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MultivariateGaussianComptonProfile

Description

The fitted function for y-Space converted values is as described by G. Romanelli. [1].

\[J(y) = \frac{1}{\sqrt{2\pi} \sigma_{x} \sigma_{y} \sigma_{z}} \frac{2}{\pi} \int_{0}^{1} d(\cos \theta) \int_{0}^{\frac{\pi}{2}} d \phi S^{2}(\theta, \phi) \exp \left( -\frac{y^{2}} {2 S^{2}(\theta, \phi)} \right)\]

Where \(S^{2}(\theta, \phi)\) is given by:

\[\frac{1}{S^{2}(\theta, \phi)} = \frac{\sin^{2}\theta \cos^{2}\phi}{\sigma_{x}^{2}} + \frac{\sin^{2}\theta \sin^{2}\phi}{\sigma_{y}^{2}} + \frac{\cos^{2}\theta}{\sigma_{z}^{2}}\]

The \(A_{3}\) Final State Effects (FSE) correction is applied as an additive correction expressed as:

\[-A_{3}(q)\frac{d^{3}}{dy^{3}}J(y) = \frac{\sigma_{x}^{4} + \sigma_{x}^{4} + \sigma_{x}^{4}} {9 \sqrt{2 \pi} \sigma_{x} \sigma_{y} \sigma_{z} q} \int_{0}^{1} d(\cos \theta) \int_{0}^{\frac{\pi}{2}} d \phi \left[ \frac{y^{3}}{S^{2}(\theta, \phi)^{4}} -3 \frac{y}{S^{2}(\theta, \phi)^{2}} \right] S^{2}(\theta, \phi) \exp \left( -\frac{y^{2}} {2 S^{2}(\theta, \phi)} \right)\]

Attributes (non-fitting parameters)

Name	Type	Default	Description
IntegrationSteps	Integer	256	Length of each dimension of integration grid (must be even)

Properties (fitting parameters)

Name	Default	Description
Mass	0.0	Atomic mass (amu)
Intensity	1.0	Gaussian intensity parameter
SigmaX	1.0	Sigma X parameter
SigmaY	1.0	Sigma Y parameter
SigmaZ	1.0	Sigma Z parameter

References

[1]

7. Romanelli, On the quantum contributions to phase transitions in Water probed by inelastic neutron scattering

Categories: FitFunctions | General

Source

C++ header: MultivariateGaussianComptonProfile.h

C++ source: MultivariateGaussianComptonProfile.cpp