Dielectrics Final

8/7/2019 Dielectrics Final

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3. DIELECTRICS

Introduction

Generally insulators are wide band gap materials with no free charges. Hence they prevent the

flow of electric current through them when an electric field is applied. However, the applied electric

field polarizes the material by creating electric dipoles. An electric dipole comprises equal positive

and equal negative charges bound to each other that are separated by a small distance. If q is the

magnitude of each charge and d is the separation between them then the electric dipole moment of the

dipole is given by = q.d. is a vector quantity which is directed from the negative to positivecharge. The unit of electric dipole moment is Debye; 1 Debye = 3.33x10

30Coulomb-meter.

An electric dipole is capable of producing its own electric field. The field due to a dipole of

moment at a point at a vector distance x from the center of the dipole isE(r) = [3(.r)r - r2] / 40.r5. Therefore the total electric field in a solid dielectric is the vector sumof the applied field and the fields due to the individual dipoles.

The dielectric materials are characterized by a large relative permittivity, r which is also

known as dielectric constant. The capacity of a capacitor is increased when a dielectric material fills

the space between the plates of the capacitor.

Effect of dielectric material on a capacitor:

Consider a parallel plate capacitor with two metal plates of area A separated in vacuum by a

distance d. When a battery source of voltage V volts is connected across these plates, an electric field

E =V/d V.m-1

arises from the charges. If Q be the charge density (charge/unit area) on the plates then

dE

++++++++++++++++

++++++++++++++++++++++++

V


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Q = 0 E .. (1) Where 0 = 8.854 x 10-12

Farad. m-2

is the

permittivity of free space. Q is a source of electric flux lines between the plates and the electric flux

density (also known as electric displacement) is D = Q = 0E. (2)

Now if a dielectric medium is introduced so as to fill the space between the plates then the field

polarizes the medium i.e., dipoles are created in the dielectric medium. Inside the material all dipole

ends of opposite charge will cancel. But there will be uncompensated surface charges, negative at the

top and positive at the bottom of the dielectric. These surface charges on the dielectric will attract and

hold corresponding charges of opposite sign on the plates. Let QBbe the density of charges on the

plates held by the charges on the dielectric. This neutralization of charges on the plates tends to

decrease the electric field between the plates. To maintain the field according to E =V/d, additional

charges flow from the battery to the plates. Thus the total charge density on the plates after introducing

the dielectric material is Q = Q + QB ..(3)

The charge QB is held up and doesnt contribute to the field.

Now the electric flux density is D = Q = E = 0. r.E (4)

Where is the permittivity of the medium and r= / 0 is the relative permittivity of the medium

which is a dimensionless quantity of the dielectric medium. Using eqn (3),

D =Q= Q + QB = 0E + P (5) where the bound charge density is called the

polarization P. This is identical with the induced dipole moment per unit volume.

Thus the Polarization P may be defined as the (induced) dipole moment per unit volume of the

dielectric medium in the presence of an electric field.

Comparing eqns. (4) & (5) , 0.r.E = 0E + P. Rearranging this, we have,

0(r 1) E = P. (or) P / 0E = = (r 1) .. (6)

= (r 1) is called electric susceptibility of the dielectric. As 0E = Q is the free charge density and

P = QB is the bound charge density, is a measure of the ratio of bound charge density to free charge

density.

Thus the presence of the dielectric increases the capacity of the capacitor to store charge by the

factorr, as Q = (0rE) = rQ [Using eqns. (1) & (4) ] where Q is the density of charge stored in

the capacitor without using the dielectric medium.

If N is the number of electric dipoles per unit volume induced by the electric field and if they

are identical each with a dipole moment , then polarization P is given by


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P = N , the induced dipole moment per unit volume. But the induced dipole moment isproportional to the applied field E. So, = E, where is known as the polarizability of the dipole. is of three kinds depending on the basic microscopic mechanisms of polarization. They are (1)

electronic, (2) ionic and (3) orientational polarizations. There is a fourth type of mechanism called

space charge polarization which is generally not considered.

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MECHANISMS OF POLARIZATION: (Microscopic explanation for the origin of Dielectric Polarization)

In a dielectric material, the macroscopic polarization P = N = NE ( is the polarizability ofthe dipole) is created mainly by three types of microscopic polarizations namely, (1) electronic , (2)

ionic and (3) orientational polarizations.

