Voigt profile

 ( Continuous Distributions ) 

,

             ~~~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)|
 kurtosis   =(not defined)|
 entropy    =|
 mgf        =(not defined)|
 char       =$e^\{-\backslash gamma|t|-\backslash sigma^2\; t^2/2\}$
     

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.

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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, $G(x;\backslash sigma)$ is the centered Gaussian profile:

$$

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

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

$$

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

The defining integral can be evaluated as:

$$

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

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

$$

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

In the limiting cases of $\backslash sigma=0$ and $\backslash gamma\; =0$ then $V(x;\backslash sigma,\backslash gamma)$ simplifies to $L(x;\backslash gamma)$ and $G(x;\backslash sigma)$, respectively.

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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 (damping of the emission or absorption, as considered in the original article by Waldemar Voigt 1912Voigt, Woldemar, 1912, ''Das Gesetz der Intensitätsverteilung innerhalb der Linien eines Gasspektrums'', Sitzungsbericht der Bayerischen Akademie der Wissenschaften, 25, 603). Voigt profiles are common in many branches of spectroscopy and diffraction. Due to the expense of computing the Faddeeva function, the Voigt profile is sometimes approximated using a pseudo-Voigt profile.

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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 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.

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Cumulative distribution function Using the above definition for z , the cumulative distribution function (CDF) can be found as follows:

$F(x\_0;\backslash mu,\backslash sigma)$

=\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:

$$

\int e^{-z^2}\left1-\operatorname{erf}(-iz)\right\,dz,

which may be solved to yield

$$

\frac{1}{\sqrt{\pi}}\int w(z)\,dz = \frac{\operatorname{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 of Voigt:

$F(x;\backslash mu,\backslash sigma)=\backslash operatorname\{Re\}\backslash left[\backslash frac\{1\}\{2\}+$

\frac{\operatorname{erf}(z)}{2} +\frac{iz^2}{\pi}\,_2F_2\left(1,1;\frac{3}{2},2;-z^2\right)\right].

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The uncentered Voigt profile If the Gaussian profile is centered at $\backslash mu\_G$ and the Lorentzian profile is centered at $\backslash mu\_L$, the convolution is centered at $\backslash mu\_V\; =\; \backslash mu\_G+\backslash mu\_L$ 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 probability density function is simply offset from the centered profile by $\backslash mu\_V$:

$$

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

where:

$$

The mode and median are both located at $\backslash mu\_V$.

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Derivatives Using the definition above for $z$ and $x\_\{c\}=x-\backslash mu\_\{V\}$, the first and second derivatives can be expressed in terms of the Faddeeva function as

$$

\begin{aligned}

\frac{\partial}{\partial x} V(x_{c};\sigma,\gamma) &=
     

= -\frac{x_{c}}{\sigma^2} \frac{\operatorname{Re}\leftw(z)\right}{\sigma\sqrt{2\pi}}+\frac{\gamma}{\sigma^2} \frac{\operatorname{Im}\leftw(z)\right}{\sigma\sqrt{2\pi}} \\ &= \frac{1}{\sigma^{3}\sqrt{2\pi}}\cdot\left(\gamma\cdot\operatorname{Im}\leftw(z)\right-x_{c}\cdot\operatorname{Re}\leftw(z)\right\right) \end{aligned} and

$$

\begin{aligned}

\frac{\partial^2}{\left(\partial x\right)^2} V(x_{c};\sigma,\gamma)
     

+\frac{\gamma}{\sigma^4}\frac{1}{\pi} \\ &= -\frac{1}{\sigma^{5}\sqrt{2\pi}}\cdot\left(\gamma\cdot\left(2x_{c}\cdot\operatorname{Im}\leftw(z)\right - \sigma\cdot\sqrt{\frac{2}{\pi}}\right) + \left(\gamma^{2} + \sigma^{2} - x_{c}^{2}\right)\cdot\operatorname{Re}\leftw(z)\right\right), \end{aligned} respectively.

Often, one or multiple Voigt profiles and/or their respective derivatives need to be fitted to a measured signal by means of non-linear least squares, e.g., in spectroscopy. Then, further partial derivatives can be utilised to accelerate computations. Instead of approximating the Jacobian matrix with respect to the parameters $\backslash mu\_\{V\}$, $\backslash sigma$, and $\backslash gamma$ with the aid of finite differences, the corresponding analytical expressions can be applied. With $\backslash operatorname\{Re\}\backslash leftw(z)\backslash right\; =\; \backslash Re\_\{w\}$ and $\backslash operatorname\{Im\}\backslash leftw(z)\backslash right\; =\; \backslash Im\_\{w\}$, these are given by:

$$

\begin{align} \frac{\partial V}{\partial \mu_{V}} = -\frac{\partial V}{\partial x} = \frac{1}{\sigma^{3}\sqrt{2\pi}}\cdot\left(x_{c}\cdot\Re_{w} - \gamma\cdot\Im_{w}\right) \end{align}

$$

\begin{align} \frac{\partial V}{\partial \sigma} = \frac{1}{\sigma^{4}\sqrt{2\pi}}\cdot\left(\left(x_{c}^{2} - \gamma^{2}-\sigma^{2}\right)\cdot\Re_{w} - 2x_{c}\gamma\cdot\Im_{w} + \gamma\sigma\cdot\sqrt{\frac{2}{\pi}}\right) \end{align}

$$

\begin{align} \frac{\partial V}{\partial \gamma} = -\frac{1}{\sigma^{3}\sqrt{2\pi}}\cdot\left(\sigma\cdot\sqrt{\frac{2}{\pi}} - x_{c}\cdot\Im_{w} - \gamma\cdot\Re_{w}\right) \end{align}

for the original voigt profile $V$;

$$

\begin{align} \frac{\partial V'}{\partial \mu_{V}} = -\frac{\partial V'}{\partial x} = -\frac{\partial^{2} V}{\left(\partial x\right)^{2}} = \frac{1}{\sigma^{5}\sqrt{2\pi}}\cdot\left(\gamma\cdot\left(2x_{c}\cdot\Im_{w} - \sigma\cdot\sqrt{\frac{2}{\pi}}\right) + \left(\gamma^{2} + \sigma^{2} - x_{c}^{2}\right)\cdot\Re_{w}\right) \end{align}

$$

\begin{align} \frac{\partial V'}{\partial \sigma} = \frac{3}{\sigma^{6}\sqrt{2\pi}}\cdot\left(-\gamma\sigma x_{c}\cdot\frac{2\sqrt{2}}{3\sqrt{\pi}} + \left(x_{c}^{2} - \frac{\gamma^{2}}{3} - \sigma^{2}\right)\cdot\gamma\cdot\Im_{w} + \left(\gamma^{2} + \sigma^{2} - \frac{x_{c}^{2}}{3}\right)\cdot x_{c}\cdot\Re_{w}\right) \end{align}

$$

\begin{align} \frac{\partial V'}{\partial \gamma} = \frac{1}{\sigma^{5}\sqrt{2\pi}}\cdot\left(x_{c}\cdot\left(\sigma\cdot\sqrt{\frac{2}{\pi}} - 2\gamma\cdot\Re_{w}\right) + \left(\gamma^{2} + \sigma^{2} - x_{c}^{2}\right)\cdot\Im_{w}\right) \end{align}

for the first order partial derivative $V\text{'}\; =\; \backslash frac\{\backslash partial\; V\}\{\backslash partial\; x\}$; and

