The Voigt profile (named after
Woldemar Voigt
Woldemar Voigt (; 2 September 1850 – 13 December 1919) was a German physicist, who taught at the Georg August University of Göttingen. Voigt eventually went on to head the Mathematical Physics Department at Göttingen and was succeeded in ...
) is a
probability distribution given by a
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of a
Cauchy-Lorentz distribution and a
Gaussian distribution. It is often used in analyzing data from
spectroscopy or
diffraction.
Definition
Without loss of generality, we can consider only centered profiles, which peak at zero. The Voigt profile is then
:
where ''x'' is the shift from the line center,
is the centered Gaussian profile:
:
and
is the centered Lorentzian profile:
:
The defining integral can be evaluated as:
:
where Re
'w''(''z'')is the real part of the
Faddeeva function evaluated for
:
In the limiting cases of
and
then
simplifies to
and
, respectively.
History and applications
In spectroscopy, a Voigt profile results from the convolution of two broadening mechanisms, one of which alone would produce a Gaussian profile (usually, as a result of the
Doppler broadening
In atomic physics, Doppler broadening is broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting (or absorbing) particles result in different Dop ...
), and the other would produce a Lorentzian profile. Voigt profiles are common in many branches of spectroscopy and
diffraction. Due to the expense of computing the
Faddeeva function, the Voigt profile is sometimes approximated using a pseudo-Voigt profile.
Properties
The Voigt profile is normalized:
:
since it is a convolution of normalized profiles. The Lorentzian profile has no moments (other than the zeroth), and so the
moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
for the
Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
is not defined. It follows that the Voigt profile will not have a moment-generating function either, but the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
for the
Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
is well defined, as is the characteristic function for the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
. The
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
for the (centered) Voigt profile will then be the product of the two:
:
Since normal distributions and Cauchy distributions are
stable distributions
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be sta ...
, they are each closed under
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
(up to change of scale), and it follows that the Voigt distributions are also closed under convolution.
Cumulative distribution function
Using the above definition for ''z'' , the cumulative distribution function (CDF) can be found as follows:
:
Substituting the definition of the
Faddeeva function (scaled complex
error function) yields for the indefinite integral:
:
which may be solved to yield
:
where
is a
hypergeometric function. In order for the function to approach zero as ''x'' approaches negative infinity (as the CDF must do), an integration constant of 1/2 must be added. This gives for the CDF of Voigt:
:
The uncentered Voigt profile
If the Gaussian profile is centered at
and the Lorentzian profile is centered at
, the convolution is centered at
and the characteristic function is:
:
The probability density function is simply offset from the centered profile by
:
:
where:
:
The mode and median are both located at
.
Derivatives
Using the definition above for
and
, the first and second derivatives can be expressed in terms of the
Faddeeva function as
:
and
:
respectively.
Often, one or multiple Voigt profiles and/or their respective derivatives need to be fitted to a measured signal by means of
Non-linear least squares
Non-linear least squares is the form of least squares analysis used to fit a set of ''m'' observations with a model that is non-linear in ''n'' unknown parameters (''m'' ≥ ''n''). It is used in some forms of nonlinear regression. The ...
, e.g., in
spectroscopy. Then, further partial derivatives can be utilised to accelerate computations. Instead of approximating the
Jacobian matrix with respect to the parameters
,
, and
with the aid of
finite differences
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
, the corresponding analytical expressions can be applied. With
and
, these are given by:
:
:
:
for the original voigt profile
;
:
:
:
for the first order partial derivative
; and
:
:
:
for the second order partial derivative
. Since
and
play a relatively similar role in the calculation of
, their respective partial derivatives also look quite similar in terms of their structure, although they result in totally different derivative profiles. Indeed, the partial derivatives with respect to
and
show more similarity since both are width parameters. All these derivatives involve only simple operations (multiplications and additions) because the computationally expensive
and
are readily obtained when computing
. Such a reuse of previous calculations allows for a derivation at minimum costs. This is not the case for
finite difference gradient approximation as it requires the evaluation of
for each gradient respectively.
Voigt functions
The Voigt functions ''U'', ''V'', and ''H'' (sometimes called the line broadening function) are defined by
:
:
where
:
erfc is the
complementary error function
In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as:
:\operatorname z = \frac\int_0^z e^\,\mathrm dt.
This integral is a special (non-elementar ...
, and ''w''(''z'') is the
Faddeeva function.
Relation to Voigt profile
:
with
:
and
:
Numeric approximations
Tepper-García Function
The Tepper-García function, named after German-Mexican Astrophysicist
Thor Tepper-García, is a combination of an exponential function and rational functions that approximates the line broadening function
over a wide range of its parameters.
It is obtained from a truncated power series expansion of the exact line broadening function.
In its most computationally efficient form, the Tepper-García function can be expressed as
:
where
,
, and
.
Thus the line broadening function can be viewed, to first order, as a pure Gaussian function plus a correction factor that depends linearly on the microscopic properties of the absorbing medium (encoded in
); however, as a result of the early truncation in the series expansion, the error in the approximation is still of order
, i.e.
. This approximation has a relative accuracy of
:
over the full wavelength range of
, provided that
.
In addition to its high accuracy, the function
is easy to implement as well as computationally fast. It is widely used in the field of quasar absorption line analysis.
Pseudo-Voigt approximation
The pseudo-Voigt profile (or pseudo-Voigt function) is an approximation of the Voigt profile ''V''(''x'') using a
linear combination of a
Gaussian curve
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real constants , and non-zero . It is ...
''G''(''x'') and a
Lorentzian curve ''L''(''x'') instead of their
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
.
The pseudo-Voigt function is often used for calculations of experimental
spectral line shape Spectral line shape describes the form of a feature, observed in spectroscopy, corresponding to an energy change in an atom, molecule or ion. This shape is also referred to as the spectral line profile. Ideal line shapes include Lorentzian, Gaussi ...
s.
The mathematical definition of the normalized pseudo-Voigt profile is given by
:
with
.
is a function of
full width at half maximum
In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve mea ...
(FWHM) parameter.
There are several possible choices for the
parameter. A simple formula, accurate to 1%, is
:
where now,
is a function of Lorentz (
), Gaussian (
) and total (
)
Full width at half maximum
In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve mea ...
(FWHM) parameters. The total FWHM (
) parameter is described by:
:
The width of the Voigt profile
The
full width at half maximum
In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve mea ...
(FWHM) of the Voigt profile can be found from the
widths of the associated Gaussian and Lorentzian widths. The FWHM of the Gaussian profile
is
:
The FWHM of the Lorentzian profile is
:
An approximate relation (accurate to within about 1.2%) between the widths of the Voigt, Gaussian, and Lorentzian profiles is:
:
By construction, this expression is exact for a pure Gaussian or Lorentzian.
A better approximation with an accuracy of 0.02% is given by
(originally found by Kielkopf
)
:
Again, this expression is exact for a pure Gaussian or Lorentzian.
In the same publication,
a slightly more precise (within 0.012%), yet significantly more complicated expression can be found.
References
External links
* http://jugit.fz-juelich.de/mlz/libcerf, numeric C library for complex error functions, provides a function ''voigt(x, sigma, gamma)'' with approximately 13–14 digits precision.
*The original article is : Voigt, Woldemar, 1912,
''Das Gesetz der Intensitätsverteilung innerhalb der Linien eines Gasspektrums
'', Sitzungsbericht der Bayerischen Akademie der Wissenschaften, 25, 603 (see also:
http://publikationen.badw.de/de/003395768)
{{ProbDistributions, continuous-infinite
Continuous distributions
Spectroscopy
Special functions
Probability distributions with non-finite variance