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The Voigt profile is a spectral line profile found in all branches of spectroscopy for which the spectral line broadening mechanism consists of independent contributions from mechanisms producing both a Doppler profile and a Lorentzian profile. All normalized line profiles can be considered to be probability distributions. The Doppler profile is essentially a normal distribution and a Lorentzian profile is essentially a Cauchy distribution. Without loss of generality, we can consider only centered profiles whose mean is zero. The Voigt profile is the convolution of a Lorentzian profile and a Doppler profile:
- <math>
V(f,\sigma,\gamma)=\int_{-\infty}^\infty D(f',\sigma)L(f-f',\gamma) df'
<math>
where f is frequency from line center, D(f,σ) is the Doppler profile:
- <math>
D(f,\sigma)\equiv\frac{e^{-f^2/2\sigma^2}}{\sigma \sqrt{2\pi}}
<math>
and L(f,γ) is the Lorentzian profile:
- <math>
L(f,\gamma)\equiv\frac{\gamma}{\pi(f^2-\gamma^2)}
.<math>
The defining integral can be evaluated as:
- <math>
V(f,\sigma,\gamma)=\frac{\textrm{Re}[w(z)]}{\sigma\sqrt{2 \pi}}
<math>
where Re[w(z)] is the real part of the complex error function of z and
- <math>
z=\frac{f+i\gamma}{\sigma\sqrt{2}}
.<math>
Properties
The Voigt profile is normalized:
- <math>
\int_{-\infty}^\infty V(f,\sigma,\gamma)df = 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
functon 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:
- <math>
\varphi_f(t,\sigma,\gamma) = E(e^{itf}) = e^{-\sigma^2t^2/2 - |\gamma t|}
.<math>
See also
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