• Static dielectric constant of solid and Liquids

Exibiting only electronic polarizability – elemental solid dielectric

e.g. diamond, Ge, S etc. (materials consisting of single type atom)

Exibiting electronic & ionic polarizabilities – ionic non-polar dielectric – ionic crystal like alkali halides; NaCl, KCl, KBr etc. (materials contain more than one atom but no permanent dipole)

Possess orientational as well as electronic & ionic polarizabilities – polar dielectric (except of molecule and gas which can rotate freely in some solid like nitrobenzene, C6H5NO2 )

Dielectric in alternating field

Does dielectric constant depends on frequency ?

• YESAll polarization mechanisms respond to an electrical field by shifting masses around.

• This means that masses must be accelerated and de-accelerated, and this will always take some time.

• So we must expect that the (mechanical) response to a field will depend on the frequency of the electrical field; on how often per second it changes its sign.

If the frequency is very large, no mechanical system will be able to follow. We thus expect that at very large frequencies all polarization mechanisms will "die out", i.e. there is no response to a high frequency field. This means that the dielectric constant r will approach 1 for .

Electronic polarizability and frequency of the applied field [SO Pillai, page 764]

• In presence of an alternating filed, the electron cloud would execute a simple harmonic motion which will be given by,

0i tE e

2

02

2 20 0

0

0

2

30

2 exp( ) (1)

where 2 damping factor6

in which is permiablity of free space,

c=velocity of light, =natural frequency,

ef=restoring force constant= .

4

d x dxm b fx eE i t

dt dt

eb

mc

R

• Equation (1) can be written as

20

2

2 0

0

20

2 20

2exp( ) 0 (2)

Assumea solution of the form x=Aexp(iωt)

substitute it in (2) to get

2-A exp( ) exp( ) exp( ) exp( ) 0

or,

A= where 2

( )

eEd x b dx fx i t

dt m dt m m

eEb fi t Ai i t A i t i t

m m m

eEfmmb

im

natural frequency of vibration

0

2 20

20

2 20

2

2 20

exp( ), ( ) exp( )

2( )

exp( )( )

2( )

and

2( )

e

ee

eEi t

mx t A i tb

im

e Ei t

mp ex tb

im

ep mE b

im

Q: Can we get their real and complex part ?

Induced Dipole moment:

Polarizability:

Thus Electronic polarizability is a complex quantity.

µe

µe

2

2 20

22 20

2 2 2 20 0

2 22 0

/ //2 2

2 2 20 2

2 22/ 0

2( )

2( )

2 2( ) ( )

2( )

,4

( )

( ),

(

e

e e e

e

em

bi

m

e bi

m m

b bi i

m m

bi

e mor i

bmm

e

m

2 2

2 2 20 2

2//

2 222 2 20 2

4)

2

4( )

e

bm

e band

bmm

Real and complex part of electronic polarizability

Real Part

Imaginary part

Variation of real part and complex part of electronic polarizability with frequency of the applied field

22 2

/ //02 22

00

2/ //

00

/ //0

/ //0

1.If 0, ; 0

2.If , 0;2

3. If , both and are positive

4.If , is negativeand s positive

e e

e e

e e

e e

e e

m m

e

b

i

Self Study [SO Pillai, page 764]

• Find out value of αe’ for = - 0.

• Find out max. and min. value of αe’.

• Find out full width at half maximum for αe’’.

Real part of polarization P

0

/ //0

/ // // /0

/ //

exp( )

( ) cos( ) sin( )

cos( ) sin( ) cos( ) sin( )

Re[ ( )] cos( ) sin( )

e e

e e

e e e e

e e

P N E N E i t

NE i t i t

NE t t i t t

P t t t

Obtain the complete expression for Re[P(t)] and Im[P(t)]

Real and imaginary part of dielectric constant

/ //

0 0

/ /// //

0

/ /// //

0 0

/ //

( )

( )1

Thus, 1 and

Find out complete expression for and .

e e

e er r r

e er r

r r

N iP

E

N ii

N N

Ionic Polarizability

Ionic polarizability• In alternating field the ionic polarization is analogous to the

electronic polarization. The ionic polarizability of a molecule is complex and can be expressed as

2/ //

2 20

0

13 130

12

( )

1 1 1where, reduced mass

Natural frequency of vibration (lies in the

range 10 radian/sec, e.g. for NaCl, =3.2 10 rad/sec)

i i i

i

ei

bi

M M

Ionic polarizability in static field: A particular case of interest

• In case of static filed ω=0. Therefore,

2

20

1 1i

i

e

M M

Find ionic polarizability in a static field for NaCl ion.

Dielectric constant of ionic crystal• Total polarization of an ionic dielectric crystal is the sum of

ionic and electronic polarization. Thus in a static field ω=0 :

0

2

2 20 0 0

2 2

2 20 0 0 0

2

20 0

(1 ) 1

1 1 1 11

1 1 11

1 1 1(0) ( )

r e i

i

ri

r ri

N

Ne

m M M

Ne Ne

m M M

Ne

M M

εr(∞)=n2 is the dielectric constant at a frequency at which ionic polarizability vanishes but electronic polarizability remains operative. This happens in optical frequencies. εr(0)=static dielectric constant; contains contribution form both electronic and ionic polarizability.

Complex Dielectric constant of non-polar solids:

In alternating field; the field experienced by single atom in dielectric (local field/internal field) is given by

EL(t)=Eex(t) +P(t)/3ε0

On using the contribution of two types of polarizabilities: electronic and ionic (net dipole moment is zero)

i.e. α* = αe* + αi

*

Hence the polarization at any instant is given by

P(t) = NReal{(αe* + αi

*) E0L eiwt}

Finally for ac field the relation between complex dielectric constant and polarizabilty is given by Clausius-

Mosotti equation:

(εr* -1)/ (εr

* +2)= N/3ε0 [αe* + αi*]

As, εr*= εr

’ – iεr”