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Spectroscopy Functions

Gaussian (Area)

The Gaussian
or normal
peak with a_{0} as the peak area is defined as follows:

a_{0} = Area

a_{1} = Center (mean,mode,median)

a_{2} = Width (SD)

Built in model: Gauss

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

Gaussian (Amplitude)

The Gaussian
or normal
peak with a_{0} as the peak amplitude is defined as follows:

a_{0} = Amplitude

a_{1} = Center (mean,mode,median)

a_{2} = Width (SD)

Built in model: Gauss[amp]

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

Lorentzian (Area)

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

a_{0} = Area

a_{1} = Center (mode)

a_{2} = Width

Built in model: Lorentz

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

Lorentzian (Amplitude)

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

a_{0} = Amplitude

a_{1} = Center (mode)

a_{2} = Width

Built in model: Lorentz[amp]

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

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

a_{3} = proportional
to Lorentzian/Gaussian Width ratio

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

a_{3} = proportional
to Lorentzian/Gaussian Width ratio

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})

Generalized Voigt - Gaussian Ä Student's t (Area)

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

a_{0} = Area

a_{1} = Center

a_{2} = Gaussian Width

a_{3} = Lorentzian Width

a_{4}
= Student's t nu (1=Lorentzian, Infinite=Gaussian)

Built in model: Gauss<S>

User-defined peaks and view functions: Gauss[S]i[amp](x,a_{0},a_{1},a_{2},a_{3})
(Warning: computed as integral, very slow!)

Generalized Voigt - Lorentzian Ä Student's t (Area)

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

a_{0} = Area

a_{1} = Center

a_{2} = Lorentzian Width

a_{3} = Gaussian Width

a_{4}
= Student's t nu (1=Lorentzian, Infinite=Gaussian)

Built in model:Lorentz<S>

User-defined peaks and view functions: Lorentz[S]i(x,a_{0},a_{1},a_{2},a_{3})
(Warning: computed as integral, very slow!)

Pearson VII (Area)

The Pearson
VII symmetric model with a_{0} as the peak area is defined as follows:

a_{0} = Area

a_{1} = Center (mode)

a_{2} = FWHM

a_{3} = Shape
(1=Lorentzian, infinity=Gaussian)

Built in model: PearsonVII

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

Pearson VII (Amplitude)

The Pearson
VII symmetric model with a_{0} as the peak amplitude is defined as follows:

a_{0} = Amplitude

a_{1} = Center (mode)

a_{2} = FWHM

a_{3} = Shape (1=Lorentzian,
infinity=Gaussian)

Built in model:PearsonVII[amp]

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

Sum Gaussian and Lorentzian (Area)

a_{0} = Area

a_{1} = Center (mode)

a_{2} = Width (FWHM)

a_{3} = Fraction
Gaussian (0<a_{3}<1)

Built in model: GLSum

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

Sum Gaussian and Lorentzian (Amplitude)

a_{0} = Amplitude

a_{1} = Center (mode)

a_{2} = Width (FWHM)

a_{3} = Fraction
Gaussian (0<a_{3}<1)

Built in model: GLSum[amp]

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

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 a_{3} varies from 0 to
1, with 0 being a pure Lorentzian and 1 being a pure Gaussian.

Gaussian and Lorentzian "Cross Product" (Area)

a_{0} = Area

a_{1} = Center (mode)

a_{2} = Width

a_{3} = Fraction
Gaussian (0<a_{3}<1)

Built in model: GLProd

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

Gaussian and Lorentzian "Cross Product" (Amplitude)

a_{0} = Amplitude

a_{1} = Center (mode)

a_{2} = Width

a_{3} = Fraction
Gaussian (0<a_{3}<1)

Built in model: GLProd[amp]

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

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 a_{3}
parameter varies from 0 to 1. Here though, the pure Lorentzian occurs with a_{3}=1 and the pure
Gaussian with an a_{3} =0. Another difference is that the degree of Lorentzian character is not
a linear function of a_{3}.

Constrained Gaussian (Area)

The Constrained
Gaussian with a_{0} as the peak area is defined as follows:

a_{0} = Area

a_{1} = Center (mode)

a_{2} = width 1 (frequency invariant)

a_{3} = width
2 (frequency dependent)

Built in model: GaussCnstr

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

Constrained Gaussian (Amplitude)

The Constrained
Gaussian with a_{0} as the peak amplitude is defined as follows:

a_{0} = Amplitude

a_{1} = Center (mode)

a_{2} = width 1 (frequency invariant)

a_{3} = width 2 (frequency
dependent)

Built in model:GaussCnstr[amp]

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

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 a_{2} width represents a constant line spread function, the width of each
peak due to effects which have no frequency or energy dependence. The a_{3} 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
a_{2} and a_{3} 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.

a_{0} = Amplitude (photopeak)

a_{1} = Center (energy photopeak and edge)

a_{2} = width (photopeak and edge smearing)

a_{3} = calibration (MeV/channels)

a_{4}
= edge magnitude (as fraction of a_{0})

m_{e} = mass electron
(.511004116)

Built in model:GammaRay

User-defined peaks and view functions: GammaRay(x,a_{0},a_{1},a_{2},a_{3},a_{4})

Compton Edge

The Compton Edge model is defined as follows:

a_{0} = Amplitude edge
magnitude

a_{1} = Center (energy edge)

a_{2} = width (edge smearing)

a_{3} = calibration (MeV/channels)

m_{e} = mass electron
(.511004116)

Built in model:ComptonEdge

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

Gaussian First Derivative

a_{0} = Area

a_{1} = Center (mean,mode,median)

a_{2} = Width (SD)

Built in model: D1Gauss[amp]

User-defined peaks and view functions: D1Gauss[amp](x,a_{0},a_{1},a_{2})

Gaussian Second Derivative

a_{0} = Area

a_{1} = Center (mean,mode,median)

a_{2} = Width (SD)

Built in model: D2Gauss[amp]

User-defined peaks and view functions: D2Gauss[amp](x,a_{0},a_{1},a_{2})

Lorentzian First Derivative

a_{0} = Area

a_{1} = Center (mode)

a_{2} = Width

Built in model: D1Lor[amp]

User-defined peaks and view functions: D1Lor[amp](x,a_{0},a_{1},a_{2})

Lorentzian Second Derivative

a_{0} = Area

a_{1} = Center (mode)

a_{2} = Width

Built in model: D2Lor[amp]

User-defined peaks and view functions: D2Lor[amp](x,a_{0},a_{1},a_{2})