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
Support
PDF
CDF	(complicated - see text)
Mean	(not defined)
Median
Mode
Variance	(not defined)
Skewness	(not defined)
Ex. kurtosis	(not defined)
MGF	(not defined)
CF

The Voigt profile (named after Woldemar Voigt) is a probability distributions 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 \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}}.

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:

\int _{-\infty }^{\infty }V(x;\sigma ,\gamma )\,\mathrm {d} x=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 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 )=\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 of Voigt:

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 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^[1] 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 ).

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)=\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.^[2]^[3]^[4]^[5] A simple formula, accurate to 1%, is^[6]^[7]

\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}}}.

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

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.

References

↑ 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
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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: 79–83. doi:10.1107/S0021889887087090.
↑ 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.

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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] 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

[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. Retrieved 2014-07-31.

[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: 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. Retrieved 2009-04-01.

Probability distributions
List
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	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic 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 Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-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 asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

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	$e^{-\gamma\|t\|-\sigma^2 t^2/2}$