log(e − e ) − log x
dx dx
1
= coth x −
x
= L(x)
L(x) is called Langevin’s function
Since x is a small quantity
1 x
coth x ≈ +
x 3
x
∴ L(x) =
3
pE
=
3kT
So polarization
pE
P = Np
3kT
p E
=N
3kT
So that the orientation polarisability per molecule is
p
α =
3kT
If o is the sum of electronic and ionic polarisability, then the total polarisability is
𝛼 =𝛼 +𝛼
𝑝
=𝛼 +
3𝑘𝑇
This is called Langevin – Debye equation

Complex Dielectric Constant

Permittivity is a physical quantity that describes how an electric field affects a
dielectric medium. It is determined by the ability of a material to polarise in response to
the field.
 If the dielectric material is subjected to an alternating electric field, the polarization P
and the displacement D are vary periodically with time. In general P and D may lag behind
the phase relative to the electric field.
Let the electric field applied be
E = Eo sin t
Then the electric displacement can be written as
D = Do cos (t  )
= Do cos t cos  + Do sin t sin 
= D1 cos t + D2 sin t
For most dielectric Do  Eo

But the ratio is generally frequency dependent. Therefore we introduce two frequency
dependent dielectric constants.
D
ε =
E
Do cos 
=
E
D
ε =
E
Do sin 
=
E
The dielectric constant is given by
ε = ε − iε
The corresponding displacement vector is
D=εE e

Also tan 𝛿 =

Since ’ and ’’ are frequency dependent,  is also frequency dependent

This imaginary part of permittivity describes the energy loss from an AC signal as it
passes through the dielectric. The real part of permittivity is also called the dielectric
constant or relative permittivity. the term sin  is called the loss factor (or power factor)
and  the loss angle (or phase angle)
The energy dissipated per unit volume per second is given by
ω
W= ε ε′′
2
Plasma Oscillations
Plasma is called the fourth state of matter. It consists of electrons and positive ions.
Under normal conditions, there are always equal numbers positive ions and electrons. So
the charge density  = 0. Hence there is no large scale electric field in plasma.
Now imagine that all of the electrons are displaced to the right by a small amount x,
while the positive ions are held fixed. The displacement of the electrons to the right leaves
an excess of positive charge on the left side of the plasma slab and an excess of negative
charge on the right side. This charges produce an electric field pointing toward the right.
This pulls the electrons back toward their original locations. However, the electric force on
the electrons causes them to accelerate and gain kinetic energy. So they will overshoot
their original positions. This situation is similar to a mass on a horizontal frictionless surface
connected to a horizontal spring. The spring provides a restoring force on the mass. This
tries to bring back the mass to its equilibrium position. This produces a simple harmonic
motion.
The frequency with which the electrons execute simple harmonic motion is called
the electron plasma frequency.
Expression for Plasma Frequency
Consider a plasma slab in which all of the electrons are displaced to the right by a
small amount x, while the positive ions are held fixed. This produces an excess of positive
charge on the left side of the plasma slab and an excess of negative charge on the right side.
This charges produce an electric field E. By Gauss’s law
Q
E=
ε A
Let n be the number density of electrons in the plasma and  be the charge density
∴ Q= ρV
= ne Ax
ne Ax
E =
ε A
ne x
=
ε
Using Newton’s second law, the force on each electrons is
d x
m = −eE
dt
−ne x
=
ε
Where m is the mass of the electron
d x ne x
m + =0
dt ε
d x ne x
+ =0
dt mε
Thus the oscillations are simple harmonic
Hence the angular frequency of electron plasma oscillations is

ne
ω =
mε

Frequency of electron plasma oscillation is

ω
 =
2π

1 ne
=
2π m ε

Plasmons
A plasmon is a quantum of plasma oscillation. ie, Plasmons are collective oscillations of the
free electron gas density. Just as light consists of photons, the plasma oscillation consists of
plasmons. The energy of a Plasmon is given by
E = h