$$

\begin{align} \frac{\partial V }{\partial \mu_{V}} = -\frac{\partial V}{\partial x} = -\frac{\partial^{3} V}{\left(\partial x\right)^{3}} = -\frac{3}{\sigma^{7}\sqrt{2\pi}}\cdot\left(\left(x_{c}^{2} - \frac{\gamma^{2}}{3} - \sigma^{2}\right)\cdot\gamma\cdot\Im_{w} + \left(\gamma^{2} + \sigma^{2} - \frac{x_{c}^{2}}{3}\right)\cdot x_{c}\cdot\Re_{w} - \gamma\sigma x_{c}\cdot\frac{2\sqrt{2}}{3\sqrt{\pi}}\right) \end{align}

$$

\begin{align} & \frac{\partial V''}{\partial \sigma} = -\frac{1}{\sigma^{8}\sqrt{2\pi}}\cdot \\ & \left(\left(-3\gamma x_{c}\sigma^{2} + \gamma x_{c}^{3} - \gamma^{3} x_{c}\right)\cdot 4\cdot\Im_{w} + \left(\left(2x_{c}^{2} - 2\gamma^{2} - \sigma^{2}\right)\cdot 3\sigma^{2} + 6\gamma^{2} x_{c}^{2} - x_{c}^{4} - \gamma^{4}\right)\cdot\Re_{w} + \left(\gamma^{2} + 5\sigma^{2} - 3x_{c}^{2}\right)\cdot\gamma\sigma\cdot\sqrt{\frac{2}{\pi}}\right) \end{align}

$$

\begin{align} \frac{\partial V''}{\partial \gamma} = -\frac{3}{\sigma^{7}\sqrt{2\pi}}\cdot\left(\left(\gamma^{2} + \sigma^{2} - \frac{x_{c}^{2}}{3}\right)\cdot x_{c}\cdot\Im_{w} + \left(\frac{\gamma^{2}}{3} + \sigma^{2} - x_{c}^{2}\right)\cdot \gamma\cdot\Re_{w} + \left(x_{c}^{2} - \gamma^{2} - 2\sigma^{2} \right)\cdot\sigma\cdot\frac{\sqrt{2}}{3\sqrt{\pi}}\right) \end{align}

for the second order partial derivative $V\text{'}\text{'}\; =\; \backslash frac\{\backslash partial^\{2\}\; V\}\{\backslash left(\backslash partial\; x\backslash right)^\{2\}\}$. Since $\backslash mu\_\{V\}$ and $\backslash gamma$ play a relatively similar role in the calculation of $z$, their respective partial derivatives also look quite similar in terms of their structure, although they result in totally different derivative profiles. Indeed, the partial derivatives with respect to $\backslash sigma$ and $\backslash gamma$ show more similarity since both are width parameters. All these derivatives involve only simple operations (multiplications and additions) because the computationally expensive $\backslash Re\_\{w\}$ and $\backslash Im\_\{w\}$ are readily obtained when computing $w\backslash left(z\backslash right)$. Such a reuse of previous calculations allows for a derivation at minimum costs. This is not the case for finite difference gradient approximation as it requires the evaluation of $w\backslash left(z\backslash right)$ for each gradient respectively.

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

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

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

where

$z\; =\; \backslash frac\{1-ix\}\{2\backslash sqrt\; t\},$

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

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Relation to Voigt profile

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

with Gaussian sigma relative variables $u\; =\; \backslash frac\{x\}\{\backslash sqrt\; 2\backslash ,\; \backslash sigma\}$ and $a\; =\; \backslash frac\{\backslash gamma\}\{\backslash sqrt\; 2\backslash ,\backslash sigma\}.$

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Numeric approximations

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Tepper-García Function The Tepper-García function, named after Mexican-born, German-Australian Astrophysicist Thor Tepper-García, is a combination of an exponential function and rational functions that approximates the line broadening function $H(a,u)$ over a wide range of its parameters. It is obtained from a truncated power series expansion of the exact line broadening function.

In its most computationally efficient form, the Tepper-García function can be expressed as

$$

T(a,u) = R - \left(a /\sqrt{\pi} P \right) ~\leftR^2 \, , where $P\; \backslash equiv\; u^2$, $Q\; \backslash equiv\; 3\; /\; (2\; P)$, and $R\; \backslash equiv\; e^\{-P\}$.