There is a fourth type of mechanism called space charge polarization. It is due to the

accumulation of charges at the interfaces of different phases in a multiphase solid dielectric. It is

usually observed when one phase has higher resistivity than the other. However this is generally negle

(1) MECHANISM OF ELECTRONIC POLARIZATIONThis is the only type of polarization observed in monoatomic gases such as rare gases (He, Ne, Ar etc).

Consider an atom of radius R and atomic number Z so that the charge on the nucleus is +Ze and that

on the electronic charge cloud is Ze. The center of the positive nucleus and the center of the spherical

electronic charge cloud in the atom coincide giving rise to a zero dipole moment. When an externalstatic electric field E is applied, the center of the electronic charge cloud is displaced relative to the

center of the nucleus due to the Lorentz force. The Lorentz force is given by F = Ze.E . (1)

But this displacement gives rise to an attractive Coulomb force between the displaced electronic

charge (within the radius equal to the displacement) and the nucleus. When the total force is zero (i.e.,

when the Coulomb force is exactly equal but opposite to the Lorentz force), a new equilibrium is

established in the presence of the applied field E where x is the new equilibrium separation between

the centers of the nucleus and the electronic charge cloud.

This results in the formation of an electric dipole of moment e = Ze.x . (1)The induced electronic dipole moment e is directly proportional to the field E and hence, theelectronic dipole moment of the atom e = eE .. (2)


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where the constant of proportio nality e is known as the electronic

polarizability of the atom which is a characteristic of the atom. Thus the displacement x is

proportional to the applied field E.Assuming that the charge density of electrons within the atom is uniform,

Charge density of electrons in the atom =1

- .( )4 o

Ze xR 34

3

Ze

R

. (3)

Therefore, electronic charge q within the sphere of radius x = 3

3

4

4 3

3

Ze

R

q =3

3ZexR

(4)

Coulomb force between q & nucleus =' 3

2 2 3

1 . 1. = . .( )

4 4o o

Ze q Ze Zex

x x R

=3

1- .( )

4 oZe x

R

As the total force must be zero when the atom is in equilibrium, it implies that

Lorentz force (Ze.E) is equal and opposite to the Coulomb force.

From the eqns (2) and (4),

We have,( )

2

3

04

Ze xZeE

R

=

Or the displacement3

04 R ExZe

= (5)

x

E


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By definition of an electric dipole, the induced electronic dipole moment

e = Ze.x =3

0.4 Ze R E

Ze

(using eqn5)

i.e., e = 4 0R3E (6)

Comparing eqn(6) with eqn (1) we have electronic polarizability

e=40R3...(7)

As eproportional to R3

, it implies that e is directly proportional to volume of atom.

Also , it is independent of temperature as there is no temperature term is the expansion for e .

Though the theory was approximate, it gives the theoretical interpretation e for the rare gas atom He

, Ne , Ar , Kr, Xe in the increasing order of radii

is, 40 40 40 40 400.18 10 ,0.35 10 ,1.43 10 ,2.18 10 ,3.54 10 F/m2 verifying the correctness of theory.

Thus e depends on the electronic structure of the atoms. As the electronic structure is independent

of temperature, e is independent of temperature. This is found to be correct experimentally if there

are eN atoms per unit volume. Then the induced electronic polarization is

3

0(4 )e e e e e eP N N E N R E = = =

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2. MECHANISM OF IONIC POLARISATION:

Consider an ion pair such as NaCl .in an applied static electric field, the +ve and ve ions are

displaced slightly in opposite direction. Let x1, x2 be the displacement of +ve and ve ions. Let m1 and

m2 be their respective masses. If q be the charge on the ions, then the Lorentz force causing the

separation is given by Fl = qE -------------(1)

Due to the change in the equilibrium separation a restoring force Fracts on the ions. Let 0 be

the angular frequency of vibration of the ionic molecule.

Then, Fr= -k1x1 =-k2x2 --------------- (2)

Where k1 and k2 are the force constants which depends on the masses m1 and m2 and the frequency 0 .

i.e K1 = m102

and k2 = m202

Fr = m102x1 (or) x1 = Fr/m10

2]

Similarly Fr= m202x2 (or) x2 = Fr/m20

2


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Therefore (x1 + x2) = - 20 1 2

1 1rF

m m

+

--------------------------------- (3)

But in the new equilibrium in the presence of applied field E, Lorentz force is equal and opposite to

restoring force. Therefore - Fr = Fl = qE. Using this in eqn. (3),

Therefore (x1 + x2 ) = 20 1 2

1 1qE

m m

+

=2

0

qE

1

where M is the reduced mass of the ion pair given by1 2

1 1 1

m m

= +

.

