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Voigt

Voigt (Area, Traditional Form)

The traditional Voigt peak with a_{0} as the peak area is defined as follows:

a_{0} = Area

a_{1} = Center (mode)

a_{2} = proportional to Gaussian Width (Gaussian SD=a_{2}/sqrt(2))

a_{3} = proportional
to Lorentzian/Gaussian Width ratio (Lorentzian width=a_{2}*a_{3})

Built in model: Voigt

User-defined peaks and view functions: Voigt(x,a_{0},a_{1},a_{2},a_{3})

Voigt (Amplitude, Traditional)

The traditional Voigt peak with a_{0} as the peak amplitude is defined as follows:

a_{0} = Amplitude

a_{1} = Center (mode)

a_{2} = proportional to Gaussian Width (Gaussian SD=a_{2}/sqrt(2))

a_{3} = proportional
to Lorentzian/Gaussian Width ratio (Lorentzian width=a_{2}*a_{3})

Built in model: Voigt[amp]

User-defined peaks and view functions: Voigt[amp](x,a_{0},a_{1},a_{2},a_{3})

Voigt (Area, Gaussian and Lorentzian Widths)

The Voigt peak fitting the two widths directly with a_{0} as the peak area is defined as follows:

a_{0} = Area

a_{1} = Center (mode)

a_{2} = Gaussian Width (SD)

a_{3} = Lorentzian Width

Built in model: VoigtGL

User-defined peaks and view functions: VoigtGL(x,a_{0},a_{1},a_{2},a_{3})

Voigt (Amplitude, Gaussian and Lorentzian Widths)

The Voigt peak fitting the two widths directly with a_{0} as the peak amplitude is defined as
follows:

a_{0} = Amplitude

a_{1} = Center (mode)

a_{2} = Gaussian Width (SD)

a_{3} = Lorentzian Width

Built in model: VoigtGL[amp]

User-defined peaks and view functions: VoigtGL[amp](x,a_{0},a_{1},a_{2},a_{3})

PFChrom offers amplitude and area forms for a Voigt parameterization that directly computes the Gaussian
and Lorentzian widths. This enables you to get a standard error and confidence limits for the computation
of each of the widths. This is also useful since PFChrom allows you to set the individual a_{2}
and a_{3} parameters as varying or shared. In the traditional form of the Voigt, sharing a_{3}
means a single ratio of Gaussian to Lorentzian character across all peaks in a spectrum.

Computation of Voigt Function

The Voigt functions are shown containing complex math since the convolution integrals lack non-complex closed form solutions. PFChrom computes exact Voigt functions to |e|<1e-14 (to at least 14 significant figures).

For many years, the difficulty of computing the Voigt function resulted in a host of approximations, functions which combined the Gaussian and Lorentzian, or statistical functions which varied between the two shapes (such as the Pearson VII).

In PFChrom you should fit the Voigt directly unless you need to fit one of these approximations for historical reasons.

Spectral Fitting Considerations

When fitting spectral peaks, the x variable must be proportional to frequency, wave number, or energy. The y variable must be a quantitative measure. As such, you must convert wavelengths to wave number and transmission to absorption prior to fitting.

Depending on the extent of a Voigt’s Lorentzian character, it is not uncommon for some portion of the fitted Voigt's area to lay outside the range of the spectrum being fitted. When this occurs, analytical peak areas will be greater than the measured areas which report a numeric integration using the lower and upper x limits of the data being fitted.

True Non-Linear Deconvolution

Although “deconvolution” seems to be inappropriately applied to all peak fitting, with the Voigt a true
deconvolution does occur, and in a manner that no noise is introduced into the analysis. Another way to
state the Voigt is Lorentz(a_{0},a_{1},a_{3}) Ä
Gauss(1,0,a_{3}). As such, the fitted parameters in the G/L forms directly produce the deconvolved
Lorentzian and instrument response function width. Because convolution of an instrument response function
is area invariant, one assumes that Lorentz(a_{0},a_{1},a_{3}) represents the
true peak (as measured by a perfect instrument).

Suggested References

Further information on convolution, the Voigt function, and spectral line broadening can be found in Peter A. Jansson, Deconvolution with Applications in Spectroscopy, Academic Press, 1984, ISBN 0-12-380220-2.