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:
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 storage part and the loss part of the permittivity (here: ) can be calculated as
The maximum of the loss part lies at
The Havriliak–Negami relaxation can be expressed as a superposition of individual Debye relaxations
with the distribution function
if the argument of the arctangent is positive, else
The first logarithmic moment of this distribution, the average logarithmic relaxation time is
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. Unfortunately, there are no numerical algorithms available for the computation of such functions with the required parameters.
- 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.
- Zorn, R. (1999). "Applicability of Distribution Functions for the Havriliak–Negami Spectral Function". Journal of Polymer Science Part B 37 (10): 1043–1044. Bibcode:1999JPoSB..37.1043Z. doi:10.1002/(SICI)1099-0488(19990515)37:10<1043::AID-POLB9>3.3.CO;2-8.
- Zorn, R. (2002). "Logarithmic moments of relaxation time distributions". Journal of Chemical Physics 116 (8): 3204–3209. Bibcode:2002JChPh.116.3204Z. doi:10.1063/1.1446035.
- Schönhals, A. (1991). "Fast calculation of the time dependent dielectric permittivity for the Havriliak-Negami function". Acta Polymerica 42: 149–151.
- Hilfer, J. (2002). "H-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems". Physical Review E 65: 061510. doi:10.1103/physreve.65.061510.