3.7

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