Complex susceptibility functions

Another condition arises because P and E are real in the time domain. Combined with the causality this establishes the following form of the complex susceptibility function... 

In Refs. 80 and 81 it is shown that the Mittag-Leffier function is the exact relaxation function for an underlying fractal time random walk process, and that this function directly leads to the Cole-Cole behavior [82] for the complex susceptibility, which is broadly used to describe experimental results. Furthermore, the Mittag-Leffier function can be decomposed into single Debye processes, the relaxation time distribution of which is given by a mod-... 

A. Spectral Function and Complex Susceptibility (General Expressions)... 

Expression of Complex Susceptibility Through Spectral Function... 

X = X + i%" I co = 2jicv cop = 2 e JnN/m 9 Complex susceptibility Current precessional shift of a dipole moment Angular frequency of radiation Plasma frequency Period of q(t) function (dimensional quantity)... 

In our approach [1, 2] termed the dynamic method the complex susceptibility x = x — ix" is determined by a law of undamped motion of a dipole in a given potential well and by dissipation mechanism often described as stosszahlansatz in the underlying kinetic or Boltzmann equation. In this review we shall refer to this (dynamic) method as the ACF method, since it is actually based on calculation of the spectrum of the dipolar autocorrelation function (ACF). Actually we use a one-particle approximation, in which the form of an employed potential well (being in many cases rectangular or close to it) is taken a priori. Correlation of the particles coordinates is characterized implicitly by the Kirkwood correlation factor g, its value being taken from the experimental data. The ACF method is simple and effective, because we do not employ the stochastic equations of motions. This feature distinguishes our method from other well-known approaches—for example, from those described in books [13, 14]. 

Third, the expression for the spectral function pertinent to the HO model is derived in detail using the ACF method. Some general results given in GT and VIG (and also in Section II) are confirmed by calculations, in which an undamped harmonic law of motion of the bounded charged particles is used explicitly. The complex susceptibility, depending on a type of a collision model,... 

Substituting this relation into Eq. (31), we finally derive unambiguous relation between the complex susceptibility and spectral function relevant to the Gross collision model12 ... 

We employ the following equations Eq. (142) for the complex susceptibility X, Eq. (141) for the complex permittivity , and Eq. (136) for the absorption coefficient a. In (142) we substitute the spectral functions (132) for the PL-RP approximation and (133) for the hybrid model, respectively. In Table IIIB and IIIC the following fitted parameters and estimated quantities are listed the proportion r of rotators, Eqs. (112) and (127) the mean number m of reflections of a dipole from the walls of the rectangular well during its lifetime x, Eqs. (118)... 

The spectral function L(z) determines the complex susceptibility % of the medium,35... 

Let us calculate the broadband spectra of liquid water H20 and D20. The adopted experimental data are presented in Table XII. In accord with the scheme (238), we use Eq. (249) for the complex susceptibility x and use Eqs. (242) and (243) for the modified spectral function R(z). All other expressions used in these calculations are the same as were employed in Section V. 

Validity of our formulas for the resonance lines, which express the complex susceptibility through the spectral function, could be confirmed as follows. We have obtained an exact coincidence of the equations (353), (370), (371), which were (i) directly calculated here in terms of the harmonic oscillator model and (ii) derived in GT and VIG (see also Section II, A.6) by using a general linear-response theory. 

Some graphic examples justifying these statements are presented in Figures 4.12 and 4.13, where the components of two nonlinear complex susceptibilities are plotted as the functions of the parameter ct. For a given sample, a in a natural way serves as a dimensionless inverse temperature. In those figures, the solid lines correspond to the above-proposed asymptotic formulas where we retain the terms, including a 3. The circles show the results of numerical solutions obtained by the method described in Ref. 67. Note that even at ct 5 the accuracy is still rather high. 

Note that the relationship between the complex susceptibility and correlation function (14), together with Eq. (134) leads directly to the requirement that Ca = 0. 

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