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{{Probability distribution|
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  name      =(Centered) Voigt |
  type      =density|
  pdf_image  =[[Image:Voigt distributionPDF.png|325px|Plot of the centered Voigt profile for four cases]]<br /><small>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&nbsp;=0) and the Lorentzian (σ&nbsp;=0) profiles respectively. </small>|
  cdf_image  =[[Image:Voigt distributionCDF.png|325px|Centered Voigt CDF.]]|
  parameters =<math>\gamma,\sigma>0</math>|
  support    =<math>x\in(-\infty,\infty)</math>|
  pdf        =<math>\frac{\Re[w(z)]}{\sigma\sqrt{2\pi}},
              ~~~z=\frac{x+i\gamma}{\sigma\sqrt{2}}</math>|
  cdf        =(complicated - see text)|
  mean      =(not defined)|
  median    =<math>0</math>|
  mode      =<math>0</math>|
  variance  =(not defined)|
  skewness  =(not defined)|
  kurtosis  =(not defined)|
  entropy    =|
  mgf        =(not defined)|
  char      =<math>e^{-\gamma|t|-\sigma^2 t^2/2}</math>
}}


In [[spectroscopy]], the '''Voigt profile''' (named after [[Woldemar Voigt]]) is a [[spectral line|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 [[Powder diffraction|diffraction]]. Due to the computational expense of the convolution operation, the Voigt profile is often approximated using a pseudo-Voigt profile.
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All normalized line profiles can be considered to be [[probability distribution]]s. The Gaussian profile is equivalent to a [[Normal distribution|Gaussian or normal distribution]] and a Lorentzian profile is equivalent to a [[Cauchy distribution|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:
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  V(x;\sigma,\gamma)=\int_{-\infty}^\infty G(x';\sigma)L(x-x';\gamma)\, dx'
 
</math>
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where ''x'' is frequency from line center, <math>G(x;\sigma)</math> is the centered Gaussian profile:
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:<math>
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  G(x;\sigma)\equiv\frac{e^{-x^2/(2\sigma^2)}}{\sigma \sqrt{2\pi}}
 
</math>
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and <math>L(x;\gamma)</math> is the centered Lorentzian profile:
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  L(x;\gamma)\equiv\frac{\gamma}{\pi(x^2+\gamma^2)}.
 
</math>
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The defining integral can be evaluated as:
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:<math>
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   V(x;\sigma,\gamma)=\frac{\textrm{Re}[w(z)]}{\sigma\sqrt{2 \pi}}
 
</math>
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where Re[''w(z)''&nbsp;] is the real part of the [[Faddeeva function]] evaluated for
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:<math>
z=\frac{x+i\gamma}{\sigma\sqrt{2}}.
</math>


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The Voigt profile is normalized:
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:<math>
  \int_{-\infty}^\infty V(x;\sigma,\gamma)\,dx = 1
</math>
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 (probability theory)|characteristic function]] for the [[Cauchy distribution]] is well defined, as is the characteristic function for the [[normal distribution]].  The [[characteristic function (probability theory)|characteristic function]] for the (centered) Voigt profile will then be the product of the two:


:<math>
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  \varphi_f(t;\sigma,\gamma) = E(e^{ixt}) = e^{-\sigma^2t^2/2 - \gamma |t|}.
 
</math>
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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.
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=== Cumulative distribution function ===
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Using the above definition for ''z''&nbsp;, the CDF can be found as follows:
 
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:<math>F(x_0;\mu,\sigma)
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=\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)
</math>


Substituting the definition of the [[Faddeeva function]] (scaled complex [[error function]]) yields for the indefinite integral:
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:<math>
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\frac{1}{\sqrt{\pi}}\int w(z)\,dz =\frac{1}{\sqrt{\pi}}
\int e^{-z^2}\left[1-\mathrm{erf}(-iz)\right]\,dz
</math>


