Havriliak–Negami relaxation is an empirical modification of the Debye relaxation model, accounting for the asymmetry and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymers,[1] by adding two exponential parameters to the Debye equation:
where is the permittivity at the high frequency limit, where is the static, low frequency permittivity, and is the characteristic relaxation time of the medium. The exponents and describe the asymmetry and broadness of the corresponding spectra.
Depending on application, the Fourier transform of the stretched exponential function can be a viable alternative that has one parameter less.
The inverse Fourier transform of the Havriliak-Negami function (the corresponding time-domain relaxation function) can be numerically calculated.[4] It can be shown that the series expansions involved are special cases of the Fox-Wright function.[5] In particular, in the time-domain the corresponding of can be represented as
where is the Dirac delta function and
is a special instance of the Fox-Wright function and, precisely, it is the three parameters Mittag-Leffler function[6] also known as the Prabhakar function. The function can be numerically evaluated, for instance, by means of a Matlab code
.[7]
References
^
Havriliak, S.; Negami, S. (1967). "A complex plane representation of dielectric and mechanical relaxation processes in some polymers". Polymer. 8: 161–210. doi:10.1016/0032-3861(67)90021-3.
^Schönhals, A. (1991). "Fast calculation of the time dependent dielectric permittivity for the Havriliak-Negami function". Acta Polymerica. 42: 149–151.
^Gorenflo, Rudolf; Kilbas, Anatoly A.; Mainardi, Francesco; Rogosin, Sergei V. (2014). Springer (ed.). Mittag-Leffler Functions, Related Topics and Applications. ISBN978-3-662-43929-6.