Ionic dipole moment, i = q(x1 + x2)

=2

2

0

q E

------------------(5)

If there are N number of ion pairs per unit volume, then,

Ionic polarization Pi= Ni =2

2

0

Nq E

.

(or)2

2

0

i i

Nq E P N E

= = as i = iE

Ionic polarizability,2

2

0

i

q

=

This equation implies that, i is independent of temperature. It depends on the force constant.

Increasing ionic size and separation reduces the binding between ions of opposite charge. Hence it will

increase the polarizability.

---------------- --------------------------------------- ----------------------------

ORIENTATIONAL (OR) MOLECULAR POLARIZATION: (DIPOLAR POLARIZATION):

Polar molecules such as water, alcohols, CO, HCl, NH3Cl etc., have no center of symmetry. Each

such molecule is found to possess permanent dipole moment in the absence of an electric field. But

when we consider the collection of large number of such identical molecular dipoles, they are

randomly aligned due to the thermal energy. Hence, in the absence of an external applied field, the net

dipole moment or polarization in any direction is zero. This happens, when the molecules are non-


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interacting (nearly) and are free to move. But when a static electric field is applied along a certain

direction, the different dipoles are partially rotated by the field to different extents. This results in a net

polarization Po in the direction of the field and it vanishes when the applied field is removed. This is

called as Orientational Polarization. If there are N number of molecular dipoles, then P0 = N0E

where 0 is the orientational polarizability of each molecules.

The potential energy of a dipole of moment when it is placed in an electric field is E is

U = -.E = -Ecos where is the angle which the dipole makes with the applied field

E From Boltzmann statistics, the probability the a dipole makes an angle with respect to

E is given by exp[-U/kT] = exp[cos /kT].(1)

No. of dipoles making an angle with field E, is n = n 0 exp[ cosE /kT], where n0 is a

constant

The orientations of different dipoles varies from = 0 to =

No. of dipoles inclined at angles between and d + with field E, dn = d(n0 exp ( cosE /kT))

i.e dn = - n0 ( / ) exp( cos / )sin E kT E kT d .(2)

if there are N dipoles (i.e., polar molecules) per unit volume , then the orientational polarization ,

P0=0 0

( ) .( ( cos ) ) /( ) N N dn dn

=

0

0

0

0

cos( ) exp sin ( cos )

cosexp sin

E E N n d

kT kT E E

n dkT kT

=

= 0

0

cos. cos .exp( )sin

cosexp( )sin

EN d

kT

Ed

kT

Substituting , cos sinE

and d dxkT

= = = , we have

p=

1

1

1

1

. .

.

x

x

N x e dx

e dx


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= N . ( )L

where ( ) [coth( ) 1/ ]L = , is known as Langevin function.

L ()

Langevins function L( ) varies with 1 as shown in the figure.

0

L () 1 as and L () = /3 when is very small such that


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In general P=N E. Considering all types of polarizations, the total polarization P is given by

( ) ( )1 1

.n m

e i oj kj k

P N E = =

= + +

where ( )e j is the electronic polarizability of the jth atom in a molecule and the summation is

carried out over all the n atoms of the molecule . Similarly, ( )ki represents the ionic polarizability

of the k th ion pair in a molecule and the summation is over all the m ion pairs present in one

molecule. o is the orientational polarizability of each molecule.

As electronic and ionic polarizations are independent of temperature, whether the total

polarization will depend on temperature or not depends on whether it has molecular dipoles with

permanent moments or not.

The susceptibility of the system is given by

01 1

/ [ ( ) ( ) ] ( 1)n m

e i o r j ko

N p E j k

= == = + + =

Thus susceptibility and relative permittivityr of a system will vary inversely with temperature

when the system has permanent dipoles which exhibit orientational polarization, as 2 / 3oo kT =

where o is electric dipole moment of each molecular dipole.

The variation of dielectric constant (i.e., relative permittivity)r

or the susceptibility 1r

= ,

as a function of temperature is shown for various gases.

