Havriliak-Negami relaxation
Encyclopedia
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, 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 (Occam's razor
).
For
the Havriliak–Negami equation reduces to the Cole–Cole equation, for 
to the Cole–Davidson equation.
The storage part
and the loss part 
of the permittivity (here: 
) can be calculated as
and
with
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
where
if the argument of the arctangent is positive, else
The first logarithmic moment of this distribution, the average logarithmic relaxation time is
where
is the digamma function and 
the Euler constant.
Asymmetry
Asymmetry is the absence of, or a violation of, symmetry.-In organisms:Due to how cells divide in organisms, asymmetry in organisms is fairly usual in at least one dimension, with biological symmetry also being common in at least one dimension....
and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
s, by adding two exponential
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
parameters to the Debye equation:
where
is the permittivityPermittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
at the high frequency limit,
where is the static, low frequency permittivity, and is the characteristic relaxation timeRelaxation time
In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium.Each relaxation process can be characterized by a relaxation time τ...
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 (Occam's razor
Occam's razor
Occam's razor, also known as Ockham's razor, and sometimes expressed in Latin as lex parsimoniae , is a principle that generally recommends from among competing hypotheses selecting the one that makes the fewest new assumptions.-Overview:The principle is often summarized as "simpler explanations...
).
For
the Havriliak–Negami equation reduces to the Cole–Cole equation, for to the Cole–Davidson equation.The storage part
and the loss part of the permittivity (here: ) can be calculated asand
with
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
where
if the argument of the arctangent is positive, else
The first logarithmic moment of this distribution, the average logarithmic relaxation time is
where
is the digamma function and the Euler constant.