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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

Categories: FitFunctions | General

Source

C++ header: MultivariateGaussianComptonProfile.h

C++ source: MultivariateGaussianComptonProfile.cpp