(rvs T for Nitro methane) ( vs 1/T)

CH3Cl

CH2Cl2

CCl4

CH4

1/T (per K)


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It is found that while methane did not show any temperature variation, methyl chloride showed

the inverse dependence ofon temperature. This is because methane is non-polar because of its

center of symmetry .On the other hand methyl chloride has no center of symmetry and hence it is a

polar substance having a permanent dipole moment. So its value inversely varies with temperature

Nitro methane is a polar liquid. It is a a solid below -30 degree C. The dielectric constant of nitro

methane is small at low temperature and it does not vary with temperature till the melting point is

reached. When the temperature becomes equal to its melting point, suddenly,r value rises to a

high value. This is because, in the solid state, the dipoles are held rigidly so that there is no

orientational Polarization. Once they attain the liquid state, the molecule become free to rotate and

hence show a large value ofr . On increasing the temperature further, r values inversely a

temperature.

In solids, Polyethylene terephthalate (PET), Polyvinyl chloride, and carbonates are

polar showing orientational polarization.

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INTERNAL FIELD OR LOCAL FIELD (LORENTZ INTERNAL FIELD):

In a solid medium, an electric dipole experiences an electric field greater than

applied electric field E. This is because field due to the surrounding dipoles also

Have to be taken into account.

Let P be the polarization in a dielectric medium due to an applied static field

E the actual field experiences at any point (such as O) is due to the external

applied field E plus the combined effect of all the dipoles produced by the field

on the point. Lorenz assumed that:

1. The effect of most dipoles away from the point O is given macroscopicPolarization P provided they are sufficiently away.

2. The effect due to dipoles in the vicinity of O is zero if the material has cubicsymmetry about O.

Consider a parallel plate capacitor filled with the given dielectric medium

in the space between the plates.


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Imagine that a small spherical cavity of radius r is cut with the point O

as center at which the local field is to be calculated.The field at O is Eloc = E 1+ E2 + E 3+ E 4

where E 1 is the field due to charges on the plate.

As D = Q = 0 E + P , the field due to all charges on the plates is,

0 0 0

D Q P E

= = +

i.e 10

PE E

= + ----------------------------- (1)

Where E is the applied field and P is the polarization of the dielectric.

E2 is the field due to charges on the surfaces of the dielectric. As P=Q B.

The field due to the bound charge density QB is0 0

BQ P

= . But charges

of opposite polarities are induced on the surface of the dielectric. Hence, this

field is in the direction opposite to the applied field, E.

Hence, 20

PE

= --------------------------------- (2)

E3 is the field due to dipole interactions of the material within the small

sphere of radius r when the material is replaced. For cubic or spherical symmetry of the material this

field is zero.

i.e E3=0 ----------------------- (3) (for cubic crystal)


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E4 is the field due to polarization charges appearing on the surface of the cavity.

Consider an elementary ring of area dA lying between and + d. Let y is its radius and dy be the

thickness. Then 2 2 ( sin )( )dA ydy r rd = =

i.e 22 sindA r d =

The polarization charge density at angle is P.cos

and hence the radial Field at O

due to these polarization charges. =2

0

cos .

4

P dA

r

Therefore field at O due to these polarization charges along the direction of the

applied field = (2

0

cos .

4

P dA

r

).cos

Therefore total field due to all Polarization charges on the spherical

cavity in the direction parallel to the applied field2

4 2

00

cos

4

P dAE

r

= ------------ (5)

Substituting for dA from (4) , and evaluating the integral, one finds,

4

03PE

= -------------------(6)

Therefore total local field at O is

1 2 3 4

0 0 0

. , ( ) 03

lo c

lo c

E E E E E

P P P i e E E

= + + +

= + + +

or03

loc

PE E

= + This eqn is valid for crystals with cubic symmetry .

In general the local field is given by0

loc

PE E

= + where, is internal field constant.

= 1/3 for cubic materials.


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Clausius -_Mossatti relation:

Consider a dielectric with N molecules per unit volume. If the molecules do not possess permanent

dipole moments then the induced polarization is proportional to the local electric field.

i.e. P = N = N locE ---------------------

(1)

Where = locE is the induced dipole moment on each atom; the Polarizability

and locE the local field given by

0

loc

PEE = +

------------------------------------ (2)

But polarization 0 ( 1)rP E= ------------------ (3)

Substituting this in eqn-(2) we get

0

0

[ ( 1) ] 11

3 3

r rloc

E E E E

= + = +

i.e.2

3

rlocE E

+ =

------------------------------ (4)

Substituting this in eqn (1) we have

2

3

r

loc P N E N E

+

= = -----------------------

(5)

Comparing equations (5) and (3),

2

3

rN E +

= 0 ( 1)r E

Canceling E and rearranging

0

1

3 2

r

r

N =

+---------------------------------------- (6)

