MultivariateGaussianComptonProfile.rst

MultivariateGaussianComptonProfile

.. index:: 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::

   IntegrationSteps;Integer;256;Length of each dimension of integration grid (must be even)

.. properties::

References

[1]	Romanelli, On the quantum contributions to phase transitions in Water probed by inelastic neutron scattering

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