Download D (t-t

Lecture 6
1. The dielectric response functions. Superposition principle.
2. The complex dielectric permittivity. Loss factor.
3. The complex dielectric permittivity and the complex
conductivity
4. The Kronig-Kramers relations
5. The dielectric relaxation.
1
PHENOMENOLOGICAL THEORY OF LEANER
DIELECTRIC IN TIME-DEPENDENT FIELDS
The dielectric response functions. Superposition principle.
A leaner dielectric is a dielectric for which the superposition principle is
valid, i.e. the polarization at a time to due to an a electric field with a
time-dependence that can be written as a sum E(t)+E’(t), is given by
the sum of the polarization’s P(to) and P’(to) due to the fields E(t) and
E’(t) separately. Most dielectrics are linear when the field strength is
not too high.
The superposition principle makes it possible to describe the
polarization due to an electric field with arbitrary time dependence,
with the help of so-called response functions.
Let us consider the changes of electric field from value E1 to a value E2
at a moment t‘ :
E (t)  E1  ( E2  E1 )S(t  t' )
(6.1)
2
where S is the unit-step function:
S(t)  0
S(t)=1
t  0

t>0 
(6.2)
The time-dependent field given by (6.1) can be considered as the
superposition of a static field, E2, and time-dependent field given by:
E(t)  (E1 - E2 )1  S (t  t )
P2
P
(6.3)
Therefore, we find from the
superposition principle that
the polarization at times t  t’
due to the field given by
eqn.(6.1) is the equilibrium
polarization E2 for the static
field E2 and the response of
the field change E1-E2 (see fig.
6.1).
P1
0
E2
E
E1
0
t'
t
Figure 6.1
3
For a linear dielectric this response will be proportional to E1-E2, so
that the total polarization is given by:
P(t)  E2   E1  E2  t  t,
t  t
(6.4)
Here (t) is called the step-response function or decay function
of the polarization. For simplicity let us rewrite this expression in much
convenient form:
P(t)  P(0) ( t )  E( 0 ) ( t )
(6.5)
At t=0 (t=t’ in 6.4)
( 0 )  1
(6.6)
In principle, both a monotonously decreasing and oscillating behavior
of (t-t’) are possible. For high values of t, P will approximate the
equilibrium value of the polarization connected with the static field E2.
From this it follows that
(  )  0
(6.7)
4
Let us consider the case of block function. For t1-t<tt1 the field
strength is equal to E1 , and for t t1-t and t> t1 it equal to zero. This
block function can be considered as the superposition of two fields with
unit-step time dependence:


 
(6.8)
E( t )  E1 S t t1  t  E1 S t t1
The resulting polarization for t  t1 can be considered as the
superposition of the effects of both unit-step functions:
P(t)  E1 ( t  t1  t )  ( t  t1 )
(6.9)
An arbitrary time dependence of E can be approximated by splitting it
up in a number of block functions E=Ei for ti - t < t  ti. The effect of
one of these block functions is given by (6.9). Since the effects of all
block functions may again be superimposed, we have:
P (t )    E i   (t  ti  t )   (t  ti )
(6.10)
i
In the limit increasing the number of block functions (6.10) can be
written in the integral form:
5
   (t  t' ) 
P (t )    E (t ' ) 
dt' =

t



t
(6.11)
t
=   E (t ' ) (t  t ' )dt'

  p (t  t ' )   p (t  t ' )
where,
called pulse-response function of
polarization. The equation (6.11) gives the general expression for the
polarization in the case of a time-dependent Maxwell field.
Let us consider now the time dependence of the dielectric displacement
D for a time dependent electric field E.
D(t)  E(t)  4P( t )
(6.12)
For the linear dielectrics the dielectric displacement is a linear function
of the electric field strength and the polarization, and for those
dielectrics where the superposition principle holds for P, it will also hold
for D.
Thus, we can write for D analogously to (6.11):
t
D(t )    E(t ')D (t  t ')dt'