This Equation is known as Clausius Mossatti relation

This expression is a simple way of relating the microscopic quantity with the macroscopic quantity,

r. If = density,

M=molecular weight and 0N Avogadro number


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Then 0

0

1

3 2

r

r

N

M

= +

Or 0

0

1

3 2

r

r

N M

= +

------------------------- ( 7)

Here 0N is the molar polarizability

If the material consists of different types of molecules then the eqn -6 becomes

0

1

3 2

i i

i r

r

N

= +

where i is the polarizability ofthi molecule and iN is the number per unit volume of

thi type

of molecule

NOTE:

If all the three types of polarizations are present eqn-(7) can be written as

2

0

0

3 1

3 2

e i

r

r

NkT M

+ + = +

where e and i are the total electronic polarizability and total

ionic polarizability per molecule and2

3kT

= 0 is the orientational polarizability per molecule.

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Dielectric loss:

An electric dipole tends to align itself in the direction of a static field. If it is placed in an alternating

field of frequency , it tends to follow the field and be in phase with it. However, viscous forces due

to the interaction of a dipole with other dipoles of the medium tend to prevent this. This leads to

dielectric loss, which appears as heat. Because of the possibility of a phase difference between the

applied field E and the instantaneous polarization P the relative permittivity becomes a complex

quantity and is expressed in terms of a real; and imaginary part, i.e. complex relative permittivity*

r r ri = -------------------------------------- (1)

It can be shown that the imaginary part gives rise to absorption of energy by the material from the

alternating field.

Expressing the instantaneous field E(t) as


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E(t) =0 0cos .Re[ ]

i tt eE E

= --------------------------(2)

Instantaneous electric displacement is

D(t) =*

0 00 0cos Re[ ]

i t

r rt eE E

= ---------------------------(3)

the current density J(t)= dD/dt=*

00[Re( )]

iwt

r

d

dteE

Using eqn (1) J(t) = 00 [Re( ) ]iwt

r r

di

dteE

= Re[( )( ) ]i t

oo r ri eE

= Re[ ( )(cos sin )]oo r r

i i t i t E +

= [ cos sin ]oo r rt tE --------------------------

(5)

This expression indicates that the current density has two components; one component is in

phase with field E(t) while the other component is out of phase by / 2 with the field . The

amount of energy absorbed from the field per unit volume per unit time is given by

W(t)=0

1( ) ( )

T

J t E t dtT where 2 /T = is the time period of the alternating field .

Substituting for J(t) from (5) and using0

( ) cos , E t t E =

W(t)=!! !2

00

0

/ ( cos sin ) cosT

r rT t t tdt E

=!!2

002 rE

. Thus W(t)!!

r . This energy W(t) represents the energy

dissipated in the medium and is termed as dielectric loss. We find that dielectric loss is

proportional to the imaginary part of the relative permittivity.

It is customary to characterize the dielectric loss at a particular frequency at a given

temperature by a factor called loss tangent tan , defined as

tan =

!!

!(6)r

r

. is known as loss angle.


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Physical meaning of and tan

Consider the equation for instantaneous current density J(t).

J(t)= [ cos sin ] (5)oo r

t trE

If the material is an ideal dielectric material without any dielectric loss, then 0r =

and J(t)=0 00

sin cos( 90)o r

t tE E = + ie. The current leads the field by 90

degrees. Under these circumstances, tan 0r

r

= =

.

But if the material is lossy, then there is a current component in phase with the field.

The resulting current is the resultant of the two components and it will no longer lead the field

by90 but by (90- ) as indicated in the fig..The ratio of in-phase current amplitude, Jin to J

the out of phase current Jout gives tanr

r

=

. Jout

E

Jin

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Schering Bridge:

Schering bridge is used to determine the capacitance of

capacitors. Also it is very useful in studying the relative

permittivity of dielectric materials as a function of frequency. In

the circuit diagram1c represents the unknown capacitance

which is to be determined and R1 is the leakage resistance of this

capacitor. R3 and R4 are non-inductive resistances of which

3R is variable. 2c is a standard capacitor which has very low

losses (such a standard mica capacitor).4c is the variable

capacitor parallel to which the resistance4R is connected . D is


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a null detector which shows null deflection or null signal when the bridge is balanced.

