Kramers-Kronig关系

t t
(5.4)
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:
E2
E (t) E1 ( E2 E1 )S(t t')
(5.1)
2
where S is the unit-step function:
S(t) 0 S(t)= 1 t 0 t>0
(5.2)
The time-dependent field given by (5.1) can be considered as the superposition of a static field, E2, and time-dependent field given by:
P (t ) E i (t ti t ) (t ti )
i
(5.10)
In the limit increasing the number of block functions (5.10) can be written in the integral form:
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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:
P(t) E1 ( t t1 t ) ( t t1 )
(5.9)
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 (5.9). Since the effects of all block functions may again be superimposed, we have:
4. The Kramers-Kronig relations
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PHENOMENOLOGICAL THEORY OF LINEAR DIELECTRIC IN TIME-DEPENDENT FIELDS
2 The dielectric response functions. Superposition principle.
(5.15)
The unit step function in (5.14) 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) (5.16) ( t t ' ) lim D
D(t ) E(t ')D (t t ')dt'

t
(5.13)
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D (t t ') D (t t ') with
The relation between p and D is the following:
D (t t ')
1 4

1E1 S t t1 t E1 S t t1




(5.8)
The resulting polarization for t t1 can be considered as the superposition of the effects of both unit-step functions:
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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 ,
P(t) P(0) ( t ) E( 0 )( t )
At t=0 (t=t‟ in 5.4)
(5.5)
( 0 ) 1
(5.6)
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
Lecture 5
1. The dielectric response functions. Superposition principle. 2. The complex dielectric permittivity. Loss factor.
3. The complex dielectric permittivity and the complex conductivity
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(t t' ) P (t ) E (t ' ) dt' = t
t
(5.11)
= E (t ' ) (t t ' )dt'

t
p (t t ' ) p (t t ' ) where, called pulse-response function of polarization. The equation (5.11) 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.
P 2
E(t) (E1 - E 2 ) 1 S (t t )
(5.3)
P
P 1
0
E2 E E1
0 t' t
Therefore, we find from the superposition principle that the polarization at times t t‟ due to the field given by eqn.(5.1) is the equilibrium polarization E2 for the static field E2 and the response of the field change E1-E2 (see fig. 5.1). Figure 5.1

p (t t ')
(5.14)
Taking the negative derivative of (5.14) we can get the relation between p and D :
D (t t ')
(t t ') 4 p (t t ')
( ) 0
(5.7)
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Let us consider the case of block function. For t1-t<tt1 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:
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.
D(t) E(t) 4P(t )
(5.12)
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 (5.11):