(6.13)
6
with   D (t  t ' )  D (t  t ' )
The relation between p and D is the following:
 D (t  t ') 
1

1  S (t  t ') 
4

 p (t  t ')
(6.14)
Taking the negative derivative of (6.14) we can get the relation
between p and D :
 (t  t ') 4
D (t  t ') 

 (t  t ')

 p
(6.15)
The unit step function in (6.14) implies that there is an instantaneous
decrease of the function D (t-t’), from the value D (0)=1 to a limit
value given by:
(   1)
(6.16)

(
t

t
'
)

lim D
t  t'

In contrast the step-response function of the polarization cannot show,
in principle, such an instantaneous decrease, since any change of the
polarization is connected with the motion of any kind of microscopic
particles, that cannot be infinitely fast.
7
However, in the case of orientation polarization we can neglect the
time necessary for the intermolecular motions by which the induced
polarization adapts itself to the field strength. In this approximation,
the induced polarization is given at any time t:
Pin (t)  E(t)(    1 ) / 4
(6.17)
where  is the dielectric constant of induced polarization. We can
rewrite (6.12) in the following way:
D(t )    E (t )  4Por (t )
(6.18)
It is then useful to introduce response functions por and por describing
the behavior of the orientation polarization for a time dependent field
and to consider there relationship with D and D respectively.

s   or
 D (t  t ' )  1  S (t  t ' ) 
 p (t  t ' )
s
s
(6.19)

s   or
D (t  t ' )   (t  t ' ) 
p ( t  t ' )
s
s
(6.20)
8
From (6.19) that now (6.16) no longer holds, but should be changed by:
lim
t t '
( t  t' ) 
D
s  
s
(6.21)
From comparison of (6.19) and (6.20) with (6.14) and (6.15) one can
obtain the expressions for the response functions of the polarization in
the case that the time necessary for the intramolecular motion
connected with the induced polarization can be neglected.
  1
 s    or
 p ( t  t' ) 
1  S ( t  t' ) 
 p ( t  t' )
s  1
s  1
  1
 s    or
 p ( t  t' ) 
 ( t  t' ) 
 p ( t  t' )
s  1
s  1
(6.22)
(6.23)
As was expected, the assumption that the induced polarization follows
the electric field without any delay leads to the occurrence of a unitstep function in the expression for response function of orientation
polarization. From (6.16) it follows:
(6.24)
s  
lim ( t  t' )    1
t  t'
p
s
9
The complex dielectric permittivity.
Laplace and Fourier Transforms.
Let us consider the time dependence of the dielectric displacement D
for a time dependent Electric field:
t
D(t )    E(t ')D (t  t ')dt'

Applying to the left and right parts the Laplace transform and taking
into account the theorem of deconvolution we can obtain:
D * ( )   * ( ) E * ( ),
where
(6.25)

 * (s)   s   D (t ) exp( st )dt   s L[ D (t )]
(6.26)
0
(s=+i; 0 and we’ll write instead of s in all Laplace transforms i).
10
Taking into account the relation (6.20) we can rewrite (6.26) in the
following way:
(6.27)
 * (i )     ( s    ) L[ por (t )]
From another side complex dielectric permittivity can be written in the
following form:
 * (  )  ' (  )  i" (  )
(6.28)
The equation (6.27) justifies the use of the symbol  for the dielectric
constant of induced polarization, since for infinite frequency the
Laplace transform vanishes, and the expression becomes equal to .
The real part of complex dielectric permittivity ’() is associated with
real part of Laplace transform of orientation pulse-response function:

 ' ( )       por (t ) cos  dt
(6.29)
0
and the imaginary part of complex dielectric permittivity ’’() is
associated with the negative imaginary part of the Laplace transform
of orientation pulse-response function:
11

