Voigt profile

Under the Voigt profile or the Voigt function (after Woldemar Voigt ) is the convolution of a Gaussian curve with a Lorentzian curve.

Mathematical Description

Corresponds to the standard deviation of a Gaussian distribution. In spectroscopy, it is referred to as the Doppler width. is known as the half width at half maximum of the Lorentzian distribution, in spectroscopy as a distribution. The Voigt profile arises from the convolution of Gaussian profile with the Lorentzian profile. The Voigt profile is in each case as the Gaussian and Lorentz profile normalized to 1 (area under the profiles ).

Numeric representation

Of the convolution integral, there is no analytical solution, but it can be expressed as a real part of the function Faddeeva (scaled complex error function, plasma dispersion function), are available for the sufficiently good approximations:

Is defined herein as

The width of the Voigt profile

The FWHM of the Voigt profile can be determined from the widths of the involved Lorentz and Gaussian curves. Well known is the width of the Gaussian profile:

And the width of the Lorentz profile:

Then, to a first approximation:

With c0 = 2.0056 and c1 = 1.0593. This estimate has a relative error of about 2.4 % for values ​​between 0 and 10 above equation gives for the limit values ​​and the correct behavior.

An alternative approximation is the formula by Olivero and Longbothum.

Which is given with an accuracy of 0.02%.

Properties

The Voigt function is invariant with respect to convolution, ie, the convolution of a Voigt function with a Voigt function gives back a Voigt function. The line widths of the Gaussian or Lorentzian component arising thereby:

Approximation by pseudo - Voigt profile

The pseudo -Voigt Profile (or the pseudo - Voigt function ) is an approximation function of the Voigt profile V (x ) comprising a convolution of a Gaussian curve G (x) with a Lorentzian curve L ( x). The Pseudo -Voigt Profile this convolution is replaced by a linear combination of Gaussian and Lorentz curve.

The pseudo -Voigt function, is often used for regression analysis of X-ray diffraction profiles.

Mathematical definition:

Here, the half width of the pseudo -Voigt function.

Examples

For a large ratio between the pressure and Doppler broadening of the Voigt profile with the Lorentzian profile is almost identical. Only directly at the center of the line there is a slight rounding of the convolution with the Gaussian curve. Is 1, the central part of the line is dominated by the Gaussian profile, and is then called Doppler core. However, the exterior is through the much slower decaying Lorentz profile are referred to this area as the damping wings. In the case is almost a Gaussian profile from the Voigt profile. The logarithmic representation ( the Gaussian curve will appear as a parable ) but shows that far from the line center still emerges the Lorentz profile, but then at a very low level.

The case meets throughout terrestrial conditions, which about the spectral lines existing in the Earth's atmosphere are molecules subject. The case or even requires low pressures and high temperatures, as they are mostly characteristic of stellar atmospheres.