diff --git a/docs/source/methods/physics.rst b/docs/source/methods/physics.rst index 247eb98846..af595202d9 100644 --- a/docs/source/methods/physics.rst +++ b/docs/source/methods/physics.rst @@ -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 +++++++++++++++++++++++++++++++++++++++++++++++++++++