Thus the line broadening function can be viewed, to first order, as a pure Gaussian function plus a correction factor that depends linearly on the microscopic properties of the absorbing medium (encoded in $a$); however, as a result of the early truncation in the series expansion, the error in the approximation is still of order $a$, i.e. $H(a,u)\; \backslash approx\; T(a,u)\; +\; \backslash mathcal\{O\}(a)$. This approximation has a relative accuracy of

$$

\epsilon \equiv \frac{\vert H(a,u) - T(a,u) \vert}{H(a,u)} \lesssim 10^{-4} over the full wavelength range of $H(a,u)$, provided that $a\; \backslash lesssim\; 10^\{-4\}$. /ref>

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

There are several possible choices for the $\backslash eta$ 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, $\backslash eta$ 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^{1/5}.

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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

$f\_\backslash mathrm\{G\}=2\backslash sigma\backslash sqrt\{2\backslash ln(2)\}.$

The FWHM of the Lorentzian profile is

$f\_\backslash mathrm\{L\}=2\backslash gamma.$

An approximate relation (accurate to within about 1.2%) between the widths of the Voigt, Gaussian, and Lorentzian profiles is:

$f\_\backslash mathrm\{V\}\backslash approx\; f\_\backslash mathrm\{L\}/2+\backslash sqrt\{f\_\backslash mathrm\{L\}^2/4+f\_\backslash mathrm\{G\}^2\}.$

By construction, this expression is exact for a pure Gaussian or Lorentzian.

A better approximation with an accuracy of 0.02% is given by Olivero and Longbothum (originally found by Kielkopf)

$f\_\backslash mathrm\{V\}\backslash approx\; 0.5343\; f\_\backslash mathrm\{L\}+\backslash sqrt\{0.2169f\_\backslash mathrm\{L\}^2+f\_\backslash mathrm\{G\}^2\}.$

In the same publication, a slightly more precise (within 0.012%), yet significantly more complicated expression can be found.

In the near-Gaussian case $f\_\backslash mathrm\{L\}\backslash ll\; f\_\backslash mathrm\{G\}$, the half width can be computed from a Maclaurin series in $f\_\backslash mathrm\{G\}/f\_\backslash mathrm\{L\}$. Recursion formulae and tables of the coefficients can be found in the literature.Y. Wang, B. Zhou, R. Zhao, B. Wang, Q. Liu, M. Dai, Super-Accuracy Calculation for the Half Width of a Voigt Profile, Mathematics, 10 (2022), no. 210 (14 pages). https://doi.org/10.3390/math10020210 /ref> In the near-Lorentzian case $f\_\backslash mathrm\{G\}\backslash ll\; f\_\backslash mathrm\{L\}$, the half width can be expressed as an asymptotic series in $(f\_\backslash mathrm\{L\}/f\_\backslash mathrm\{G\})^2$, with coefficients again given through a recursion.

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Other related functions In particle physics, a relativistic version of the Voigt function is used. It is a convolution of a Relativistic Breit–Wigner distribution and a Gaussian distribution.

An asymmetric pseudo-Voigt function, also known as Martinelli function, can be constructed similarly to the split normal distribution by having different widths on each side of the peak position:

$V\_p(x,f)\; =\; \backslash eta\; \backslash cdot\; L(x,f)\; +\; (1\; -\; \backslash eta)\; \backslash cdot\; G(x,f)$

with being the weight of the Lorentzian and the width $f$ being a split function ($f=f\_1$for and $f=f\_2$ for $x\backslash geq\; 0$). In the limit $f\_1\; \backslash rightarrow\; f\_2$, this function returns to a symmetric pseudo Voigt function. This function has been used to model elastic scattering on resonant inelastic X-ray scattering instruments.

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External links

• numeric C library for complex error functions, provides a function voigt(x, sigma, gamma) with approximately 13–14 digits precision.

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