First1c is connected without inserting the dielectric medium for which constant and loss

factor tan are to be determined. The bridge balanced by adjusting34

andc R . Under the

balanced condition, the impedance of the different elements are in the ratio,

31

2 4

ZZ

Z Z= -------------- (1) Where Z is the impedance

1 1

1

iZ R

C

=

is the impedance of first arm

42 3 3 4

2 4 4

, ,1

Ri Z Z R Z

C i R C

= = = +

where is the frequency of a.c

substituting for the impedance in equation 1 and cross multiplying

41 3

1 4 4 21

Ri iR R

C i R C C

= +

rearranging, ( ) ( )31 4 4 41 2

1Ri

R R i i R C C C

= +

expanding, 3 3 4 441 41 2 2

R R R C R R R i i

C C C = +

Equating the real parts , 3 4 41 42

R R CR RC

= (or) 41 32

CR RC

= ------------------ (2)

Equating the imaginary parts

34 41 2

1 2 3

( )RR R

or C C C C R

= = ------------------------ (3)

The loss tangent tan is given by

4 2 41 1 3

3 2

tan .R C C

C R RR C

= =

i. e 4 4ta n C R = --------------------------------(4)

Equation (3) gives the unknown capacitance of the capacitor C1 . to determine the dielectric

constant and tan of the given dielectric material it is inserted between the plates of capacitor C1.Its

shape and size are such that it fills the space between the plates exactly .once again the bridge is


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balanced and the capacitance is determined as 1C . The ratio of 1C to C1 gives the dielectric constant

of the material at frequency . i.e., ( )11

r

C

C

=

At this frequency the loss factor is 4 4tan C R = . If R4 has a fixed value, the value of 4C is calibrated to read tan directly. Similarly, the value of R3 is calibrated to read C1 directly .

As tan ,r rr

=

also can be found as a function of frequency.

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FREQUENCY DEPENDENCE OF POLARISATION AND DIELECTRIC CONSTANT:

The total polarization P of a dielectric material may contain contribution from all the three

types of polarization namely electronic, ionic and orientational polarization.

P = NE = 0 (r-1)E

Here and rare total polarizability and total relative permittivity respectively of the medium

including all the three contributions ; = e + i + o and r= (r)e + (r)I + (r)o

When such a dielectric material is subjected to an alternating electric field, the dipoles tend to follow

the variation of the applied field. But due to the effect of viscous forces they are unable to follow the

field in-phase at certain frequency ranges. Due to the possibility of a phase difference, P, e and r

become complex quantities. The complex relative permittivity r*

is defined by r*=r - ir.

At certain characteristic frequencies, the effect of viscous forces becomes predominant and hence, the

dipoles absorb energy from the applied a.c field to overcome the viscous forces. This results in the

reduction ofr around these frequencies while rbecomes significant. r corresponds to the

dissipation of energy by the material.

At very low frequencies, all dipoles are able to follow the variation of the applied field , without

any phase difference and hence there is no dielectric loss . So the relative permittivity is purely real ie.

r*=r at low frequencies between 0 and 10

6Hz.

As the molecular dipoles are heavier, orientational polarization occurs very slowly and their

response time is characterized by certain relaxation times of the order of=10-10 to 10-6s.When the

frequency of applied a.c field becomes closer to 1/ (i.e. 106 to 1010 s-1, corresponding to microwave

frequencies), effect of viscous forces is significant. Hence in order to overcome the viscous effect, the


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dipoles absorb energy from the applied field leading to dielectric loss (ie. dissipation of energy in the

form of heat) in this frequency interval .

r decreases while rbecomes significant having maximum at =1/ .If the frequency of field is

increased further the molecular dipoles can no longer keep up with the applied field and so it is no

longer oriented by it. Hence, beyond = 1/ , polarization and dielectric constants have contributions

from electronic and ionic polarization only.

At frequencies higher than 1010

Hz, ionic and electronic polarizations show resonances

which occur at about 1012

and 1015

Hz respectively in the IR and UV of the electromagnetic spectrum

respectively. In the frequencies corresponding to these resonances, r decreases while r increases

.Above IR frequencies but below UV frequencies (1015

Hz) only electronic polarization occurs.

Above these frequencies, none of the charges or dipoles is able to show a dielectric response.

At the resonance frequencies of ionic dipoles and electronic dipoles also r shows peak values.

Note:

1) Generally in solids, the inter atomic forces seriously restrict the orientational polarization. So it is

relatively unimportant in solids. 2) As the refractive index n = r, in the visible and IR regions,

the dielectric constant may be obtained from square of refractive index.


