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# GramCharlierComptonProfile¶

## Description¶

The GramCharlierComptonProfile function calculates the Compton profile of a nucleus using a Gram-Charlier approximation convoluted with an instrument resolution function. The Gram-Charlier expansion of the Neutron Compton profile, $$J(y)$$ is given by  as an expansion of Hermite polynomials,

$J(y) = \frac{e^{-y^2/2\sigma^2}}{\sqrt{2\pi}\sigma}\left[ 1+ \sum_{n=2}^{\infty}\frac{a_n}{2^{2n}n!}H_{2n}\left(\frac{y}{\sqrt{2}\sigma}\right)\right]\label{a}$

where, $$\sigma$$ is the standard deviation (Gaussian width parameter), $$a_n$$ the hermite coefficients and $$H_n$$ the Hermite polynomial terms. As well as the even polynomial terms, a third order factor is included of the form,

$\frac{A}{\sqrt{2\pi} \sigma} \times FSE \times \exp(-z^2) \times H_3 (z) \label{b}$

where $$z=y/\sqrt{2\pi\sigma^2}$$ and $$FSE$$ is an input ampltiude scaling parameter. The Hermite coefficients, $$a_n$$, are supplied to the function in the parameters $$C_0$$, $$C_2$$ and $$C_4$$. The attribute HermiteCoeffs may be used to determine which polynomial terms are active, e.g “1 0 1” will cause $$C_0$$ and $$C_4$$ to be active.

The instrument resolution, $$R_M$$, is approximated by a Voigt function.

Name

Type

Default

Description

HermiteCoeffs

## Properties (fitting parameters)¶

Name

Default

Description

Mass

0.0

Atomic mass (amu)

Width

1.0

Gaussian width parameter

FSECoeff

1.0

FSE coefficient k