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Voigt


Voigt (Area, Traditional Form)

The traditional Voigt peak with a0 as the peak area is defined as follows:

Voigt1.png

a0 = Area

a1 = Center (mode)

a2 = proportional to Gaussian Width (Gaussian SD=a2/sqrt(2))

a3 = proportional to Lorentzian/Gaussian Width ratio (Lorentzian width=a2*a3)

 

Built in model: Voigt

User-defined peaks and view functions: Voigt(x,a0,a1,a2,a3)

 

Voigt (Amplitude, Traditional)

The traditional Voigt peak with a0 as the peak amplitude is defined as follows:

Voigt2.png

a0 = Amplitude

a1 = Center (mode)

a2 = proportional to Gaussian Width (Gaussian SD=a2/sqrt(2))

a3 = proportional to Lorentzian/Gaussian Width ratio (Lorentzian width=a2*a3)

 

Built in model: Voigt[amp]

User-defined peaks and view functions: Voigt[amp](x,a0,a1,a2,a3)

 

v5_Voigt.png

Voigt (Area, Gaussian and Lorentzian Widths)

The Voigt peak fitting the two widths directly with a0 as the peak area is defined as follows:

Voigt3.png

a0 = Area

a1 = Center (mode)

a2 = Gaussian Width (SD)

a3 = Lorentzian Width

 

Built in model: VoigtGL

User-defined peaks and view functions: VoigtGL(x,a0,a1,a2,a3)

 

Voigt (Amplitude, Gaussian and Lorentzian Widths)

The Voigt peak fitting the two widths directly with a0 as the peak amplitude is defined as follows:

Voigt4.png

a0 = Amplitude

a1 = Center (mode)

a2 = Gaussian Width (SD)

a3 = Lorentzian Width

 

Built in model: VoigtGL[amp]

User-defined peaks and view functions: VoigtGL[amp](x,a0,a1,a2,a3)

 

v5_VoigtGL.png

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 a2 and a3 parameters as varying or shared. In the traditional form of the Voigt, sharing a3 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(a0,a1,a3) Ä Gauss(1,0,a3). 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(a0,a1,a3) 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.



c:\1pf\v5 help\home.gif Lorentzian Generalized Voigt