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PIEZO, PYRO AND FERROELECTRIC MATERIALS

Totally there are 32 crystal classes based on their crystal symmetry. Out of these 32 , 20 crystal classes

do not possess a center of symmetry . All the materials in these 20 classes are piezoelectric. Out of

these 20 classes, 10 are pyroelectric. A restricted group of pyroelectrics have the further property of

being ferroelectric. Thus all ferroelectric materials are also pyroelectric as well as piezoelectric. All

pyroelectric materials are also piezoelectric .But the converse is not true always. All piezoelectric

materials are not pyroelectric or ferroelectric and all pyroelectrics are not ferroelectrics.

Piezoelectric materials have the property of becoming electrically polarized in response to an applied

mechanical stress.

A Pyroelectric Material exhibits spontaneous polarization in the absence of an electric field and it

changes its polarization on heating.

A Ferroelectric material is one which exhibit spontaneous polarization in the absence of an electric

field and it shows a hysteresis in the polarization electric field (p vs E ) relation.

A material cannot be piezoelectric, pyroelectric or ferroelectric unless its crystalline symmetry

lacks an inversion center (ie. it does not possess centre of inversion symmetry).

Piezoelectric Effect:

When certain crystals are subjected to mechanical stresses, equal and opposite charges appear

at the two end faces resulting in polarization. The magnitude of charge developed is directly

proportional to the mechanical stresses. This polarization is due the strain induced by the stress.

Thus, the process of application of stress to a crystal which induces a strain that results in a net

polarization is known as Piezoelectric effect.

The inverse of this piezoelectric effect known as inverse piezoelectric effectproduces a strain in the

piezoelectric material when an electric field is applied to the material. Whether the strain is elongation

or contraction depends on the field direction.

Both direct and inverse piezoelectric effects find important practical applications.

The piezoelectric materials permit the conversion of mechanical energy into electric energy and vice-

versa.


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Among single crystal piezoelectric materials, quartz is notable for its temperature independence.

Natural quartz crystals have the shape of hexagonal prism with a pyramid attached to each end. A thin

slice of quartz with its faces parallel to the hexagonal basal plane will contract or expand in the

direction parallel to an electric field, which is applied perpendicular to the faces. When the frequency

of the applied a.c field matches the natural frequency of vibration of quartz, due to resonance

vibration, ultrasonic waves are produced.

Other important piezoelectric materials are

1) Rochelle salt, Barium titantate, PZT (50% lead titanate PbTiO3+50% lead zirconate, PbZrO3), lead

mate niobate etc. These are ferroelectric ceramic materials.

2) Many group -VI compound semiconductors such as ZnS, ZnSe, ZnTe, CdS, CdSe, CdTe etc are

strongly piezoelectric.

3) Group III-V semiconductors such as GaAs are weakly piezoelectric.

Applications of piezoelectric materials:

Rochelle salt, BaTiO3, PZT etc. are used as:

1) Gramophone pickups

2) Air transducers such as earphones, hearing aids, microphones.

3) Ultrasonic flaw detectors

4) Underwater sonar devices

5) High voltage generators

6) Delay lines etc.

7) The propagation of electrical Signals as acoustic surface waves in quartz and lithium niobate is the

basis of a number of signal processing applications at GHZ frequencies

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Pyroelectric effect:

Pyroelectric materials have spontaneous polarization in the absence of an external applied

electrical field. However the electric field of the material due to spontaneous polarization is not easy to

detect. But if the temperature of the material is changed, there is a sudden change of polarization and

the external field of the material can be detected for a short time during the change of temperature. If

T is the change in temperature and P is the change of of polarization then,

P = T, where is called the pyroelectric coefficient of the material.


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Significance: As it is possible to detect a charge of 10-6

C with a suitable electrometer, temperature

changes as small as 10-6

Degree C can be measured by using a pyroelectric material.

Parameters: Pyro electric materials are characterized by the factor known as figure of merit defined

by figure of merit 1/ 2r

F s

= . A good pyroelectric material should have a large value of F. Thus

general requirement for pyroelectric material are high pyro electric coefficient ; low relative

permittivity r , low density and low specific heat s. In addition, they should have a good physical

and chemical stability and low piezoelectric response.

