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