Voigt profile

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(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 frequency from 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 the 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:

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

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:

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

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:

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

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 just $f_{L}=2\gamma$ . Define φ = $f_{L}/f_{G}$ . Then the FWHM of the Voigt profile ( $f_{V}$ ) can be estimated as:

$f_{{\mathrm {V}}}\approx f_{{\mathrm {G}}}\left(1-c_{0}c_{1}+{\sqrt {\phi ^{2}+2c_{1}\phi +c_{0}^{2}c_{1}^{2}}}\right)$

where $c_{0}$ = 2.0056 and $c_{1}$ = 1.0593. This estimate will have a standard deviation of error of about 2.4 percent for values of φ between 0 and 10. Note that the above equation will be 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]

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

This approximation will be exactly correct for a pure Gaussian, but will have an error of about 0.0325 percent 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 will be centered at $\mu _{G}+\mu _{L}$ and the characteristic function will then be:

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

The mode and median will then both be 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

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

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

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

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

References

↑ 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. Cite uses deprecated parameters (help)
↑ 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

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.

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
Erlang
exponential
F
folded normal
Flory-Schulz
Fréchet
Gamma
Gamma/Gompertz
generalized inverse Gaussian
Gompertz
half-logistic
half-normal
Hotelling's T-squared
hyper-Erlang
hyperexponential
hypoexponential
inverse chi-squared (scaled inverse chi-squared)
inverse Gaussian
inverse gamma
Kolmogorov
Lévy
log-Cauchy
log-Laplace
log-logistic
log-normal
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson SU Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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