 ' ' ( )    por (t ) sin  dt
(6.30)
0
Let us now reconsider the relationship between time dependent
displacement and harmonic electric field:
E (t )  E o e it  E o cos t  iE o sin t
(6.31)
We’ll rewrite in this case the relation (6.25) in the following form:
D( t )  ' (  )E o cos  t  " E o sin  t
(6.32)
that can be presented as follows:
D(t )  D0 cos  cos  t  D0 sin  sin  t  D0 cos( t   )
where D  E
0
0
 '  "
2
2
(6.33)
 "( )
and tg ( ) =
 '( )
12
From the equation (6.32) it clearly appears that the dielectric
displacement can be considered as a superposition of two harmonic
fields with the same frequency, one in phase with electric field and
another with a phase difference

2
. The amplitudes of these fields
are given by ´E o and ” E o , respectively. Calculation of the energy
changes during one cycle of the electric field shows that the field with

a phase difference 2 with respect to the electric field gives rise to
absorption of energy.
The total amount of work exerted on the dielectric during one cycle
can be calculated in the following way:
1
W 
4
2 / 
 EdD
t 0
2 / 
2 / 


1
0 2

( E )  ' ( )  cos  t d cos  t   ' ' ( )  cos  t d sin  t 
4
t 0
t 0


1

 ' ' ( )( E 0 ) 2
4
(6.34)
13
Since the fields E and D have the same value at the end of the cycle as
at the beginning, the potential energy of the dielectric is also the same.
Therefore, the net amount of work exerted by the field on the dielectric
corresponds with absorption of energy. Since the dissipated energy is
proportional to ”, this quantity is called the loss factor.
From (6.34) we find the average energy dissipation per unit of time:
o 2

(
E
)
 o o

W
" (  ) 
D E sin 
8
8
(6.35)
where  called a loss angle.
According to the second law of thermodynamics, the amount of energy
dissipated per cycle must be always positive or zero. It means
that
    0
(6.36)
14
The complex dielectric permittivity
and complex conductivity
In a harmonic field with angular frequency  and amplitude Eo the
dissipation of energy per unit of time in a dielectric is given by (6.35):
0 2

(
E
)

W
 "( )
8
This equation holds for dielectrics that are ideal isolators. However,
most of real dielectrics show a certain conductivity , leading, in a first
approximation, to an electric current density I in phase with the electric
field:
(6.37)
I(t)  E( t )
The electric current causes dissipation of energy. According to Joule’s
law, the amount of energy dissipated during the time interval dt is
given by:
2
(6.38)
W  I  E dt  E dt
15
For a harmonic field, the energy dissipation during one cycle amounts
2  /
to:

0 2
2
(6.39)
W   ( E ) cos  tdt   ( E 0 ) 2


t 0
Hence, the average dissipation of per unit of time due to condition is:

1
W   (E 0 )2
2
o 2

(
E
)

W
 " ( )
8
(6.40)
Comparing (6.40) with (6.35), we see that if we determine ”())
from the absorption of energy in a dielectric we always obtain the sum
”()+4/ so that we must correct for the contribution 4/ due
to the conductivity of the dielectric; the reason for this is the
equivalence of the current density and the time derivative of the
dielectric displacement in Maxwell’s law:
1
curlH = ( D  4I )
c
(6.41)
16
As long as the current density is given by (6.38) there is no problem in
separating the effects of conduction and polarization, since  is a
constant that can be determined from measurements in static fields.
However, when it takes a certain time for the current to reach its
equilibrium value, the relation between the field and the current
density is given by a pulse-response function I :
t
I (t )    E(t ' )I (t  t ' )dt'
(6.42)

As for the pulse-response functions of the polarization p and of
dielectric displacement D , the pulse-response function of the current
density I is associated with a step-response function I:

