The Havriliak–Negami relaxation is an empirical modification of the Debye relaxation model in electromagnetism.
Unlike the Debye model, the Havriliak–Negami relaxation accounts 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,
is the static, low frequency permittivity, and
is the characteristic relaxation time of the medium.
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.
and the loss part
) can be calculated as and with The maximum of the loss part lies at The Havriliak–Negami relaxation can be expressed as a superposition of individual Debye relaxations with the real valued distribution function where if the argument of the arctangent is positive, else[2] Noteworthy,
becomes imaginary valued for and complex valued for The first logarithmic moment of this distribution, the average logarithmic relaxation time is where
the Euler constant.
[3] 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.
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.
can be numerically evaluated, for instance, by means of a Matlab code .