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	$x\in(-\infty,\infty)$
PDF	$\frac{\Re[w(z)]}{\sigma\sqrt{2\pi}}, ~~~z=\frac{x+i\gamma}{\sigma\sqrt{2}}$
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	$e^{-\gamma|t|-\sigma^2 t^2/2}$

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 is equivalent to a Gaussian, or normal, distribution and a Lorentzian profile is equivalent to 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:

$V(x;\sigma,\gamma) = \int_{-\infty}^\infty G(x';\sigma)L(x-x';\gamma)\, dx',$

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:

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

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

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

Properties

The Voigt profile is normalized:

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

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:

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

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

Cumulative distribution function

Using the above definition for z , the CDF can be found as follows:

$F(x_0;\mu,\sigma) =\int_{-\infty}^{x_0} \frac{\mathrm{Re}(w(z))}{\sigma\sqrt{2\pi}}\,dx =\mathrm{Re}\left(\frac{1}{\sqrt{\pi}}\int_{z(-\infty)}^{z(x_0)} w(z)\,dz\right).$

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

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

which may be solved to yield

$\frac{1}{\sqrt{\pi}}\int w(z)\,dz = \frac{\mathrm{erf}(z)}{2} +\frac{iz^2}{\pi}\,_2F_2\left(1,1;\frac{3}{2},2;-z^2\right),$

where ${}_2F_2()$ 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:

$F(x;\mu,\sigma)=\mathrm{Re}\left[\frac{1}{2}+ \frac{\mathrm{erf}(z)}{2} +\frac{iz^2}{\pi}\,_2F_2\left(1,1;\frac{3}{2},2;-z^2\right)\right].$

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

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

Define Φ = $f_L/f_G$. Then the FWHM of the Voigt profile ($f_V$) can be estimated as

$f_\mathrm{V}\approx f_\mathrm{G}\left(1-c_0c_1+\sqrt{\phi^2+2c_1\phi+c_0^2c_1^2}\right),$

where $c_0$ = 2.0056 and $c_1$ = 1.0593. This estimate has a standard deviation of error of about 2.4% for values of φ between 0 and 10. Note that the above equation is 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]

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

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

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

$\varphi_f(t;\sigma,\gamma,\mu_\mathrm{G},\mu_\mathrm{L})= e^{i(\mu_\mathrm{G}+\mu_\mathrm{L})t-\sigma^2t^2/2 - \gamma |t|}.$

The mode and median are both located at $\mu_G+\mu_L$.

Voigt functions

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

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

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

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

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

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

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

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

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

where

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

References

1. 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.
2. Temme, N. M. (2010), "Voigt function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
3. Wertheim, G. K. and Butler, M. A. and West, K. W. and Buchanan, D. N. E. (1974). "Determination of the Gaussian and Lorentzian content of experimental line shapes". Review of Scientific Instruments 45 (11): 1369–1371. doi:10.1063/1.1686503.
4. 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. Retrieved 2014-07-31.
5. Liu, Yuyan and Lin, Jieli and Huang, Guangming and Guo, Yuanqing and Duan, Chuanxi (2001). "Simple empirical analytical approximation to the Voigt profile". JOSA B 18 (5): 666–672. doi:10.1364/josab.18.000666.
6. Di Rocco, HO and 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.
7. Ida, T and Ando, M and Toraya, H (2000). "Extended pseudo-Voigt function for approximating the Voigt profile". Journal of Applied Crystallography 33 (6): 1311–1316. doi:10.1107/s0021889800010219.

External links

• 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.