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Add some WMP theory documentation

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Sterling Harper 2016-04-03 12:53:42 -04:00
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docs/source/methods/cross_sections.rst
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@ -63,6 +63,53 @@ Other Methods
A good survey of other energy grid techniques, including unionized energy grids,
can be found in a paper by Leppanen_.
---------------------------------
Windowed Multipole Representation
---------------------------------
In addition to the usual pointwise representation of cross sections, OpenMC
offers support for an experimental data format called windowed multipole (WMP).
This data format requires less memory than pointwise cross sections, and it
allows on-the-fly Doppler broadening to arbitrary temperature.
The multipole method was introduced by [Hwang]_ and the faster windowed
multipole method by [Josey]_. In the multipole format, cross section resonances
are represented by poles, `p_j`, and residues, `r_j`, in the complex plane. The
0 K cross sections in the resolved resonance region can be computed by summing
up a contribution from each pole:
.. math::
\sigma(E, T=0\text{K}) = \frac{1}{E} \sum_j \text{Re} \left[
\frac{i r_j}{\sqrt{E} - p_j} \right]
Assuming free-gas thermal motion, cross sections in the multipole form can be
analytically Doppler broadened to give (after some approximation) the form:
.. math::
\sigma(E, T) = \frac{1}{2 E \sqrt{\xi}} \sum_j \text{Re} \left[i r_j
\sqrt{\pi} W(z) \right]
.. math::
z = \frac{\sqrt{E} - p_j}{2 \sqrt{\xi}}
.. math::
\xi = \frac{k_B T}{4 A}
where `T` is the temperature of the resonant scatterer, `k_B` is the Boltzmann
constant, `A` is the mass of the target nucleus, and `W` is the Faddeeva
function.
.. math::
W(z) = e^{-z^2} \text{erfc}(-iz)
It is prohibitively expensive to evaluate a Faddeeva function for all of the
poles every time a cross section lookup is performed in Monte Carlo. To
mitigate that computational cost, the WMP method only evaluates poles within a
certain energy "window" around the incident neutron energy and accounts for the
effect of resonances outside that window with a polynomial fit.
Note that the implementation of WMP in OpenMC assumes that inelastic scattering
does not occur in the resolved resonance region.
.. only:: html
.. rubric:: References
@ -70,6 +117,14 @@ can be found in a paper by Leppanen_.
.. [Brown] Forrest B. Brown, "New Hash-based Energy Lookup Algorithm for Monte
Carlo codes," LA-UR-14-24530, Los Alamos National Laboratory (2014).
.. [Hwang] R. N. Hwag, "A Rigorous Pole Representation of Multilevel Cross
Sections and Its Practical Application," *Nucl. Sci. Eng.*, **96**,
192-209 (1987).
.. [Josey] Colin Josey, Pablo Ducru, Benoit Forget, and Kord Smith, "Windowed
Multipole for Cross Section Doppler Broadening," *J. Comp. Phys*,
**307**, 715-727 (2016). http://dx.doi.org/10.1016/j.jcp.2015.08.013
.. _MCNP: http://mcnp.lanl.gov
.. _Serpent: http://montecarlo.vtt.fi
.. _NJOY: http://t2.lanl.gov/codes.shtml