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, by adding two exponential parameters to the Debye equation:
ε ^ ( ω ) = ε ∞ + Δ ε ( 1 + ( i ω τ ) α ) β , {\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon _{\infty }+{\frac {\Delta \varepsilon }{(1+(i\omega \tau )^{\alpha })^{\beta }}},}
where ε ∞ {\displaystyle \varepsilon _{\infty }} is the permittivity at the high frequency limit, Δ ε = ε s − ε ∞ {\displaystyle \Delta \varepsilon =\varepsilon _{s}-\varepsilon _{\infty }} where ε s {\displaystyle \varepsilon _{s}} is the static, low frequency permittivity, and τ {\displaystyle \tau } is the characteristic relaxation time of the medium. The exponents α {\displaystyle \alpha } and β {\displaystyle \beta } describe the asymmetry and broadness of the corresponding spectra. The relation is named after Stephen Havriliak Jr. and Sinichi Negami, who published in 1967. Depending on application, the Fourier transform of the stretched exponential function can be a viable alternative that has one parameter less. For β = 1 {\displaystyle \beta =1} the Havriliak–Negami equation reduces to the Cole–Cole equation, for α = 1 {\displaystyle \alpha =1} to the Cole–Davidson equation.
Mathematical properties
Real and imaginary parts The storage part ε ′ {\displaystyle \varepsilon '} and the loss part ε ″ {\displaystyle \varepsilon ''} of the permittivity (here: ε ^ ( ω ) = ε ′ ( ω ) − i ε ″ ( ω ) {\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon '(\omega )-i\varepsilon ''(\omega )} with ( ± i ) 2 = − 1 {\displaystyle (\pm i)^{2}=-1} ) can be calculated as
ε ′ ( ω ) = ε ∞ + Δ ε ( 1 + 2 ( ω τ ) α cos ( π α / 2 ) + ( ω τ ) 2 α ) − β / 2 cos ( β ϕ ) {\displaystyle \varepsilon '(\omega )=\varepsilon _{\infty }+\Delta \varepsilon \left(1+2(\omega \tau )^{\alpha }\cos(\pi \alpha /2)+(\omega \tau )^{2\alpha }\right)^{-\beta /2}\cos(\beta \phi )}
and
ε ″ ( ω ) = Δ ε ( 1 + 2 ( ω τ ) α cos ( π α / 2 ) + ( ω τ ) 2 α ) − β / 2 sin ( β ϕ ) {\displaystyle \varepsilon ''(\omega )=\Delta \varepsilon \left(1+2(\omega \tau )^{\alpha }\cos(\pi \alpha /2)+(\omega \tau )^{2\alpha }\right)^{-\beta /2}\sin(\beta \phi )}
with
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