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Spectroscopy Functions


Gaussian (Area)

The Gaussian or normal peak with a0 as the peak area is defined as follows:

Gaussian1.png

a0 = Area

a1 = Center (mean,mode,median)

a2 = Width (SD)

 

Built in model: Gauss

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

 

Gaussian (Amplitude)

The Gaussian or normal peak with a0 as the peak amplitude is defined as follows:

Gaussian2.png

a0 = Amplitude

a1 = Center (mean,mode,median)

a2 = Width (SD)

 

Built in model: Gauss[amp]

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

v5_Gauss.png

Lorentzian (Area)

The Lorentzian peak with a0 as the peak area is defined as follows:

Lorentzian1.png

a0 = Area

a1 = Center (mode)

a2 = Width

 

Built in model: Lorentz

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

 

Lorentzian (Amplitude)

The Lorentzian peak with a0 as the peak amplitude is defined as follows:

Lorentzian2.png

a0 = Amplitude

a1 = Center (mode)

a2 = Width

 

Built in model: Lorentz[amp]

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

 

v5_Lorentzian.png

 

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

a3 = proportional to Lorentzian/Gaussian Width ratio

 

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

a3 = proportional to Lorentzian/Gaussian Width ratio

 

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

 

Generalized Voigt - Gaussian Student's t (Area)

This is a Generalized Voigt which can be used to check the validity of the Voigt model assumption.

a0 = Area

a1 = Center

a2 = Gaussian Width

a3 = Lorentzian Width

a4 = Student's t nu (1=Lorentzian, Infinite=Gaussian)

 

Built in model: Gauss<S>

User-defined peaks and view functions: Gauss[S]i[amp](x,a0,a1,a2,a3) (Warning: computed as integral, very slow!)

 

v5_Spectroscopy_Gauss[S]i.png

Generalized Voigt - Lorentzian Student's t (Area)

This is a Generalized Voigt which can be used to check the validity of the Voigt model assumption.

a0 = Area

a1 = Center

a2 = Lorentzian Width

a3 = Gaussian Width

a4 = Student's t nu (1=Lorentzian, Infinite=Gaussian)

 

Built in model:Lorentz<S>

User-defined peaks and view functions: Lorentz[S]i(x,a0,a1,a2,a3) (Warning: computed as integral, very slow!)

v5_Spectroscopy_Lorentz[S]i.png

Pearson VII (Area)

The Pearson VII symmetric model with a0 as the peak area is defined as follows:

Pearson VII1.png

a0 = Area

a1 = Center (mode)

a2 = FWHM

a3 = Shape (1=Lorentzian, infinity=Gaussian)

 

Built in model: PearsonVII

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

 

Pearson VII (Amplitude)

The Pearson VII symmetric model with a0 as the peak amplitude is defined as follows:

Pearson VII2.png

a0 = Amplitude

a1 = Center (mode)

a2 = FWHM

a3 = Shape (1=Lorentzian, infinity=Gaussian)

 

Built in model:PearsonVII[amp]

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

 

v5_PearsonVII.png

 

Sum Gaussian and Lorentzian (Area)

 

Spectroscopy Functions3.png

a0 = Area

a1 = Center (mode)

a2 = Width (FWHM)

a3 = Fraction Gaussian (0<a3<1)

 

Built in model: GLSum

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

 

Sum Gaussian and Lorentzian (Amplitude)

 

Spectroscopy Functions4.png

a0 = Amplitude

a1 = Center (mode)

a2 = Width (FWHM)

a3 = Fraction Gaussian (0<a3<1)

 

Built in model: GLSum[amp]

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

v5_Spectroscopy_GLSum.png

An approximation for the Voigt, this model simply sums equal FWHM Lorentzians and Gaussians. The parameter a directly computes the full-width at half-maximum (FWHM). The parameter a3 varies from 0 to 1, with 0 being a pure Lorentzian and 1 being a pure Gaussian.

