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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