Lecture 6 Dipolar ion PH611

8/7/2019 Lecture 6 Dipolar ion PH611

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Physically understanding ofdipolar

relaxation Consider what happens if the driving force (electricalfield) is suddenly switched off, after it has been constant for a sufficiently long time so that an

equilibrium distribution of dipoles could be obtained.

We expect then that the dipoles will randomize, i.e. theirdipole moment or theirpolarization will go to zero.

However, that cannot happen instantaneously. A specific dipole will have a certain orientation at the time the

field is switched off, and it will change that orientation only bysome interaction with other dipoles (or, in a solid, with phonons),

in other words upon collisions or other "violent" encounters. It willtake a characteristic time, roughly the time between collisions,before the dipole moment will have disappeared.


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Since we are discussing statistical events in this case, theindividual characteristic time for a given dipole will besmall for some, and large for others.

But there will be an average value which we will call therelaxation time Xof the system. We thus expect a smoothchange over from the polarization with field to zero withinthe relaxation time X, or a behavior as shown below

In formulas, we expect that Pdecays starting at the time ofthe switch-off according to

Po(t)=Posexp(-t/ XThis simple equation describes the behavior of a simple

system like our "ideal" dipoles very well


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Depolarization from saturation

Suppose a static filed is applied for a long time so that

it has reached to the saturation polarization, and then the

field is switched off, then

( ) exp (2)

and

exp( ) ( ) ( )(3)

o os

oso o os o

t P t P

tPdP t P t P P t

dt

X

XX X X

!

! ! !

Rate of depolarization

Because as t goes to infinity, Po (t) become zero. Hence we may write equation as

[Po() Po(t)]/ X= dPo(t)/dt


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AC dipolar polarizability

Recall: Physical understanding of dipole relaxation

On application of a static field, due to collision between dipoles,

polarization of the dipolar dielectric increases as

Here, Po(t) is the polarization at any instant of time and Pos is

Po() the saturation (equilibrium) polarization for the

instantaneous applied field. is relaxation time ortime constant.

( ) 1 exp

exp( ) ( )(1)

o os

oso os o

t P t P

tPdP t P P t

dt

X

X

X X

!

! !

Rate of Polarization


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Now apply an alternating field

Apply

The saturation polarization at any instant of

time will be given as

0( ) exp( ) (4)E t E i t[!

( ) (0) ( );

(0

) static dipolar polarizability

os o

o

P t N E t E

E

!

!

How polarizability will change on application of

oscillatory field?


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How polarizability will change on application

of

oscillatory field?

Form (1) polarizability will change as:

0

( ) ( ) ( )

( ) ( ) ( ) (0) exp( )(5)

o os o

o o os o

dP t P t P t

dt

dP t P t P t N E i t

dt

XE [

X X X

!

! !

The solution of this equation will have the same form as experienced field:

0 0( ) ( ) exp( ) (6)o t E i tE [ [!


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Desired equation of AC dipolar

polarizability Substitute (6) in (5) to obtain

0 0

0

0 0

0

( ) exp( ) (0) exp( )( ) exp( )

( ) exp( ) (0) exp( )( ) exp( )

( ) ( ) (0)

(0)( ) (7)

1

o oo

o oo

o o o

oo

N E i t N E i tdN E i t

dtN E i t N E i t

i N E i t

i

i

E [ [ E [

E [ [

X XE [ [ E [

[ E [ [

X X[XE [ E [ E

E

E [

[X

!

!

!

! Use (7) and derive expressions for

real and complex part of dielectric

Constant and polarization.


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Total Polarizability

2 2 2

2 2

0 0

2 2

2 2 2 2

0 0

ForStatic Field :

1 1

3

ForOscilatory Field:

(0)

2 2 1

e i o

e i

o

e i

e e p

m M M kT

e em mib ib i

m m

E E E E

[ [

E

E

[ [ [X[ [ [ [

! !

!

2


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Complete Frequency Dependence

of a Model Material

Note that [ is on a logarithmicscale!


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Summary


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H2O

NaCl

H2, N2

Lecture 6 Dipolar ion PH611

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