 

Gaussian and Lorentzian "Cross Product" (Area)

 

Spectroscopy Functions5.png

a0 = Area

a1 = Center (mode)

a2 = Width

a3 = Fraction Gaussian (0<a3<1)

 

Built in model: GLProd

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

 

Gaussian and Lorentzian "Cross Product" (Amplitude)

 

Spectroscopy Functions6.png

a0 = Amplitude

a1 = Center (mode)

a2 = Width

a3 = Fraction Gaussian (0<a3<1)

 

Built in model: GLProd[amp]

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

v5_Spectroscopy_GLProd.png

Another Voigt approximation, this model has been used for fitting XPS spectra. It combines the Gaussian and Lorentzian in a multiplicative format. As with the Gaussian-Lorentzian sum function, the a3 parameter varies from 0 to 1. Here though, the pure Lorentzian occurs with a3=1 and the pure Gaussian with an a3 =0. Another difference is that the degree of Lorentzian character is not a linear function of a3.

Constrained Gaussian (Area)

The Constrained Gaussian with a0 as the peak area is defined as follows:

Constrained Gaussian1.png

a0 = Area

a1 = Center (mode)

a2 = width 1 (frequency invariant)

a3 = width 2 (frequency dependent)

 

Built in model: GaussCnstr

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

 

Constrained Gaussian (Amplitude)

The Constrained Gaussian with a0 as the peak amplitude is defined as follows:

Constrained Gaussian2.png

a0 = Amplitude

a1 = Center (mode)

a2 = width 1 (frequency invariant)

a3 = width 2 (frequency dependent)

 

Built in model:GaussCnstr[amp]

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

 

v5_Spectroscopy_GaussCnstr1.png

This model has no validity for fitting a single peak. The concept of this model is to fit many peaks with only two widths. The a2 width represents a constant line spread function, the width of each peak due to effects which have no frequency or energy dependence. The a3 term simply creates a scaled width which is linearly proportional to energy. It is not a width per se, but is used to produce a unique frequency-dependent width component for each peak. When fitting constrained Gaussians, a single a2 and a3 is always fit. Widths and shapes cannot be varied.

Gamma Ray Peak (Gaussian + Compton Edge)

The Gamma Ray model combines an amplitude Gaussian with a Gaussian-smeared Compton edge function.

v5_GammaRayEqn.png

a0 = Amplitude (photopeak)

a1 = Center (energy photopeak and edge)

a2 = width (photopeak and edge smearing)

a3 = calibration (MeV/channels)

a4 = edge magnitude (as fraction of a0)

me = mass electron (.511004116)

 

Built in model:GammaRay

User-defined peaks and view functions: GammaRay(x,a0,a1,a2,a3,a4)

 

v5_Spectroscopy_GammaRay.png

Compton Edge

 

The Compton Edge model is defined as follows:

v5_ComptopnEdgeEqn.png

a0 = Amplitude edge magnitude

a1 = Center (energy edge)

a2 = width (edge smearing)

a3 = calibration (MeV/channels)

me = mass electron (.511004116)

 

Built in model:ComptonEdge

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

v5_Spectroscopy_ComptopnEdge.png

Gaussian First Derivative

 

Spectroscopy Functions7.png

a0 = Area

a1 = Center (mean,mode,median)

a2 = Width (SD)

 

Built in model: D1Gauss[amp]

User-defined peaks and view functions: D1Gauss[amp](x,a0,a1,a2)

v5_Spectroscopy_D1Gauss.png

 

Gaussian Second Derivative

 

Spectroscopy Functions8.png

a0 = Area

a1 = Center (mean,mode,median)

a2 = Width (SD)

 

Built in model: D2Gauss[amp]

User-defined peaks and view functions: D2Gauss[amp](x,a0,a1,a2)

v5_Spectroscopy_D2Gauss.png

 

Lorentzian First Derivative

 

Spectroscopy Functions9.png

a0 = Area

a1 = Center (mode)

a2 = Width

 

Built in model: D1Lor[amp]

User-defined peaks and view functions: D1Lor[amp](x,a0,a1,a2)

v5_Spectroscopy_D1Lor.png

 

Lorentzian Second Derivative

 

Spectroscopy Functions10.png

a0 = Area

a1 = Center (mode)

a2 = Width

 

Built in model: D2Lor[amp]

User-defined peaks and view functions: D2Lor[amp](x,a0,a1,a2)

v5_Spectroscopy_D2Lor.png

 



c:\1pf\v5 help\home.gif LN4 Generalized Normal Lorentzian