Which may be solved to yield:
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:<math>
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\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)
</math>


where <math>\,_2F_2()</math> 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:
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:<math>F(x;\mu,\sigma)=\mathrm{Re}\left[\frac{1}{2}+
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\frac{\mathrm{erf}(z)}{2}
+\frac{iz^2}{\pi}\,_2F_2\left(1,1;\frac{3}{2},2;-z^2\right)\right]
</math>


=== The width of the Voigt profile ===
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The [[full width at half maximum]] (FWHM) of the Voigt profile can be found from the
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widths of the associated Gaussian and Lorentzian widths. The FWHM of the Gaussian profile
is


:<math>f_\mathrm{G}=2\sigma\sqrt{2\ln(2)}.\,</math>
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The FWHM of the Lorentzian profile is just <math>f_L=2\gamma</math>. Define
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φ = <math>f_L/f_G</math>. Then the FWHM of the Voigt profile (<math>f_V</math> ) can be estimated as:


:<math>f_\mathrm{V}\approx f_\mathrm{G}\left(1-c_0c_1+\sqrt{\phi^2+2c_1\phi+c_0^2c_1^2}\right)</math>
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where <math>c_0</math> = 2.0056 and <math>c_1</math> = 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.
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A better approximation with an accuracy of 0.02% is given by<ref>{{Cite journal
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| doi = 10.1016/0022-4073(77)90161-3
 
| issn = 0022-4073
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| volume = 17
 
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| pages = 233–236
 
| last = Olivero
</ul>
| first = J.J.
| coauthors = R.L. Longbothum
| title = Empirical fits to the Voigt line width: A brief review
| journal = Journal of Quantitative Spectroscopy and Radiative Transfer
| accessdate = 2009-04-01
| date = February 1977
| url = http://www.sciencedirect.com/science/article/B6TVR-46V0D4P-K/2/6ea61622d51016571492f70ccb7df928
|bibcode = 1977JQSRT..17..233O }}</ref>
 
:<math>f_\mathrm{V}\approx 0.5346 f_\mathrm{L}+\sqrt{0.2166f_\mathrm{L}^2+f_\mathrm{G}^2}.</math>
 
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 <math>\mu_G</math> and the Lorentzian profile is centered at <math>\mu_L</math>, the convolution will be centered at <math>\mu_G+\mu_L</math> and the characteristic function will then be:
 
:<math>
\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|}.
</math>
 
The mode and median will then both be located at <math>\mu_G+\mu_L</math>.
 
==Voigt functions==
 
The '''Voigt functions'''<ref>{{dlmf|id=7.19|first=N. M. |last=Temme|title=Voigt function}}</ref> ''U'', ''V'', and ''H'' (sometimes called the '''line broadening function''') are defined by
:<math>U(x,t)+iV(x,t) = \sqrt \frac{\pi}{4t} e^{z^2} \text{erfc}(z) = \sqrt \frac{\pi}{4t} w(iz)</math>
:<math>H(a,u) = \frac{U(u/a,1/4 a^2)}{a\sqrt \pi}</math>
where
:<math>z=(1-ix)/2\sqrt t,</math>
erfc is the [[complementary error function]], and ''w''(''z'') is the [[Faddeeva function]].
 
=== Relation to Voigt Profile ===
:<math> V(x;\sigma,\gamma) = H(a,u) / ( \sqrt 2 \sqrt \pi \sigma ) </math>,
with
:<math> a = \gamma /( \sqrt 2 \sigma) </math>
and
:<math> u =  x / (\sqrt 2 \sigma ) </math>
 
==References==
<references/>
 
==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.
 
{{ProbDistributions|continuous-infinite}}
 
[[Category:Continuous distributions]]
[[Category:Spectroscopy]]
[[Category:Special functions]]
[[Category:Probability distributions with non-finite variance]]
[[Category:Probability distributions]]

Latest revision as of 02:38, 10 December 2014

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