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 black and red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively.
Cumulative distribution function
|CDF||(complicated - see text)|
|Ex. kurtosis||(not defined)|
In spectroscopy, the Voigt profile (named after Woldemar Voigt) is a line profile resulting from the convolution of two broadening mechanisms, one of which alone would produce a Gaussian profile (usually, as a result of the Doppler broadening), and the other would produce a Lorentzian profile. Voigt profiles are common in many branches of spectroscopy and diffraction. Due to the computational expense of the convolution operation, the Voigt profile is often approximated using a pseudo-Voigt profile.
All normalized line profiles can be considered to be probability distributions. The Gaussian profile has a Gaussian, or normal, distribution and a Lorentzian profile has a Lorentz, or Cauchy, distribution. Without loss of generality, we can consider only centered profiles, which peak at zero. The Voigt profile is then a convolution of a Lorentz profile and a Gaussian profile:
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
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 for the Cauchy distribution is not defined. It follows that the Voigt profile will not have a moment-generating function either, but the characteristic function for the Cauchy distribution is well defined, 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 normal distributions and Cauchy distributions are stable distributions, they are each closed under 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:
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 width of the Voigt profile
The full width at half maximum (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
Define . Then the FWHM of the Voigt profile can be estimated as
where and . This estimate has a standard deviation of error of about 2.4% for values of between 0 and 10. Note that the above equation is exactly correct in the limit of and , that is for pure Gaussian and Lorentzian profiles.
A rough approximation for the relation between the widths of the Voigt, Gaussian, and Lorentzian profiles is:
A better approximation with an accuracy of 0.02% is given by
This approximation is exactly correct for a pure Gaussian, but has an error of about 0.000305% for a pure Lorentzian profile.
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 mode and median are both located at .
The Voigt functions U, V, and H (sometimes called the line broadening function) are defined by
Relation to Voigt profile
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 G(x) and a Lorentzian curve L(x) instead of their convolution.
The pseudo-Voigt function is often used for calculations of experimental spectral line shapes.
The mathematical definition of the normalized pseudo-Voigt profile is given by
- with .
is a function of full width at half maximum (FWHM) parameter.
where now, is a function of Lorentz (), Gaussian () and total () Full width at half maximum (FWHM) parameters. The total FWHM () parameter is described by:
- Olivero, J. J.; R. L. Longbothum (February 1977). "Empirical fits to the Voigt line width: A brief review". 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.
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- Liu Y, Lin J, Huang G, Guo Y, Duan C (2001). "Simple empirical analytical approximation to the Voigt profile". JOSA B. 18 (5): 666–672. Bibcode:2001JOSAB..18..666L. doi:10.1364/josab.18.000666.
- Di Rocco HO & Cruzado A (2012). "The Voigt Profile as a Sum of a Gaussian and a Lorentzian Functions, when the Weight Coefficient Depends Only on the Widths Ratio". Acta Physica Polonica A. 122 (4): 666–669. doi:10.12693/APhysPolA.122.666. ISSN 0587-4246.
- Ida T, Ando M, Toraya H (2000). "Extended pseudo-Voigt function for approximating the Voigt profile". Journal of Applied Crystallography. 33 (6): 1311–1316. doi:10.1107/s0021889800010219.
- P. Thompson, D. E. Cox and J. B. Hastings (1987). "Rietveld refinement of Debye-Scherrer synchrotron X-ray data from Al2O3". Journal of Applied Crystallography. 20: 79–83. doi:10.1107/S0021889887087090.
- http://apps.jcns.fz-juelich.de/libcerf, numeric C library for complex error functions, provides a function voigt(x, sigma, gamma) with approximately 13–14 digits precision.