Debye relaxation susceptibility

Implicit in (9.40) is the assumption that co is small compared with lattice vibrational frequencies. The susceptibility in the frequency region where Debye relaxation is the dominant mode of polarization is therefore... 

Complex dielectric susceptibility data such as those in Figure 15.6 provide a detailed view of the dynamics of polar nanodomains in rls. They define relaxation frequencies, /, corresponding to the e (T) peak temperatures Tm, characteristic relaxation times, r = 1/tu (where uj = 2nf is the angular frequency), and a measure of the interaction among nanodomains as represented by the deviation of the relaxation process from a Debye relaxation. Analysis of data on pmn and other rls clearly shows that their dipolar relaxations cannot be described by a single relaxation time represented by the Debye expression... 

The frequency dependence of the linear susceptibility (4.102) is determined by a superposition of the Debye relaxation modes as... 

P (z) are the Legendre polynomials [51] which now constitute the appropriate basis set), Eq. (132) may be solved to yield the corresponding results for rotation in space, namely, the aftereffect function [Eq. (123)] and the complex susceptibility [Eq. (11)], with x and Xo from Eqs. (81) and (84), respectively. Apparently as in normal diffusion, the results differ from the corresponding two-dimensional analogs only by a factor 2/3 in Xo and the appropriate definition of the Debye relaxation time. 

Similar heterogeneous model has been used to develop a relaxation function by Chamberlin and Kingsbury (1994), who consider the localized normal modes to be involved in the relaxation process. Localized (domains) regions are assumed to be present between Tg and T. They are described as dynamically correlated domains (DCD). A Gaussian distribution of the domain sizes has been assumed, with each domain characterized by a Debye relaxation time. Expressions for the dielectric susceptibility have been derived and used to fit the experimental susceptibilities of salol, glycerol and many other substances with remarkable agreement over 13 decades of frequency (even when only one adjustable parameter is employed). 

These formulae for the parallel and perpendicular susceptibilities, Eqs. (4.60d) and (4.80), were written down without derivation by Shliomis and Raikher for the Debye relaxation of ferrofluid particles [19]. Note that in that paper there appears to be a transcription error where n is written as Xi and vice versa. 

Thus, at low frequencies the Gross susceptibility (280) reduces to the Debye relaxation spectrum. 

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-... 

The next issue to concern us will be anomalous relaxation in which the smearing out of a relaxation spectrum (i.e., the deviation of complex susceptibility from its Debye form) is associated with the concept of a relaxation time distribution. As is well known, this concept implies an assembly of dipoles with a continuous distribution of relaxation times of Eq. (379). 

Thus, the chain of transitions considered above is effectively reduced to exponential, Debye-like relaxation with the mean relaxation time (x). In other words, the concept of a relaxation time distribution implies Debye-like relaxation of a system. However, it is evident that the relaxation will become nonexponential, should a system be characterized by a complex susceptibility of, say, Cole-Cole type. 

As for the previous example described in Sect. 3.2.1, the relaxation of the magnetization has been studied using combined ac (Fig. 6a) and dc (Fig. 6b) measurements. In order to extract the relaxation time of the system (r), the obtained frequency dependence of the in-phase x and out-of-phase x" susceptibilities and furthermore the Cole-Cole plots (x" vs. x plot) were fitted simultaneously to a generalized Debye model (solid lines in Figs. 6a and 6b). The fact that the found a parameters of this model are less than 0.06, indicates that the system is close to a pure Debye model with hence a single relaxation time. This indication is confirmed by the quasi-exponential decay of the magnetization observed between 1.8 and 0.8 K (Fig. 6b). 

The real and imaginary parts of the susceptibility of HAT6 are shown in Fig. 7 (a and b), and the temperature dependence of e is shown in Fig. 7(c) [38]. The frequency response can be explained by assuming a Debye law with, surprisingly, a single relaxation time (=5 x 10 s). This process is most probably due to the rotation of the side chain around the oxygen bond. Numerical simulations have shown this mode to have a relatively low activation energy (effective overall thermal dipole can appear. 

Figure 7. (a) The real part of the susceptibility e as a function of frequency a for HAT6. The top curve corresponds to r=362 K, the middle curve to T=384.2 K, and the lower curve to T=311.7 K. Solid lines are data, crosses are fits using simple Debye theory, (b) The frequency dependence of the imaginary part (bottom curve) of the dielectric constant for HAT6. Solid lines are data and crosses are theory fits using simple Debye theory with a weakly temperature dependent relaxation time of =5 x 10" s [38], Top curve r=384.2 K, middle curve T= 362 K, lower curve T=311.7 K (in (a) and (b) the old nomenclature K and D(, instead of Cr and Col is used), (c) The temperature dependence of the real part e(T) as a function of T at two frequencies +, 10 Hz , 10 Hz. Note that despite the typical dipolar relaxation law observed in (a) and (b), the temperature dependence only exhibits small variations which are related to structural order. 

A similar result was obtained by Debye for liquids and gases in which there is no potential barrier to inhibit the dipole motion. In this case the relaxation time results from the viscosity of the medium and is given by T = i r r]R /kT, where R is the radius of the molecule and the susceptibility is obtained from Equation 23.24 as... 

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