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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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Greetings! I am Myrtle Shroyer. I am a meter reader. One of the issues she loves most is to do aerobics and now she is attempting to make cash with it. California is our beginning place.
Here is my web page ... meal delivery service (official source)