Havriliak–Negami relaxation

The storage part ${\displaystyle \varepsilon '}$ and the loss part ${\displaystyle \varepsilon ''}$ of the permittivity (here: ${\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon '(\omega )-i\varepsilon ''(\omega )}$ with ${\displaystyle (\pm i)^{2}=-1}$) can be calculated as

${\displaystyle \varepsilon '(\omega )=\varepsilon _{\infty }+\Delta \varepsilon \left(1+2(\omega \tau )^{\alpha }\cos(\pi \alpha /2)+(\omega \tau )^{2\alpha }\right)^{-\beta /2}\cos(\beta \phi )}$

and

${\displaystyle \varepsilon ''(\omega )=\Delta \varepsilon \left(1+2(\omega \tau )^{\alpha }\cos(\pi \alpha /2)+(\omega \tau )^{2\alpha }\right)^{-\beta /2}\sin(\beta \phi )}$

with

${\displaystyle \phi =\arctan \left({(\omega \tau )^{\alpha }\sin(\pi \alpha /2) \over 1+(\omega \tau )^{\alpha }\cos(\pi \alpha /2)}\right)}$

The maximum of the loss part lies at

${\displaystyle \omega _{\rm {max}}=\left({\sin \left({\pi \alpha \over 2(\beta +1)}\right) \over \sin \left({\pi \alpha \beta \over 2(\beta +1)}\right)}\right)^{1/\alpha }\tau ^{-1}}$

The Havriliak–Negami relaxation can be expressed as a superposition of individual Debye relaxations

${\displaystyle {{\hat {\varepsilon }}(\omega )-\epsilon _{\infty } \over \Delta \varepsilon }=\int _{-\infty }^{\infty }{1 \over 1+i\omega \tau _{D}}g(\ln \tau _{D})d\ln \tau _{D}}$

with the real valued distribution function

${\displaystyle g(\ln \tau _{D})={1 \over \pi }{(\tau _{D}/\tau )^{\alpha \beta }\sin(\beta \theta ) \over ((\tau _{D}/\tau )^{2\alpha }+2(\tau _{D}/\tau )^{\alpha }\cos(\pi \alpha )+1)^{\beta /2}}}$

where

${\displaystyle \theta =\arctan \left({\sin(\pi \alpha ) \over (\tau _{D}/\tau )^{\alpha }+\cos(\pi \alpha )}\right)}$

if the argument of the arctangent is positive, else[2]

${\displaystyle \theta =\arctan \left({\sin(\pi \alpha ) \over (\tau _{D}/\tau )^{\alpha }+\cos(\pi \alpha )}\right)+\pi }$

Noteworthy, ${\displaystyle g(\ln \tau )}$ becomes imaginary valued for

${\displaystyle {{\hat {\varepsilon }}(\omega )-\epsilon _{\infty } \over \Delta \varepsilon }={(i\omega \tau )^{\alpha \beta } \over (1+(i\omega \tau )^{\alpha })^{\beta }}}$

and complex valued for

${\displaystyle {{\hat {\varepsilon }}(\omega )-\epsilon _{\infty } \over \Delta \varepsilon }={1 \over (1-(\omega \tau )^{2\alpha })^{\beta }}}$

The first logarithmic moment of this distribution, the average logarithmic relaxation time is

${\displaystyle \langle \ln \tau _{D}\rangle =\ln \tau +{\Psi (\beta )+{\rm {Eu}} \over \alpha }}$

where ${\displaystyle \Psi }$ is the digamma function and ${\displaystyle {\rm {Eu}}}$ 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.[5] In particular, in the time-domain the corresponding of ${\displaystyle {\hat {\varepsilon }}(\omega )}$ can be represented as

${\displaystyle X(t)=\varepsilon _{\infty }\delta (t)+{\frac {\Delta \varepsilon }{\tau }}\left({\frac {t}{\tau }}\right)^{\alpha \beta -1}E_{\alpha ,\alpha \beta }^{\beta }(-(t/\tau )^{\alpha }),}$

where ${\displaystyle \delta (t)}$ is the Dirac delta function and

${\displaystyle E_{\alpha ,\beta }^{\gamma }(z)={\frac {1}{\Gamma (\gamma )}}\sum _{k=0}^{\infty }{\frac {\Gamma (\gamma +k)z^{k}}{k!\Gamma (\alpha k+\beta )}}}$

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 ${\displaystyle E_{\alpha ,\beta }^{\gamma }(z)}$ can be numerically evaluated, for instance, by means of a Matlab code .[7]