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Added description of ACE Law 44 in documentation.

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Paul Romano 2012-07-25 18:38:07 -04:00
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docs/source/methods/physics.rst
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@ -290,6 +290,8 @@ and the incoming energy:
where :math:`A` is the mass of the target nucleus measured in neutron masses.
.. _ace-law-4:
ACE Law 4 - Continuous Tabular Distribution
+++++++++++++++++++++++++++++++++++++++++++
@ -457,6 +459,68 @@ derivation [Watt]_.
ACE Law 44 - Kalbach-Mann Correlated Scattering
+++++++++++++++++++++++++++++++++++++++++++++++
This law is very similar to ACE Law 4 except now the outgoing angle of the
neutron is correlated to the outgoing energy and is not sampled from a separate
distribution. For each incident neutron energy :math:`E_i` tabulated, there is
an array of precompoung factors :math:`R_{i,j}` and angular distribution slopes
:math:`A_{i,j}` corresponding to each outgoing energy bin :math:`j` in addition
to the outgoing energies and distribution functions as in ACE Law 4.
The calculation of the outgoing energy of the neutron proceeds exactly the same
as in the algorithm described in :ref:`ace-law-4`. In that algorithm, we found
an interpolation factor :math:`f`, statistically sampled an incoming energy bin
:math:`\ell`, and sampled an outgoing energy bin :math:`j` based on the
tabulated cumulative distribution function. Once the outgoing energy has been
determined with equation :eq:`ace-law-4-energy`, we then need to calculate the
outgoing angle based on the tabulated Kalbach-Mann parameters. These parameters
themselves are subject to either histogram or linear-linear interpolation on the
outgoing energy grid. For histogram interpolation, the parameters are
.. math::
:label: KM-parameters-histogram
R = R_{\ell,j} \\
A = A_{\ell,j}
If linear-linear interpolation is specified, the parameters are
.. math::
:label: KM-parameters-linlin
R = R_{\ell,j} + \frac{\hat{E} - E_{\ell,j}}{E_{\ell,j+1} - E_{\ell,j}} (
R_{\ell,j+1} - R_{\ell,j} ) \\
A = A_{\ell,j} + \frac{\hat{E} - E_{\ell,j}}{E_{\ell,j+1} - E_{\ell,j}} (
A_{\ell,j+1} - A_{\ell,j} )
where :math:`\hat{E}` is defined in equation :eq:`energy-linlin`. With the
parameters determined, the probability distribution function for the cosine of
the scattering angle is
.. math::
:label: KM-pdf-angle
p(\mu) d\mu = \frac{A}{2 \sinh (A)} \left [ \cosh (A\mu) + R \sinh (A\mu)
\right ] d\mu
The rules for sampling this probability distribution function can be derived
based on rules C39 and C40 in the `Monte Carlo Sampler`_. First, we sample two
random numbers :math:`\xi_3, \xi_4` on the unit interval. If :math:`\xi_3 > R`
then the outgoing angle is
.. math::
:label: KM-angle-1
\mu = \frac{1}{A} \ln \left ( T + \sqrt{T^2 + 1} \right )
where :math:`T = (2 \xi_4 - 1) \sinh (A)`. If :math:`\xi_3 \le R`, then the
outgoing angle is
.. math::
:label: KM-angle-2
\mu = \frac{1}{A} \ln \left ( \xi_4 e^A + (1 - \xi_4) e^{-A} \right )
ACE Law 61 - Correlated Energy and Angle Distribution
+++++++++++++++++++++++++++++++++++++++++++++++++++++