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Updated cross_sections.rst to include a description of the MG mode options

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122
docs/source/methods/cross_sections.rst
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@ -1,16 +1,20 @@
.. _methods_cross_sections:
============================
Cross Section Representation
============================
=============================
Cross Section Representations
=============================
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.
----------------------
Continuous-Energy Data
----------------------
The data governing the interaction of neutrons with
various nuclei for continous-energy problems 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,
@ -24,7 +28,6 @@ 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
-------------------
@ -48,7 +51,6 @@ 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
@ -58,11 +60,102 @@ 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_.
----------------
Multi-Group Data
----------------
The data governing the interaction of neutrons with various nuclei or materials
are represented using a multi-group library format specific to the OpenMC code.
The format is described in the MGXS library specification_
The data itself can be prepared via multiple paths including: generation via
NJOY_ and TRANSX_, or directly from a continuous-energy OpenMC calculation by
use of the Python API as is shown in the Python API example_ notebooks. This
multi-group library consists of library meta-data (such as the energy group
structure) and multiple `xsdata` objects which contains the required microscopic
or macroscopic multi-group data.
At a minimum, the library must contain the absorption cross section
(:math:`\sigma_{a,g}`) and a scattering matrix. If the problem is an eigenvalue
problem then all fissionable materials must also contain either fission spectrum
data (:math:`\chi{g'}`) and a fission production cross section
(:math:`\nu\sigma_{f,g}`), or, a fission production matrix cross section
(:math:`\nu\sigma_{f,g\arrow\g'}`). If fission or energy release from fission
tallies are requested by the user, then the library must also contain the
fission cross section (:math:`\sigma_{f,g}`) or the fission energy release
cross section (:math:`\kappa\sigma_{f,g}`).
After a scattering collision, the outgoing neutron experiences a change in both
energy and angle. The probability of a neutron resulting in a given outgoing
energy group (`g'`) given a certain incoming energy group (`g`) is provided
by the scattering matrix cross sections themselves. The angular information,
however, can be expressed either via Legendre expansion of the neutron's
change-in-angle (:math:`\mu`), a tabular representation of the probability of
a neutron experiencing a given :math:`\mu`, or a histogram representation of the
probability of a neutron experiencing a given :math:`\mu`. The formats used to
represent these are described in the library format specification_.
Unlike the continuous-energy mode, the multi-group mode does not explicitly
track neutrons produced from scattering multiplication (i.e., :math:`(n,xn)`)
reactions. These are instead accounted for by adjusting the weight of the
neutron after the collision such that the correct total weight is maintained.
The information for how to adjust this weight is optionally provided by the
`multiplicity` data which exists as a group-wise matrix. This data represents
the average number of neutrons emitted from a scattering reaction, given a
scattering reaction has occurred:
.. math::
multiplicity_{g \arrow g'} = \frac{\nu_{scatter}\sigma_{s,g \arrow g'}}{
\sigma_{s,g \arrow g'}}
This data is provided as a group-wise matrix since the probability of producing
multiple neutrons in a scattering reaction depends on both the incoming energy,
`g`, and the sampled outgoing energy, `g'`.
If this scattering multiplication information is not provided in the library
then no weight adjustment will be performed. This is equivalent to neglecting
any additional neutrons produced in scattering multiplication reactions.
However, this assumption will result in a loss of accuracy since the total
neutron population would not be conserved. This reduction in accuracy due to
the loss in neutron conservation can be mitigated by reducing the absorption
cross section as needed to maintain neutron conservation. This adjustment can
be done when generating the library, or by OpenMC. To have OpenMC perform the
adjustment, the total cross section (:math:`\sigma_{t,g}`) must be provided.
With this information, OpenMC will then adjust the absorption cross section as
follows:
.. math::
\sigma_{a,g} = \sigma_{t,g} - \sum_{g'}{\nu_{scatter}\sigma_{s,g \arrow g'}}
The above method is the same as is typically done with most deterministic methods.
Note that this method is less accurate than using the scattering multiplication
weight adjustment since simply reducing the absorption cross section does not
include any information about the outgoing energy of the neutrons produced in
these reactions.
All of the data discussed in this section can be provided to the code
independent of the neutron's direction of motion (i.e., isotropic), or the data
can be provided as a tabular distribution of the polar and azimuthal neutron
direction angles. The isotropic representation is the most commonly used,
however inaccuracies are to be expected especially near material interfaces
where a material has a very large cross sections relative to the other material
(as can be expected in the resonance range). The angular representation can be
used to minimize this error.
Finally, the above options for representing the physics do not have to be
consistent across the problem. The number of groups and the structure, however,
does have to be consistent across the data sets. That is to say that each
microscopic or macroscopic data set does not have to apply the same scattering
expansion, treatment of multiplicity or angular representation of the cross
sections. This allows flexibility for the model to use highly anisotropic
scattering information in the water while the fuel can be simulated with linear
or even isotropic scattering.
.. only:: html
.. rubric:: References
@ -75,3 +168,6 @@ can be found in a paper by Leppanen_.
.. _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
.. _specification: ENTER LINK
.. _TRANSX: ENTER LINK
.. _example: ENTER LINK

6
docs/source/methods/introduction.rst
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@ -56,9 +56,9 @@ following steps:
combined to produce material-specific cross section data.
- In a fixed source problem, source sites are sampled from the specified
source. In an eigenvalue problem, source sites are sampled from some initial
source distribution or from a source file. The source sites consist of
coordinates, a direction, and an energy.
source. In an eigenvalue problem, source sites are sampled from some
initial source distribution or from a source file. The source sites
consist of coordinates, a direction, and an energy.
Once initialization is complete, the actual transport simulation can
proceed. The life of a single particle will proceed as follows: