[1283.4.1] The Debye relaxation model describes the electric relaxation of dipoles after switching an applied electric field [21]. [1283.4.2] The normalized relaxation function , which corresponds to the polarization, obeys the Debye law
(1) |
[page 1284, §0] with the relaxation time and initial condition . [1284.0.1] The response function, i.e. the dynamical dielectric susceptibility, is related to the relaxation function via [17]
(2) |
[1284.0.2] In the following discussions we focus on Laplace transformed quantities. [1284.0.3] We use the Laplace transformation of
(3) |
where and is the frequency. [1284.0.4] Rewriting equation (2) in frequency space and using leads to
(4) |
the well known Debye susceptibility. [1284.0.5] In experiments one measures not the normalized quantity , but instead
(5) |
where and are the dynamical susceptibilities at low, respectively high frequencies.
[1284.1.1] The Debye model is not able to describe the experimental data well, because experimental relaxation peaks are broader and asymmetric. [1284.1.2] For this reason other fitting functions were proposed such as the Cole-Cole [22], Cole-Davidson [23] and Havriliak-Negami [16] expressions. [1284.1.3] Their normalized forms have typically 2 or 3 parameters (see table 1) and they were introduced purely phenomenologically to fit the data. [1284.1.4] This can be considered as a drawback. [1284.1.5] These functions with three parameters are able to fit the data over a range of at most decades (Havriliak-Negami). [1284.1.6] To fit a broader frequency range several functions are commonly added. [1284.1.7] A combination of one Havriliak-Negami expression plus one Cole-Cole form results in 6 fit parameters.
functional form |
number of parameters |
|
---|---|---|
Cole-Cole | 2 | |
Cole-Davidson | 2 | |
Havriliak-Negami | 3 |