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 is the real part of the Faddeeva function evaluated for
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, 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.
Properties
The Voigt profile is normalized: since it is a convolution of normalized profiles. The Lorentzian profile has no moments, 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 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, and it follows that the Voigt distributions are also closed under convolution.
Using the above definition for z, the cumulative distribution function can be found as follows: Substituting the definition of the Faddeeva 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, 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 mode and median are both located at.
Derivative profile
The first and second derivative profiles can be expressed in terms of the Faddeeva function as follows: using the above definition for z.
Voigt functions
The Voigt functionsU, V, and H are defined by where erfc is the complementary error function, and w is the Faddeeva function.
Relation to Voigt profile
with and
Numeric approximations
Pseudo-Voigt approximation
The pseudo-Voigt profile is an approximation of the Voigt profile V using a linear combination of a Gaussian curveG and a Lorentzian curveL 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 is a function of full width at half maximum 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 parameters. The total FWHM parameter is described by:
The width of the Voigt profile
The full width at half maximum 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 A rough approximation for the relation between the widths of the Voigt, Gaussian, and Lorentzian profiles is: This approximation is exactly correct for a pure Gaussian. 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.