Materials: Triglycine sulphate known as (TGS) is the most widely used pyroelectric material because

of its better pyro electric characteristic. It has a high figure of merit (2 to 3) and is good absorber of

radiation in the wavelength range 8 to 14 m. Pyroelectric ceramics based on

1. Barium titanate doped with strontium (F = 0.53 to 2.3 )

2. Lead zirconium titanate (PZT) system (F=.12 to 1.85) etc are also useful.

Application Of pyro electric materials:

1. The pyroelectric effect is used to make very good infra-red detectors which can operate at room

temperature. [IR detections based on semiconductors require cooling below RT for their efficient

operation]

2. They are used in fire alarms and burglar alarms.

3. They are used in the construction of pyroelectric image tubes for use in the dark.

4. Minute changes in temperature can be measured accurately.

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Ferro electric materials:

These are dielectric materials which show spontaneous polarization even in the absence of

an external applied field below a certain characteristic transition temperature known as curie

temperature. They are characterized by their hysteresis in their polarization versus applied field ( P vs

E ) relation, similar to magnetic hysteresis. Thus the name ferro electricity has arisen in analogy with

the hysteresis behavior of ferromagnetic materials. [Similar to ferromagnetic materials, ferroelectric

materials also have domains below Curie temperature. Within each domain all electric dipole moments

are aligned in the same direction. The domain structure of Ferroelectric materials is observable under

polarized light below Curie temperature.]


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In the ferroelectric phase, the ferroelectric materials have a very large value of relative

permittivity r . It shows a maximum at Curie temperature and above this transition temperature, the

spontaneous polarization is destroyed by thermal disorder. Above Curie temperature, it becomes para

electric and shows an inverse dependence of susceptibility orr

on temperature.

i.e., electric susceptibility,c

C

T T =

,= r -1, where C is a constant known as Curie constant.

Ferro electric materials are an important class of materials not only because of their ferro

electric behavior but also for their useful piezoelectric, bi-refringent and electro-optic properties which

are exploited in several devices.

Materials:

A large group of Ferro electric materials belong to crystals whose structure is based on the

mineral pervoskite ,CaTiO3 .they have general formula ABO3. The best known Ferroelectric material

with pervoskite structure is barium titanate, BaTiO3. Other useful materials belonging to this group are

niobates and tantalates; mixed compounds such as lead titanate, lead zirconate (known as PZT) etc.

Another group belongs to Rochelle salt (sodium potassium salt of tartaric acid,

NaKC4H4.H2O).Other members of this group are obtained by replacing K by NH4, Rb or Tl or

replacing Na by Li.

Another important group of materials are Dihydrogen Phosphates and arsenates of the alkali

metals. Most important among these are Potassium Dihydrogen Phosphate (KH2PO4, called KDP) and

Ammonium Dihydrogen Phosphate (NH4H2PO4, known as ADP).The ferroelectric material Triglycine

sulphate is an important pyroelectric material.

Ferro Electric Behavior Of Barium Titanate, BaTiO3 :

Barium Titanate becomes ferroelectric below 393K.In the non-ferroelectric state (above

393K), Barium Titanate has a cubic pervoskite structure. In this structure Ba2+

ions occupy

corners of the cube, O2-

ions occupy the face centers and the Ti4+

ion occupies the body centre

position. BaTiO3

shows three structural phase transitions below 393K.

When barium titanate is cooled below 393K it becomes ferroelectric at 393K and its structure

changes from cubic to tetragonal. The direction of spontaneous polarization is along [001] direction.


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Its structure becomes orthorhombic when it is

cooled below 278K. At and below 193K its

structure changes to rhombohedral. When it is

having orthorhombic and rhombohedral structures

the directions of spontaneous polarization are along

[011] and [111] directions respectively.

The spontaneous polarization of BaTiO3 measured

along [001] direction changes with temperature as

shown in the figure.

MECHANISM OF SPONTANEOUS POLARISATION IN BaTiO3:

Below 393K, spontaneous polarization of BaTiO3 arises from a

coupling between ionic and electronic dipole moments.

1. Due to the rattling motion (vibratory motion of Ti4+

ions owing

to their high charge, an ionic dipole moment is created in the direction

of displacement.

Ba2+ O2- 2. The electric field associated with the ionic dipole causes

Ti4+

electronic polarization on some oxygen ions in the opposite directions.

3. The field produced by O2-

ions tends to increase the Ti4+

displacement in the same direction. Thus the central Ti4+

ion is displaced away from its equilibrium

body- centre position and hence it occupies an off-centered position.

4. The local field at O2-

and the field due to oxygen dipoles displace the Ti4+

ions which lead to a

change in structure. As a result of change in structure from cubic pervoskite to tetragonal (a = b

Dielectrics Final

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