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 black and red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively.
Cumulative distribution function
Parameters	$\gamma ,\sigma >0$
Support	$x\in (-\mathrm{\infty },\mathrm{\infty })$
PDF	$\frac{\mathrm{\Re }[w(z)]}{\sigma \sqrt{2\pi }},z=\frac{x+i\gamma }{\sigma \sqrt{2}}$
CDF	(complicated - see text)
Mean	(not defined)
Median	$0$
Mode	$0$
Variance	(not defined)
Skewness	(not defined)
Ex. kurtosis	(not defined)
MGF	(not defined)
CF	$e^{-\gamma \left|t\right|-{\sigma }^2{t}^2/2}$

The Voigt profile (named after Woldemar Voigt) is a probability distribution given by a 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

$V(x;\sigma ,\gamma )\equiv \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}G(x^{\prime };\sigma )L(x-x^{\prime };\gamma )d{x}^{\prime },$

where x is the shift from the line center, $G(x;\sigma )$ is the centered Gaussian profile:

$G(x;\sigma )\equiv \frac{e^{-x^2/(2{\sigma }^2)}}{\sigma \sqrt{2\pi }},$

and $L(x;\gamma )$ is the centered Lorentzian profile:

$L(x;\gamma )\equiv \frac{\gamma }{\pi (x^2+{\gamma }^2)}.$

The defining integral can be evaluated as:

$V(x;\sigma ,\gamma )=\frac{\mathrm{\Re }[w(z)]}{\sigma \sqrt{2\pi }},$

where Re[w(z)] is the real part of the Faddeeva function evaluated for

$z=\frac{x+i\gamma }{\sigma \sqrt{2}}.$

In the limiting cases of $\sigma =0$ and $\gamma =0$ then $V(x;\sigma ,\gamma )$ simplifies to $L(x;\gamma )$ and $G(x;\sigma )$, 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), 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 often approximated using a pseudo-Voigt profile.

Properties

The Voigt profile is normalized:

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}V(x;\sigma ,\gamma )dx=1,$

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:

${\phi }_f(t;\sigma ,\gamma )=E(e^{ixt})=e^{-{\sigma }^2{t}^2/2-\gamma \left|t\right|}.$

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:

$F(x_0;\mu ,\sigma )=\underset{-\mathrm{\infty }}{\overset{x_0}{\int }}\frac{\mathrm{\Re }(w(z))}{\sigma \sqrt{2\pi }}dx=\mathrm{\Re }(\frac{1}{\sqrt{\pi }}\underset{z(-\mathrm{\infty })}{\overset{z(x_0)}{\int }}w(z)dz).$

Substituting the definition of the Faddeeva function (scaled complex error function) yields for the indefinite integral:

$\frac{1}{\sqrt{\pi }}\int w(z)dz=\frac{1}{\sqrt{\pi }}\int e^{-z^2}[1-\mathrm{erf}(-iz)]dz,$

which may be solved to yield

$\frac{1}{\sqrt{\pi }}\int w(z)dz=\frac{\mathrm{erf}(z)}{2}+\frac{i{z}^2}{\pi }{}_2{F}_2(1,1;\frac{3}{2},2;-z^2),$

where ${}_2{F}_2$ 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:

$F(x;\mu ,\sigma )=\mathrm{\Re }[\frac{1}{2}+\frac{\mathrm{erf}(z)}{2}+\frac{i{z}^2}{\pi }{}_2{F}_2(1,1;\frac{3}{2},2;-z^2)].$

The uncentered Voigt profile

If the Gaussian profile is centered at ${\mu }_G$ and the Lorentzian profile is centered at ${\mu }_L$, the convolution is centered at ${\mu }_G+{\mu }_L$ and the characteristic function is

${\phi }_f(t;\sigma ,\gamma ,{\mu }_{\mathrm{G}},{\mu }_{\mathrm{L}})=e^{i({\mu }_{\mathrm{G}}+{\mu }_{\mathrm{L}})t-{\sigma }^2{t}^2/2-\gamma \left|t\right|}.$

The mode and median are both located at ${\mu }_G+{\mu }_L$.

