# Voigt profile

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.

Parameters Probability density functionPlot 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 ${\displaystyle \gamma ,\sigma >0}$ ${\displaystyle x\in (-\infty ,\infty )}$ ${\displaystyle {\frac {\Re [w(z)]}{\sigma {\sqrt {2\pi }}}},~~~z={\frac {x+i\gamma }{\sigma {\sqrt {2}}}}}$ (complicated - see text) (not defined) ${\displaystyle 0}$ ${\displaystyle 0}$ (not defined) (not defined) (not defined) (not defined) ${\displaystyle e^{-\gamma |t|-\sigma ^{2}t^{2}/2}}$

## Definition

Without loss of generality, we can consider only centered profiles, which peak at zero. The Voigt profile is then

${\displaystyle V(x;\sigma ,\gamma )\equiv \int _{-\infty }^{\infty }G(x';\sigma )L(x-x';\gamma )\,dx',}$

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

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

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

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

The defining integral can be evaluated as:

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

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

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

## 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 computational expense of the convolution operation, the Voigt profile is often approximated using a pseudo-Voigt profile.

## Properties

The Voigt profile is normalized:

${\displaystyle \int _{-\infty }^{\infty }V(x;\sigma ,\gamma )\,dx=1,}$

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:

${\displaystyle \varphi _{f}(t;\sigma ,\gamma )=E(e^{ixt})=e^{-\sigma ^{2}t^{2}/2-\gamma |t|}.}$

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:

${\displaystyle F(x_{0};\mu ,\sigma )=\int _{-\infty }^{x_{0}}{\frac {\operatorname {Re} (w(z))}{\sigma {\sqrt {2\pi }}}}\,dx=\operatorname {Re} \left({\frac {1}{\sqrt {\pi }}}\int _{z(-\infty )}^{z(x_{0})}w(z)\,dz\right).}$

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

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

which may be solved to yield

${\displaystyle {\frac {1}{\sqrt {\pi }}}\int w(z)\,dz={\frac {\operatorname {erf} (z)}{2}}+{\frac {iz^{2}}{\pi }}\,_{2}F_{2}\left(1,1;{\frac {3}{2}},2;-z^{2}\right),}$

where ${\displaystyle {}_{2}F_{2}}$  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:

${\displaystyle F(x;\mu ,\sigma )=\operatorname {Re} \left[{\frac {1}{2}}+{\frac {\operatorname {erf} (z)}{2}}+{\frac {iz^{2}}{\pi }}\,_{2}F_{2}\left(1,1;{\frac {3}{2}},2;-z^{2}\right)\right].}$

### The uncentered Voigt profile

If the Gaussian profile is centered at ${\displaystyle \mu _{G}}$  and the Lorentzian profile is centered at ${\displaystyle \mu _{L}}$ , the convolution is centered at ${\displaystyle \mu _{G}+\mu _{L}}$  and the characteristic function is

${\displaystyle \varphi _{f}(t;\sigma ,\gamma ,\mu _{\mathrm {G} },\mu _{\mathrm {L} })=e^{i(\mu _{\mathrm {G} }+\mu _{\mathrm {L} })t-\sigma ^{2}t^{2}/2-\gamma |t|}.}$

The mode and median are both located at ${\displaystyle \mu _{G}+\mu _{L}}$ .

## Voigt functions

The Voigt functions[1] U, V, and H (sometimes called the line broadening function) are defined by

${\displaystyle U(x,t)+iV(x,t)={\sqrt {\frac {\pi }{4t}}}e^{z^{2}}\operatorname {erfc} (z)={\sqrt {\frac {\pi }{4t}}}w(iz),}$
${\displaystyle H(a,u)={\frac {U(u/a,1/4a^{2})}{a{\sqrt {\pi }}}},}$

where

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

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

### Relation to Voigt profile

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

with

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

and

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

## 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

${\displaystyle V_{p}(x)=\eta \cdot L(x,f)+(1-\eta )\cdot G(x,f)}$  with ${\displaystyle 0<\eta <1}$ .

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

There are several possible choices for the ${\displaystyle \eta }$  parameter.[2][3][4][5] A simple formula, accurate to 1%, is[6][7]

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

where now, ${\displaystyle \eta }$  is a function of Lorentz (${\displaystyle f_{L}}$ ), Gaussian (${\displaystyle f_{G}}$ ) and total (${\displaystyle f}$ ) Full width at half maximum (FWHM) parameters. The total FWHM (${\displaystyle f}$ ) parameter is described by:

${\displaystyle f=[f_{G}^{5}+2.69269f_{G}^{4}f_{L}+2.42843f_{G}^{3}f_{L}^{2}+4.47163f_{G}^{2}f_{L}^{3}+0.07842f_{G}f_{L}^{4}+f_{L}^{5}]^{1/5}.}$

### 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

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

The FWHM of the Lorentzian profile is

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

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

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

A better approximation with an accuracy of 0.02% is given by[8]

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

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

## References

1. ^ Temme, N. M. (2010), "Voigt function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
2. ^ Wertheim GK, Butler MA, West KW, Buchanan DN (1974). "Determination of the Gaussian and Lorentzian content of experimental line shapes". Review of Scientific Instruments. 45 (11): 1369–1371. Bibcode:1974RScI...45.1369W. doi:10.1063/1.1686503.
3. ^ Sánchez-Bajo, F.; F. L. Cumbrera (August 1997). "The Use of the Pseudo-Voigt Function in the Variance Method of X-ray Line-Broadening Analysis". Journal of Applied Crystallography. 30 (4): 427–430. doi:10.1107/S0021889896015464.
4. ^ 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.
5. ^ 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.
6. ^ 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.
7. ^ 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 (2): 79–83. doi:10.1107/S0021889887087090.
8. ^ 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.