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, by adding two exponential parameters to the Debye equation:
Contents
- Real and imaginary parts
- Loss peak
- Superposition of Lorentzians
- Logarithmic moments
- Inverse Fourier transform
- References
where
Depending on application, the Fourier transform of the stretched exponential function can be a viable alternative that has one parameter less.
For
Real and imaginary parts
The storage part
and
with
Loss peak
The maximum of the loss part lies at
Superposition of Lorentzians
The Havriliak–Negami relaxation can be expressed as a superposition of individual Debye relaxations
with the distribution function
where
if the argument of the arctangent is positive, else
Logarithmic moments
The first logarithmic moment of this distribution, the average logarithmic relaxation time is
where
Inverse Fourier transform
The inverse Fourier transform of the Havriliak-Negami function (the corresponding time-domain relaxation function) can be numerically calculated. It can be shown that the series expansions involved are special cases of the Fox-Wright function. In particular, in the time-domain the corresponding of
where
is a special instance of the Fox-Wright function and, precisely, it is the three parameters Mittag-Leffler function also known as the Prabhakar function. The function