By Ronald C. Davidson

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**Sample text**

1 Sketch showing the exponential time dependence of the polarization P(t) for an ideal Debye relaxation in a dielectric material. The polarization starts to decrease after removing the application of an external electric field, E(t), at t ¼ 0. The time τD is the Debye relaxation time (see text) 18 2 Theories and Models of Ion Diffusion Fig. 2 Sketch of the frequency dependence of the real and imaginary parts of the dielectric permittivity ðε* ¼ ε0 À jε00 Þ for a dielectric material showing a Debye relaxation with relaxation time τD.

Accelerated jumps, localized jumps and long time localization and mixing of such motions are found as discussed in Chaps. 9 and 11. That is, the time scale of the power law dependence of MSD (or stretched exponential decay) is not for a single jump process. Note that the motions are just examples. Note that if the observation time was short, only localized motion might be found squared in calculation of it. An example of cooperative motions is shown in Fig. 12. The ion tends to show further successive jumps.

The Cole-Davidson (CD) function reads ε*ðωÞ ¼ ε1 þ Δε ð1 þ jω τCD Þβ ; ð2:31Þ and results from setting the parameter α ¼ 1 in the HN function. Although there are other empirical descriptions of dielectric relaxation, the Havriliak-Negami, ColeCole and Cole-Davidson functions in the frequency domain, together with the KWW function in the time domain, are most often used in the literature to describe non-Debye relaxation spectra or response function [12]. 1 21 Conductivity Relaxation Electric Modulus Formalism While in the case of dielectric relaxation the response of the system is due to bound electric charge, when analyzing conductivity relaxation dynamics in ionically conducting materials, it is important to bear in mind that it is driven by mobile electric charge carried by the ions.