Voigt profile 1

Voigt profile

(Centered) Voigt

Probability density function

Plot of the centered Voigt profile for four cases. Each case has a full width at half-maximum of very nearly 3.6. The blackand red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively.

Cumulative distribution function

Parameters

Support

PDF

CDF (complicated - see text)

Mean (not defined)

Median

Mode

Variance (not defined)

Skewness (not defined)

Ex. kurtosis (not defined)

MGF (not defined)

CF

Voigt profile 2

In spectroscopy, the Voigt profile (named after Woldemar Voigt) is a line profile resulting from the convolution oftwo broadening mechanisms, one of which alone would produce a Gaussian profile (usually, as a result of theDoppler broadening), and the other would produce a Lorentzian profile. Voigt profiles are common in manybranches of spectroscopy and diffraction. Due to the computational expense of the convolution operation, the Voigtprofile is often approximated using a pseudo-Voigt profile.All normalized line profiles can be considered to be probability distributions. The Gaussian profile is equivalent to aGaussian or normal distribution and a Lorentzian profile is equivalent to a Lorentz or Cauchy distribution. Withoutloss of generality, we can consider only centered profiles which peak at zero. The Voigt profile is then a convolutionof a Lorentz profile and a Gaussian profile:

where x is frequency from 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 complex error function of z  and

PropertiesThe Voigt profile is normalized:

since it is the convolution of normalized profiles. The Lorentzian profile has no moments (other than the zeroth) andso the moment-generating function for the Cauchy distribution is not defined. It follows that the Voigt profile willnot have a moment-generating function either, but the characteristic function for the Cauchy distribution is welldefined, as is the characteristic function for the normal distribution. The characteristic function for the (centered)Voigt profile will then be the product of the two:

Since both the normal and the Cauchy distribution are stable distributions, they are closed under convolution and itfollows that the Voigt distribution will also be closed under convolution.

Voigt profile 3

Cumulative distribution functionUsing the above definition for z , the CDF can be found as follows:

Substituting the definition of the 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 negativeinfinity (as the CDF must do), an integration constant of 1/2 must be added. This gives for the CDF:

The width of the Voigt profileThe full width at half maximum (FWHM) of the Voigt profile can be found from the widths of the associatedGaussian and Lorentzian widths. The FWHM of the Gaussian profile is

The FWHM of the Lorentzian profile is just . Define φ = . Then the FWHM of the Voigt profile( ) can be estimated as:

where = 2.0056 and = 1.0593. This estimate will have a standard deviation of error of about 2.4 percent forvalues of φ between 0 and 10. Note that the above equation will be exactly correct in the limit of φ = 0 and φ = ∞,that is for pure Gaussian and Lorentzian profiles.A better approximation with an accuracy of 0.02% is given by[1]

This approximation will be exactly correct for a pure Gaussian, but will have an error of about 0.0325 percent for apure Lorentzian profile.

The uncentered Voigt profileIf the Gaussian profile is centered at and the Lorentzian profile is centered at , the convolution will becentered at and the characteristic function will then be:

The mode and median will then both be located at .

Voigt functionsThe Voigt functions[2] U, V, and H (sometimes called the line broadening function) are defined by

Voigt profile 4

where

and erfc is the complementary error function.

Relation to Voigt Profile

,with

and

References[1] Olivero, J.J.; R.L. Longbothum (1977-02). "Empirical fits to the Voigt line width: A brief review" (http:/ / www. sciencedirect. com/ science/

article/ B6TVR-46V0D4P-K/ 2/ 6ea61622d51016571492f70ccb7df928). Journal of Quantitative Spectroscopy and Radiative Transfer 17 (2):233–236. Bibcode 1977JQSRT..17..233O. doi:10.1016/0022-4073(77)90161-3. ISSN 0022-4073. . Retrieved 2009-04-01.

[2] Temme, N. M. (2010), "Voigt function" (http:/ / dlmf. nist. gov/ 7. 19), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al.,NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248,

Article Sources and Contributors 5

Article Sources and ContributorsVoigt profile  Source: https://en.wikipedia.org/w/index.php?oldid=510383240  Contributors: Adinov, Btyner, CWenger, Estel, Evgeny, Guardian72, Kdliss, Longhair, Michael Hardy, Mike Peel,Novakyu, Oleg Alexandrov, Olegalexandrov, Omnipaedista, PAR, Qwfp, R.e.b., Samw, Shlomi Hillel, Wolf.aarons, Zolot, 15 anonymous edits

Image Sources, Licenses and ContributorsImage:Voigt_distributionPDF.png  Source: https://en.wikipedia.org/w/index.php?title=File:Voigt_distributionPDF.png  License: Public Domain  Contributors: High Contrast, PARImage:Voigt_distributionCDF.png  Source: https://en.wikipedia.org/w/index.php?title=File:Voigt_distributionCDF.png  License: Public Domain  Contributors: EugeneZelenko, High Contrast,PAR

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