Voigt profile

Different Voigt profiles each with half width 2. Special cases are the Lorentz curve (blue) and the Gauss curve (green).

The Voigt profile or the Voigt function (after Woldemar Voigt ) is understood as the convolution of a Gaussian curve with a Lorentz curve . ${\displaystyle G(x)}$ ${\displaystyle L(x)}$

Mathematical description

${\displaystyle V(x;\sigma ,\gamma )=(G*L)(x)=\int G(\tau )L(x-\tau )d\tau }$

${\displaystyle G(x;\sigma )={\frac {e^{-x^{2}/(2\sigma ^{2})}}{\sigma {\sqrt {2\pi }}}}}$

${\displaystyle L(x;\gamma )={\frac {\gamma }{\pi (x^{2}+\gamma ^{2})}}.}$

${\displaystyle \sigma }$corresponds to the standard deviation of a Gaussian distribution. In spectroscopy it is called the Doppler width. is half the width at half maximum of the Lorentz distribution, known in spectroscopy as pressure spread. The Voigt profile results from the folding of the Gaussian profile with the Lorentz profile. The Voigt profile, like the Gaussian and Lorentz profiles, is normalized to 1 (area under the profiles). ${\displaystyle \gamma }$

Numerical representation

There is no analytical solution for the convolution integral , but it can be expressed as a real part of the Faddeeva function (scaled complex error function , plasma dispersion function), for which sufficiently good approximations are available: ${\displaystyle V(x)}$ ${\displaystyle w(z)}$

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

${\displaystyle z}$ is defined here as

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

The width of the Voigt profile

The half width of the Voigt profile can be determined from the widths of the Lorentz and Gaussian curves involved. The widths of the Gaussian profile are known (fwhm: full width at half maximum),

${\displaystyle f_{\mathrm {G} }={\sqrt {8\ln(2)}}\sigma ,}$

and the Lorentz profile,

${\displaystyle f_{\mathrm {L} }=2\gamma .}$

The width of the Voigt profile is a function of and . ${\displaystyle f_{\mathrm {V} }}$ ${\displaystyle f_{\mathrm {G} }}$ ${\displaystyle f_{\mathrm {L} }}$

The simplest approximation is the symmetric interpolation formula

${\displaystyle f_{\mathrm {V} }\approx {\sqrt {f_{\mathrm {G} }^{2}+f_{\mathrm {L} }^{2}}},}$

but underestimated by up to 16%. ${\displaystyle f_{\mathrm {V} }}$

A better approximation is according to Olivero and Longbothum

${\displaystyle f_{\mathrm {V} }\approx 0{,}5346f_{\mathrm {L} }+{\sqrt {0{,}2166f_{\mathrm {L} }^{2}+f_{\mathrm {G} }^{2}}}}$

with a maximum deviation of 0.023%.

properties

The Voigt function is invariant to convolution, that is, the convolution of a Voigt function with another Voigt function results in a Voigt function again. The line widths of the Gaussian or Lorentz part result from:

${\displaystyle f_{\mathrm {G} }^{2}=\sum _{i}{(f_{\mathrm {G} }^{2})_{i}}}$

and

${\displaystyle f_{\mathrm {L} }=\sum _{i}{(f_{\mathrm {L} })_{i}}}$.

Approximation using a pseudo-Voigt profile

When comparing the Voigt profile (blue) and the pseudo-Voigt profile (magenta), hardly any differences are discernible.

The pseudo-Voigt profile (or the pseudo-Voigt function ) is an approximation function for the Voigt profile, in which the convolution is replaced by a linear combination of Gaussian and Lorentz curves. It is traditionally used for the compensation calculation of X-ray diffractometry profiles. Since an efficient and very precise implementation of the actual Voigt function is available, there is no longer any good reason to use this approximation.

Mathematical definition:

${\displaystyle V_{p}(x)=\eta \cdot L(x)+(1-\eta )\cdot G(x)\;}$   With   ${\displaystyle 0<\eta <1}$

${\displaystyle G(x)=\exp {\left[-\ln(2)\cdot \left({\frac {x-x_{0}}{w}}\right)^{2}\right]}\;}$

${\displaystyle L(x)={\frac {1}{1+({\frac {x-x_{0}}{w}})^{2}}}}$

Here is the half-width of the pseudo-Voigt function. ${\displaystyle 2w}$

Examples

With a large ratio between pressure and Doppler broadening , the Voigt profile is almost identical to the Lorentz profile. Only immediately at the center of the line is there a slight rounding due to the folding with the Gaussian curve. If it is 1, the central part of the line is dominated by the Gaussian profile, which is called the Doppler kernel . On the outside, however, the much more slowly falling Lorentz profile prevails; this area is called the damping wing . In this case , the Voigt profile becomes almost a Gaussian profile. The logarithmic representation (the Gaussian curve then appears as a parabola) shows, however, that the Lorentz profile still emerges very far from the center of the line, but then at a very low level. ${\displaystyle \gamma /\sigma \gg 1}$ ${\displaystyle \gamma /\sigma }$ ${\displaystyle \gamma /\sigma \ll 1}$

The case consistently corresponds to terrestrial conditions to which the spectral lines of the molecules in the earth's atmosphere are subjected. The fall or even presupposes low pressures and high temperatures, as they are mostly characteristic of stellar atmospheres. ${\displaystyle \gamma /\sigma \gg 1}$ ${\displaystyle \gamma /\sigma =1}$ ${\displaystyle \gamma /\sigma \ll 1}$

literature

• Woldemar Voigt: The law of the intensity distribution within the lines of a gas spectrum. Session report of the Bavarian Academy of Sciences, Volume 25, 1912, pp. 603–620, ( online ).
• Z. Shippony, WG Read, A Highly Accurate Voigt Function Algorithm. In: Journal of Quantitative Spectroscopy & Radiative Transfer. Vol. 50, No. 6, 1993, ISSN  0022-4073 , pp. 635-645, doi : 10.1016 / 0022-4073 (93) 90031-C ; Erratum: A Correction to a Highly Accurate Voigt Function Algorithm. ibid. Vol. 78, No. 2, 2003, p. 255, doi : 10.1016 / S0022-4073 (02) 00169-3 .

Individual evidence

1. ↑ Danos & Geshwind, Phys Rev91, 1159 (1953).
2. ↑ Readable from Fig. 1 in Olivero & Longbothom (1977)
3. ↑ JJ Olivero, RL Longbothum: Empirical fits to the Voigt line width: A brief review. In: Journal of Quantitative Spectroscopy & Radiative Transfer. Vol. 17, No. 2, 1977, pp. 233-236, doi : 10.1016 / 0022-4073 (77) 90161-3 .

Web links

• https://jugit.fz-juelich.de/mlz/libcerf , numerical C library for complex error functions by Steven G. Johnson and Joachim Wuttke, contains a function voigt (x, sigma, gamma) with approximately 13-digit precision.