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| '''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 [[polymer]]s,<ref>
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| {{cite journal
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| |last1=Havriliak |first1=S.
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| |last2=Negami |first2=S.
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| |year=1967
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| |title=A complex plane representation of dielectric and mechanical relaxation processes in some polymers
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| |journal=[[Polymer (journal)|Polymer]]
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| |volume=8 |issue= |pages=161–210
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| |doi=10.1016/0032-3861(67)90021-3
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| }}</ref> by adding two [[Exponential function|exponential]] parameters to the Debye equation:
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| :<math>
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| \hat{\varepsilon}(\omega) = \varepsilon_{\infty} + \frac{\Delta\varepsilon}{(1+(i\omega\tau)^{\alpha})^{\beta}},
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| </math>
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| where <math>\varepsilon_{\infty}</math> is the [[permittivity]] at the high frequency limit, <math>\Delta\varepsilon = \varepsilon_{s}-\varepsilon_{\infty}</math> where <math>\varepsilon_{s}</math> is the static, low frequency permittivity, and <math>\tau</math> is the characteristic [[relaxation time]] of the medium. The exponents <math>\alpha</math> and <math>\beta</math> describe the asymmetry and broadness of the corresponding spectra.
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| Depending on application, the Fourier transform of the [[stretched exponential function]] can be a viable alternative that has one parameter less.
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| For <math>\beta = 1</math> the Havriliak–Negami equation reduces to the [[Cole–Cole equation]], for <math>\alpha=1</math> to the [[Cole–Davidson equation]].
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| The storage part <math>\varepsilon'</math> and the loss part <math>\varepsilon''</math> of the permittivity (here: <math> \hat{\varepsilon}(\omega) = \varepsilon'(\omega) - i \varepsilon''(\omega) </math>) can be calculated as
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| :<math>
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| \varepsilon'(\omega) = \left( 1 + 2 (\omega\tau)^\alpha \cos (\pi\alpha/2) + (\omega\tau)^{2\alpha} \right)^{-\beta/2} \cos (\beta\phi)
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| </math>
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| and
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| :<math>
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| \varepsilon''(\omega) = \left( 1 + 2 (\omega\tau)^\alpha \cos (\pi\alpha/2) + (\omega\tau)^{2\alpha} \right)^{-\beta/2} \sin (\beta\phi)
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| </math>
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| with
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| :<math>
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| \phi = \arctan \left( { (\omega\tau)^\alpha \sin(\pi\alpha/2) \over
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| 1 + (\omega\tau)^\alpha \cos(\pi\alpha/2) } \right)
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| </math>
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| The maximum of the loss part lies at
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| :<math>
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| \omega_{\rm max} =
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| \left( { \sin \left( { \pi\alpha \over 2 ( \beta +1 ) } \right) \over
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| \sin \left( { \pi\alpha\beta \over 2 ( \beta +1 ) } \right) } \right) ^ {1/\alpha}
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| \tau^{-1}
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| </math>
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| The Havriliak–Negami relaxation can be expressed as a superposition of individual Debye relaxations
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| :<math>
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| { \hat{\varepsilon}(\omega) - \epsilon_\infty \over \Delta\varepsilon } = \int_{\tau_D=0}^\infty
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| { 1 \over 1 + i \omega \tau_D } g( \ln \tau_D ) d \ln \tau_D
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| </math>
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| with the distribution function
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| :<math>
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| g ( \ln \tau_D ) = { 1 \over \pi }
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| { ( \tau_D / \tau )^{\alpha\beta} \sin (\beta\theta) \over
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| ( ( \tau_D / \tau )^{2\alpha} + 2 ( \tau_D / \tau )^{\alpha} \cos (\pi\alpha) + 1 )^{\beta/2} }
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| </math>
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| where
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| :<math>
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| \theta = \arctan \left( { \sin (\pi\alpha) \over ( \tau_D / \tau )^{\alpha} + \cos (\pi\alpha) } \right)
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| </math>
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| if the argument of the arctangent is positive, else<ref>
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| {{cite journal
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| | last = Zorn |first= R.
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| | year = 1999
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| | title = Applicability of Distribution Functions for the Havriliak–Negami Spectral Function
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| | journal = [[Journal of Polymer Science Part B]]
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| | volume = 37 |issue=10 | pages = 1043–1044
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| | bibcode =1999JPoSB..37.1043Z
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| | doi = 10.1002/(SICI)1099-0488(19990515)37:10<1043::AID-POLB9>3.3.CO;2-8
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| }}</ref>
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| :<math>
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| \theta = \arctan \left( { \sin (\pi\alpha) \over ( \tau_D / \tau )^{\alpha} + \cos (\pi\alpha) } \right) + \pi
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| </math>
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| The first logarithmic moment of this distribution, the average logarithmic relaxation time is
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| :<math>
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| \langle \ln\tau_D \rangle = \ln\tau + { \Psi(\beta) + {\rm Eu} \over \alpha }
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| </math>
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| where <math>\Psi</math> is the [[digamma function]] and <math>{\rm Eu}</math> the [[Euler constant]].<ref>
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| {{cite journal
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| | last = Zorn |first= R.
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| | year = 2002
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| | title = Logarithmic moments of relaxation time distributions
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| | journal = [[Journal of Chemical Physics]]
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| | volume = 116 |issue= 8| pages = 3204–3209
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| | bibcode = 2002JChPh.116.3204Z
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| | doi=10.1063/1.1446035
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| }}</ref>
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| The inverse Fourier transform of the Havriliak-Negami function (the corresponding time-domain relaxation function) can be numerically calculated.<ref>{{cite journal
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| | author = Schönhals, A.
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| | year = 1991
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| | title = Fast calculation of the time dependent dielectric permittivity
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| for the Havriliak-Negami function
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| | journal = [[Acta Polymerica]]
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| | volume = 42
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| | pages = 149–151
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| }}</ref> It can be shown that the series expansions involved are special cases of the [[Fox-Wright function]].<ref>{{cite journal
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| | author = Hilfer, J.
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| | year = 2002
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| | title = ''H''-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems
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| | journal = [[Physical Review]] E
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| | volume = 65
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| | pages = 061510
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| }}</ref> Unfortunately, there are no numerical algorithms available for the computation of such functions with the required parameters.
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| ==References==
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| {{reflist}}
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| == See also ==
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| * [[Cole–Cole equation]]
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| * [[Dielectric spectroscopy]]
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| * [[Dipole]]
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| {{DEFAULTSORT:Havriliak-Negami relaxation}}
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| [[Category:Electric and magnetic fields in matter]]
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