Derivative profile

The first and second derivative profiles can be expressed in terms of the Faddeeva function as follows:

$\frac{\partial }{\partial x}V(x;\sigma ,\gamma )=-\frac{\mathrm{\Re }[zw(z)]}{{\sigma }^2\sqrt{\pi }}=-\frac{x}{{\sigma }^2}\frac{\mathrm{\Re }[w(z)]}{\sigma \sqrt{2\pi }}+\frac{\gamma }{{\sigma }^2}\frac{\mathrm{\Im }[w(z)]}{\sigma \sqrt{2\pi }};$

$\frac{{\partial }^2}{\partial x^2}V(x;\sigma ,\gamma )=\frac{x^2-{\gamma }^2-{\sigma }^2}{{\sigma }^4}\frac{\mathrm{\Re }[w(z)]}{\sigma \sqrt{2\pi }}-\frac{2x\gamma }{{\sigma }^4}\frac{\mathrm{\Im }[w(z)]}{\sigma \sqrt{2\pi }}+\frac{\gamma }{{\sigma }^4}\frac{1}{\pi },$

using the above definition for z.

Voigt functions

The Voigt functions U, V, and H (sometimes called the line broadening function) are defined by

$U(x,t)+iV(x,t)=\sqrt{\frac{\pi }{4t}}{e}^{z^2}\mathrm{erfc}(z)=\sqrt{\frac{\pi }{4t}}w(iz),$

$H(a,u)=\frac{U(u/a,1/4a^2)}{a\sqrt{\pi }},$

where

$z=(1-ix)/2\sqrt{t},$

erfc is the complementary error function, and w(z) is the Faddeeva function.

Relation to Voigt profile

$V(x;\sigma ,\gamma )=H(a,u)/(\sqrt{2}\sqrt{\pi }\sigma ),$

with

$a=\gamma /(\sqrt{2}\sigma )$

and

$u=x/(\sqrt{2}\sigma ).$

Numeric approximations

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 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

$V_p(x,f)=\eta \cdot L(x,f)+(1-\eta )\cdot G(x,f)$ with $0<\eta <1$.

$\eta$ is a function of full width at half maximum (FWHM) parameter.

There are several possible choices for the $\eta$ parameter. A simple formula, accurate to 1%, is

$\eta =1.36603(f_L/f)-0.47719(f_L/f{)}^2+0.11116(f_L/f{)}^3,$

where now, $\eta$ is a function of Lorentz ($f_L$), Gaussian ($f_G$) and total ($f$) Full width at half maximum (FWHM) parameters. The total FWHM ($f$) parameter is described by:

$f=[f_G^5+2.69269f_G^4{f}_L+2.42843f_G^3{f}_L^2+4.47163f_G^2{f}_L^3+0.07842f_G{f}_L^4+f_L^5{]}^{1/5}.$

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

$f_{\mathrm{G}}=2\sigma \sqrt{2\ln (2)}.$

The FWHM of the Lorentzian profile is

$f_{\mathrm{L}}=2\gamma .$

A rough approximation for the relation between the widths of the Voigt, Gaussian, and Lorentzian profiles is:

$f_{\mathrm{V}}\approx f_{\mathrm{L}}/2+\sqrt{f_{\mathrm{L}}^2/4+f_{\mathrm{G}}^2}.$

This approximation is exactly correct for a pure Gaussian.

A better approximation with an accuracy of 0.02% is given by

$f_{\mathrm{V}}\approx 0.5346f_{\mathrm{L}}+\sqrt{0.2166f_{\mathrm{L}}^2+f_{\mathrm{G}}^2}.$

This approximation is exactly correct for a pure Gaussian, but has an error of about 0.000305% for a pure Lorentzian profile.

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)