Laplace transform dielectric relaxation

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

This function is the product of the KWW and power-law dependencies. The relaxation law (25) in time domain and the HN law (21) in the frequency domain are rather generalized representations that lead to the known dielectric relaxation laws. The fact that these functions have power-law asymptotes has inspired numerous attempts to establish a relationship between their various parameters [40,41]. In this regard, the exact relationship between the parameters of (25) and the HN law (21) should be a consequence of the Laplace transform according to (14) [11,12]. However, there is currently no concrete proof that this is indeed so. Thus, the relationship between the parameters of equations (21) and (25) seems to be valid only asymptotically. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

In dielectric relaxation l = 1 so that by taking the Laplace transform of Eqs. (273)-(275) over the time variables and noting the generalized integral theorem for Laplace transforms, we then have a system of algebraic recurrence relations for the Laplace transform of cln m(t) (m = 0, 1) governing the dielectric response, namely,... 

This is the equation describing the dynamics of the longitudinal component of the magnetization obtained by the same method of truncation of the continued fraction as that employed in Section IV. This method, also used by Morita for dielectric relaxation [56], is a consequence of the final value theorem for Laplace transforms, which is... 

Dielectric relaxation measurements couple to the dynamics of the dipole moment of the sample. The dielectric permittivity is the Fourier-Laplace transform of the dipole moment autocorrelation function. 

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