Lecture 6 The dielectric response functions. Superposition principle. The complex dielectric permittivity. Loss factor. The complex dielectric permittivity and the complex conductivity The Kronig-Kramers relations The dielectric relaxation.

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‘ : (6.1)

where S is the unit-step function: (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: (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). Figure 6.1

For a linear dielectric this response will be proportional to E1-E2, so that the total polarization is given by: (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: (6.5) At t=0 (t=t’ in 6.4) (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 (6.7)

Let us consider the case of block function 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) The resulting polarization for t  t1 can be considered as the superposition of the effects of both unit-step functions: (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: (6.10) In the limit increasing the number of block functions (6.10) can be written in the integral form:

(6.11) 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. (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): (6.13)

with The relation between p and D is the following: (6.14) Taking the negative derivative of (6.14) we can get the relation between p and D : (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: (6.16) 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.

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: (6.17) where  is the dielectric constant of induced polarization. We can rewrite (6.12) in the following way: (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. (6.19) (6.20)

From (6.19) that now (6.16) no longer holds, but should be changed by: (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. (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 unit-step function in the expression for response function of orientation polarization. From (6.16) it follows: (6.24)

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: Applying to the left and right parts the Laplace transform and taking into account the theorem of deconvolution we can obtain: (6.25) where (6.26) (s=+i; 0 and we’ll write instead of s in all Laplace transforms i).

Taking into account the relation (6. 20) we can rewrite (6 Taking into account the relation (6.20) we can rewrite (6.26) in the following way: (6.27) From another side complex dielectric permittivity can be written in the following form: (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: (6.29) and the imaginary part of complex dielectric permittivity ’’() is associated with the negative imaginary part of the Laplace transform of orientation pulse-response function:

(6.30) Let us now reconsider the relationship between time dependent displacement and harmonic electric field: (6.31) We’ll rewrite in this case the relation (6.25) in the following form: (6.32) that can be presented as follows: (6.33) where and

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 . 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 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: (6.34)

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: (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 (6.36)

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): 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) 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: (6.38)

For a harmonic field, the energy dissipation during one cycle amounts to: (6.39) Hence, the average dissipation of per unit of time due to condition is: (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: (6.41)

As long as the current density is given by (6 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 : (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: (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:

(6.44) where (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): (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. The relation between D (t) and electric field can be found by substituting (6.13) and (6.42) into (6.46):

(6.47) Using (6.43) and the fact that the current step-response function I(0)=1, we can find (6.48) where D(t), the pulse-response function of the generalized dielectric displacement, is given by: (6.49) If we’ll again apply Laplace transform to the left and to the right parts of (6.47), we’ll obtain:

(6.50) where (6.51) Making the Laplace transform in (6.51) we’ll get: (6.52) Splitting up into real and negative imaginary part we arrive at: (6.53) (6.54)

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:   (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:

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: (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: (6.57) This called Hilbert transform

Let us apply this to the ´() (6.29): (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 : (6.59) and similarly: (6.60)

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: (6.61) (6.62)