lecture 6 dipolar ion ph611
TRANSCRIPT
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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