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PearsonIV

Description

The PearsonIV peak shape is used to fit the prompt pulse in time-of-flight spectra. It differs from the traditional definition of the PearsonIV distribution [1] in that the Mantid definition has the peak centre (Centre`) shifted so that it coincides with the peak maximum.

The function is defined as .. math:: frac{I}{sigma}Nleft[1 + left(frac{x - lambda - nusigma/(2m)right)^{2}}{sigma}right]^{-m}expleft(-nu arctan(frac{x - lambda - nusigma/(2m)}{sigma}) right)

where:

• \(I\) is the integrated intensity (area) of the peak (parameter name Intensity)

• \(\lambda\) is the peak centre (parameter name Centre).

• \(\nu\) is the parameter Skew

• \(m\) is the parameter Exponent (valid for \(m > 0.5\))

• \(\sigma\) is the parameter Sigma (valid for \(\sigma > 0\))

• \(N = \frac{2^{2m-2}\left|\Gamma(m+i\nu/2)\right|^2}{\pi\sigma\Gamma(2m-1)}\) is the normalisation

Attributes (non-fitting parameters)

Name	Type	Default	Description
CentreShift

Properties (fitting parameters)

Name	Default	Description
Intensity	1.0	Area under the peak.
Centre	0.0	Position of the peak maximum (note this differs from the usual definition of the PearsonIV - for which the ‘location’ parameter coincides with the maximum only for skew=0).
Sigma	1.0	One of the parameters controlling the width of peak (valid for Sigma > 0) - increasing Sigma increases the FWHM.
Exponent	1.5	One of the parameters controlling the width of peak (valid for Exponent > 0.5) - increasing Exponent decreases the FWHM.
Skew	0.0	Parameter determining the skew/asymmetry of the peak - a negative value of Skew produces a peak with centre of mass at larger x value than the peak centre/maximum.

References

[1]

Pearson, Karl. “X. Contributions to the mathematical theory of evolution.—II. Skew variation in homogeneous material.” Philosophical Transactions of the Royal Society of London.(A.) 186 (1895): 343-414.

Categories: FitFunctions | General

Source

Python: PearsonIV.py