Voigt profile
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 | <math>\gamma,\sigma>0</math> |
|---|---|
| Support | <math>x\in(-\infty,\infty)</math> |
| Probability density function (pdf) | <math>\frac{\Re[w(z)]}{\sigma\sqrt{2\pi |
| Cumulative distribution function (cdf) | {{{cdf}}} |
| Mean | {{{mean}}} |
| Median | {{{median}}} |
| Mode | {{{mode}}} |
| Variance | {{{variance}}} |
| Skewness | {{{skewness}}} |
| Excess Kurtosis | {{{kurtosis}}} |
| Entropy | {{{entropy}}} |
| mgf | {{{mgf}}} |
| Char. func. | {{{char}}} |
~~~z=\frac{x+i\gamma}{\sigma\sqrt{2}}</math>| cdf =| 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 is a spectral line profile named after Woldemar Voigt and found in all branches of spectroscopy in which a spectral line is broadened by two types of 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.
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:
- <math>
V(x;\sigma,\gamma)=\int_{-\infty}^\infty G(x';\sigma)L(x-x';\gamma)\, dx'</math>
where x is frequency from line center, <math>G(x;\sigma)</math> is the centered Gaussian profile:
- <math>
G(x;\sigma)\equiv\frac{e^{-x^2/2\sigma^2}}{\sigma \sqrt{2\pi}}</math>
and <math>L(x;\gamma)</math> is the centered Lorentzian profile:
- <math>
L(x;\gamma)\equiv\frac{\gamma}{\pi(x^2+\gamma^2)}.</math>
The defining integral can be evaluated as:
- <math>
V(x;\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{x+i\gamma}{\sigma\sqrt{2}}.</math>
Contents |
Properties
The Voigt profile is normalized:
- <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 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^{ixt}) = e^{-\sigma^2t^2/2 - \gamma |t|}.</math>
The width of the Voigt profile
The full width at half maximum (FWHM) of the Voigt profile can be found from thewidths 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>
The FWHM of the Lorentzian profile is just <math>f_L=2\gamma</math>. Define 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>
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 have the proper behavior in the limit of φ = 0 and φ = ∞.
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>.
See also
Categories
Continuous distributions | Spectroscopy