Chemistry - Voigt profile

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 which peak at zero. The Voigt profile is the convolution of a Lorentzian profile and a Doppler profile:

$V(x|\sigma,\gamma)=\int_{-\infty}^\infty D(x'|\sigma)L(x-x'|\gamma) dx'$

where x is frequency from line center, D(x |σ) is the centered Doppler profile:

$D(x|\sigma)\equiv\frac{e^{-x^2/2\sigma^2}}{\sigma \sqrt{2\pi}}$

and L(x |γ) 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 complex error function of z and

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

Contents

1 Properties

1.1 The width of the Voigt profile
1.2 The uncentered Voigt profile

2 See also

Properties

The Voigt profile is normalized:

$\int_{-\infty}^\infty V(x|\sigma,\gamma)dx = 1$

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:

$\varphi_f(t|\sigma,\gamma) = E(e^{ixt}) = e^{-\sigma^2t^2/2 - |\gamma t|} .$

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_G

$f_G=2\sigma\sqrt{2\ln(2)}$

The FWHM of the Lorentzian profile is just f_L=2γ. Define φ=f_L/f_G. Then the FWHM of the Voigt profile (f_V) can be estimated as:

$f_V=f_G\left(1-c_0c_1+\sqrt{\phi^2+2c_1\phi+c_0^2c_1^2}\right)$

where c₀=2.0056 and c₁=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 x_G and the Lorentzian profile is centered at x_L, the convolution will be centered at x_G+x_L and the characteristic function will then be:

$\varphi_f(t|\sigma,\gamma,x_G,x_L)= e^{i(x_G+x_L)t-\sigma^2t^2/2 - |\gamma t|} .$

The mode and median will then both be located at x_G+x_L.