mirror of
https://github.com/openmc-dev/openmc.git
synced 2026-07-26 05:05:30 -04:00
5c4dcff7cd
This commit responds to comments on PR #618. It also clarifies some points about the multipole algorithm that are not well documented.
159 lines
7 KiB
ReStructuredText
159 lines
7 KiB
ReStructuredText
.. _methods_cross_sections:
|
|
|
|
============================
|
|
Cross Section Representation
|
|
============================
|
|
|
|
The data governing the interaction of neutrons with various nuclei are
|
|
represented using the ACE format which is used by MCNP_ and Serpent_. ACE-format
|
|
data can be generated with the NJOY_ nuclear data processing system which
|
|
converts raw `ENDF/B data`_ into linearly-interpolable data as required by most
|
|
Monte Carlo codes. The use of a standard cross section format allows for a
|
|
direct comparison of OpenMC with other codes since the same cross section
|
|
libraries can be used.
|
|
|
|
The ACE format contains continuous-energy cross sections for the following types
|
|
of reactions: elastic scattering, fission (or first-chance fission,
|
|
second-chance fission, etc.), inelastic scattering, :math:`(n,xn)`,
|
|
:math:`(n,\gamma)`, and various other absorption reactions. For those reactions
|
|
with one or more neutrons in the exit channel, secondary angle and energy
|
|
distributions may be provided. In addition, fissionable nuclides have total,
|
|
prompt, and/or delayed :math:`\nu` as a function of energy and neutron precursor
|
|
distributions. Many nuclides also have probability tables to be used for
|
|
accurate treatment of self-shielding in the unresolved resonance range. For
|
|
bound scatterers, separate tables with :math:`S(\alpha,\beta,T)` scattering law
|
|
data can be used.
|
|
|
|
-------------------
|
|
Energy Grid Methods
|
|
-------------------
|
|
|
|
The method by which continuous energy cross sections for each nuclide in a
|
|
problem are stored as a function of energy can have a substantial effect on the
|
|
performance of a Monte Carlo simulation. Since the ACE format is based on
|
|
linearly-interpolable cross sections, each nuclide has cross sections tabulated
|
|
over a wide range of energies. Some nuclides may only have a few points
|
|
tabulated (e.g. H-1) whereas other nuclides may have hundreds or thousands of
|
|
points tabulated (e.g. U-238).
|
|
|
|
At each collision, it is necessary to sample the probability of having a
|
|
particular type of interaction whether it be elastic scattering, :math:`(n,2n)`,
|
|
level inelastic scattering, etc. This requires looking up the microscopic cross
|
|
sections for these reactions for each nuclide within the target material. Since
|
|
each nuclide has a unique energy grid, it would be necessary to search for the
|
|
appropriate index for each nuclide at every collision. This can become a very
|
|
time-consuming process, especially if there are many nuclides in a problem as
|
|
there would be for burnup calculations. Thus, there is a strong motive to
|
|
implement a method of reducing the number of energy grid searches in order to
|
|
speed up the calculation.
|
|
|
|
Logarithmic Mapping
|
|
-------------------
|
|
|
|
To speed up energy grid searches, OpenMC uses logarithmic mapping technique
|
|
[Brown]_ to limit the range of energies that must be searched for each
|
|
nuclide. The entire energy range is divided up into equal-lethargy segments, and
|
|
the bounding energies of each segment are mapped to bounding indices on each of
|
|
the nuclide energy grids. By default, OpenMC uses 8000 equal-lethargy segments
|
|
as recommended by Brown.
|
|
|
|
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, :math:`p_j`, and residues, :math:`r_j`, in the complex
|
|
plane. The 0K 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 the form:
|
|
|
|
.. math::
|
|
\sigma(E, T) = \frac{1}{2 E \sqrt{\xi}} \sum_j \text{Re} \left[i r_j
|
|
\sqrt{\pi} W_i(z) - \frac{r_j}{\sqrt{\pi}} C \left(\frac{p_j}{\sqrt{\xi}},
|
|
\frac{u}{2 \sqrt{\xi}}\right)\right]
|
|
.. math::
|
|
W_i(z) = \frac{i}{\pi} \int_{-\infty}^\infty dt \frac{e^{-t^2}}{z - t}
|
|
.. math::
|
|
C \left(\frac{p_j}{\sqrt{\xi}},\frac{u}{2 \sqrt{\xi}}\right) =
|
|
2p_j \int_0^\infty du' \frac{e^{-(u + u')^2/4\xi}}{p_j^2 - u'^2}
|
|
.. math::
|
|
z = \frac{\sqrt{E} - p_j}{2 \sqrt{\xi}}
|
|
.. math::
|
|
\xi = \frac{k_B T}{4 A}
|
|
.. math::
|
|
u = \sqrt{E}
|
|
|
|
where :math:`T` is the temperature of the resonant scatterer, :math:`k_B` is the
|
|
Boltzmann constant, :math:`A` is the mass of the target nucleus. For
|
|
:math:`E \gg k_b T/A`, the :math:`C` integral is approximately zero, simplifying
|
|
the cross section to:
|
|
|
|
.. math::
|
|
\sigma(E, T) = \frac{1}{2 E \sqrt{\xi}} \sum_j \text{Re} \left[i r_j
|
|
\sqrt{\pi} W_i(z)\right]
|
|
|
|
The :math:`W_i` integral simplifies down to an analytic form. We define the
|
|
Faddeeva function, :math:`W` as:
|
|
|
|
.. math::
|
|
W(z) = e^{-z^2} \text{Erfc}(-iz)
|
|
|
|
Through this, the integral transforms as follows:
|
|
|
|
.. math::
|
|
\text{Im} (z) > 0 : W_i(z) = W(z)
|
|
.. math::
|
|
\text{Im} (z) < 0 : W_i(z) = -W(z^*)^*
|
|
|
|
There are freely available algorithms_ to evaluate the Faddeeva function. For
|
|
many nuclides, the Faddeeva function needs to be evaluated thousands of times to
|
|
calculate a cross section. 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. This polynomial fit is then broadened exactly. This
|
|
exact broadening can make up for the removal of the :math:`C` integral, as
|
|
typically at low energies, only curve fits are used.
|
|
|
|
Note that the implementation of WMP in OpenMC currently assumes that inelastic
|
|
scattering does not occur in the resolved resonance region. This is usually,
|
|
but not always the case. Future library versions may eliminate this issue.
|
|
|
|
.. only:: html
|
|
|
|
.. rubric:: References
|
|
|
|
.. [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. Hwang, "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
|
|
.. _ENDF/B data: http://www.nndc.bnl.gov/endf
|
|
.. _Leppanen: http://dx.doi.org/10.1016/j.anucene.2009.03.019
|
|
.. _algorithms: http://ab-initio.mit.edu/wiki/index.php/Faddeeva_Package
|