$$\renewcommand\AA{\unicode{x212B}}$$

# GramCharlier¶

## Description¶

This function implements the Gram-Charlier Series A expansion. It finds its main usage in fitting broad mass peaks in Y space within Neutron Compton Scattering experiments such as on the Vesuvio instrument at ISIS. As such the expansion includes only the even numbered Hermite polynomials, up to order 10, with the exception of the 3rd order term where it is useful to include a different amplitude factor.

The function definition is given by:

$f(x) = A\frac{\exp(-z^2)}{\sqrt{2\pi\sigma^2}}(1 + \frac{C4}{(2^4(4/2)!)}H_4(z) + \frac{C6}{(2^6(6/2)!)}H_6(z) + \frac{C8}{(2^8(8/2)!)}H_8(z) + \frac{C10}{(2^10(10/2)!)}H_{10}(z)) + Afse\frac{\sigma\sqrt{2}}{12\sqrt{2\pi\sigma^2}}\exp(-z^2)H_3(z)$

where $$z=\frac{(x-X_0)}{\sqrt{2\sigma^2}}$$, $$H_n(z)$$ is the nth-order Hermite polynomial and the other parameters are defined in the properties table below.

## Properties (fitting parameters)¶

Name

Default

Description

A

0.01

Amplitude

X0

0.2

Position of the centroid

Sigma

4.0

Std. Deviation of distribution

C4

-0.005

Coefficient of 4th Hermite polynomial

C6

-0.003

Coefficient of 6th Hermite polynomial

C8

-0.002

Coefficient of 8th Hermite polynomial

C10

-0.001

Coefficient of 10th Hermite polynomial

Afse

0.01

Ampliude of final-state effects term

Categories: FitFunctions | General