I (t  t ' )    I (t  t ' )
t
(6.43)
Analogously to the relation between displacement and the electric field
(6.24) after application to the left and right parts of (6.42) the Laplace
transform and taking into account the theorem of deconvolution we can
obtain:
17
I (  )   (  )E (  )

where
*
*
  (  )   s L [  I ]   ' (  )  i " (  )
(6.44)
(6.45)
The quantity ’() gives the part of the current which is in phase with
the field and which therefore leads to absorption of energy. Hence,
this quantity is comparable with ’’(). The quantity ’’() gives the
part of the current with a phase difference of with respect to the field.
Thus, ’’() is comparable with ’()
It is possible to combine the dielectric displacement and the electric
current by defining a generalized dielectric displacement D (t):
t
D (t) = D(t) + 4  I (t' )dt'
(6.46)
-
To ensure convergence of the integral, it is necessary that E(t) approach
a limiting value zero for t=- fast enough; this corresponds with the
fact that the field has been switched on at some moment in the past.
t
t
The relation
between D (t) and electric field can be found by
I (tinto
)  (6.46):
 E(t ' )I (t  t ' )dt'
D(substituting
t )   E(t ')(6.13)
D (t and
t ')dt'(6.42)




18
t
t
t'
-


D (t) =   E (t ' )D (t  t ' )dt '4    E (t" )I (t 't" )dt ' dt"
(6.47)
Using (6.43) and the fact that the current step-response function
I(0)=1, we can find
t
t
-

D (t) =   E (t ' )D (t  t ' )dt '4  E (t ' ){1  I (t  t ' )}dt ' 
t
  E (t ' )D (t  t ' )dt '
(6.48)
-
where D(t), the pulse-response function of the generalized dielectric
displacement, is given by:
D (t  t ' )  D (t  t ' )  4[1   I (t  t ' )]
(6.49)
If we’ll again apply Laplace transform to the left and to the right parts
of (6.47), we’ll obtain:
19
D * ( )   * ( ) E * ( )
where
*    LD ( t )
(6.50)
(6.51)
Making the Laplace transform in (6.51) we’ll get:
*
4

( )
*
*
 ( )   ( ) 
i
(6.52)
Splitting up  * (  ) into real and negative imaginary part we arrive at:
'( )   '( ) 
"( )   "( ) 
4 "( )

4 '( )

(6.53)
(6.54)
20
The Kramers-Kronig relations
The Kramers-Kronig relations are ultimately a consequence of the
principle of causality - the fact that the dielectric response function
satisfies the condition:
 ( t )  0 for t  0
(6.55)
It means that there should be no reaction before action.
Let us consider again the relations for real and imaginary part of
complex dielectric permittivity:

 '( )     por (t ) cos dt
0

 ''( )   por (t ) sin  dt
0
21
Both ‘() and “() are derived from the same generating function
p(t)
and that it should be possible in principle to “eliminate” this
function and to express ‘() in terms of “().
Let us consider the properties of Hilbert transform:


sin xt
sin( x   )t
 x   dx  cos t  ( x   )t d [( x   )t ]

cos( x   )t
d [( x   )t ]
( x   )t

 sin t 
(6.56)
In this integral we ignore the imaginary contributions arising from
integration through the pole at x=. The first integral is equal to ,
the second vanishes so that we can obtain:

sin xt
(1 /  ) 
dx  cost
x



This called Hilbert transform
(6.57)
22
Let us apply this to the ´() (6.29):


sin xt
 ' ( )        (t ) 
dxdt
x 


1

1
1
    
x 

or
p

or

 p (t ) sin dxdt

(6.58)
In these manipulations we have extended the integration (6.29) to -
which is permissible in view of the causality principle. The second
integral in (6.58) is equal to “() in view of (6.30) so that one can
finally write :

 "( x)
 '( )    (1 /  ) 
dx
x 

and similarly:

 '( x)
 ''( )  (1 /  ) 
dx
x 

(6.59)
(6.60)
23
These are the Kramers-Kronig relations which express the value of
either “() or ‘() at a particular value of the frequency  in terms of
the integral transform of the other throughout the entire frequency
range (-, ). In view of what was mentioned above about the even
and odd character of these functions, one may change the range of
integration to (0, ) and thus obtain the one-sided Kramers-Kroning
integrals:

x "( x)
 ' ( )    (2 /  )  2
2 dx
x 
0
(6.61)

 ' ( x)
 ' ' ( )  (2 /  )  2
2 dx
x 
0
(6.62)
24