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8
docs/source/io_formats/mgxs_library.rst
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@ -22,9 +22,11 @@ materials.
.. _XML: http://www.w3.org/XML/
--------------------------------------
MGXS Library Specification -- mgxs.xml
--------------------------------------
.. _mgxs_lib_spec:
--------------------------
MGXS Library Specification
--------------------------
The multi-group library meta-data is contained within the groups_,
group_structure_, and inverse_velocities_ elements.

4
docs/source/methods/cmfd.rst
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@ -8,6 +8,10 @@ This page section discusses how nonlinear diffusion acceleration (NDA) using
coarse mesh finite difference (CMFD) is implemented into OpenMC. Before we get
into the theory, general notation for this section is discussed.
Note that the methods discussed in this section are written specifically for
continuous-energy mode but equivalent apply to the multi-group mode if the
particle's energy is replaced with the particle's group
--------
Notation
--------

123
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 +51,7 @@ 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
@ -58,12 +61,11 @@ 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
---------------------------------
@ -137,7 +139,98 @@ scattering does not occur in the resolved resonance region. This is usually,
but not always the case. Future library versions may eliminate this issue.
The data format used by OpenMC to represent windowed multipole data is specified
in :ref:`io_data_wmp`
in :ref:`io_data_wmp`.
----------------
Multi-Group Data
----------------
The data governing the interaction of particles with various nuclei or materials
are represented using a multi-group library format specific to the OpenMC code.
The format is described in the :ref:`mgxs_lib_spec`.
The data itself can be prepared via traditional paths or directly from a
continuous-energy OpenMC calculation by use of the Python API as is shown in the
:ref:`notebook_mgxs_part_iv` example notebook. This multi-group
library consists of 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
a fission production matrix cross section
(:math:`\nu\sigma_{f,g\rightarrow g'}`), or
both the fission spectrum data (:math:`\chi_{g'}`) and a fission production
cross section (:math:`\nu\sigma_{f,g}`), or, . 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}`) if the associated tallies are
required by the model using the library.
After a scattering collision, the outgoing particle experiences a change in both
energy and angle. The probability of a particle resulting in a given outgoing
energy group (`g'`) given a certain incoming energy group (`g`) is provided
by the scattering matrix data. The angular information can be expressed either
via Legendre expansion of the particle's change-in-angle (:math:`\mu`), a
tabular representation of the probability distribution function of :math:`\mu`,
or a histogram representation of the same PDF. The formats used to
represent these are described in the :ref:`mgxs_lib_spec`.
Unlike the continuous-energy mode, the multi-group mode does not explicitly
track particles produced from scattering multiplication (i.e., :math:`(n,xn)`)
reactions. These are instead accounted for by adjusting the weight of the
particle after the collision such that the correct total weight is maintained.
The weight adjustment factor is optionally provided by the `multiplicity` data
which is required to be provided in the form of a group-wise matrix.
This data is provided as a group-wise matrix since the probability of producing
multiple particles in a scattering reaction depends on both the incoming energy,
`g`, and the sampled outgoing energy, `g'`. This data represents the average
number of particles emitted from a scattering reaction, given a scattering
reaction has occurred:
.. math::
multiplicity_{g \rightarrow g'} = \frac{\nu_{scatter}\sigma_{s,g \rightarrow g'}}{
\sigma_{s,g \rightarrow 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 particles produced in scattering multiplication reactions.
However, this assumption will result in a loss of accuracy since the total
particle population would not be conserved. This reduction in accuracy due to
the loss in particle conservation can be mitigated by reducing the absorption
cross section as needed to maintain particle 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 \rightarrow g'}
The above method is the same as is usually done with most deterministic solvers.
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 particles produced in
these reactions.
All of the data discussed in this section can be provided to the code
independent of the particle's direction of motion (i.e., isotropic), or the data
can be provided as a tabular distribution of the polar and azimuthal particle
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

40
docs/source/methods/introduction.rst
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@ -8,7 +8,7 @@ The physical process by which a population of particles evolves over time is
governed by a number of `probability distributions`_. For instance, given a
particle traveling through some material, there is a probability distribution
for the distance it will travel until its next collision (an exponential
distribution). Then, when it collides with a nucleus, there is associated
distribution). Then, when it collides with a nucleus, there is an associated
probability of undergoing each possible reaction with that nucleus. While the
behavior of any single particle is unpredictable, the average behavior of a
large population of particles originating from the same source is well defined.
@ -45,15 +45,20 @@ following steps:
- Initialize the pseudorandom number generator.
- Read ACE format cross sections specified in the problem.
- Read the contiuous-energy or multi-group cross section data specified in
the problem.
- If using a special energy grid treatment such as a union energy grid or
lethargy bins, that must be initialized as well.
lethargy bins, that must be initialized as well in a continuous-energy
problem.
- In a multi-group problem, individual nuclide cross section information is
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:
@ -95,6 +100,10 @@ proceed. The life of a single particle will proceed as follows:
P(i) = \frac{\Sigma_{t,i}}{\Sigma_t}.
Note that the above selection of collided nuclide only applies to
continuous-energy simulations as multi-group simulations use nuclide
data which has already been combined in to material-specific data.
8. Once the specific nuclide is sampled, the random samples a reaction for
that nuclide based on the microscopic cross sections. If the microscopic
cross section for some reaction :math:`x` is :math:`\sigma_x` and the total
@ -105,13 +114,20 @@ proceed. The life of a single particle will proceed as follows:
P(x) = \frac{\sigma_x}{\sigma_t}.
Since multi-group simulations use material-specific data, the above is
performed with those material multi-group cross sections (i.e.,
macroscopic cross sections for the material) instead of microscopic
cross sections for the nuclide).
9. If the sampled reaction is elastic or inelastic scattering, the outgoing
energy and angle is sampled from the appropriate distribution. Reactions
of type :math:`(n,xn)` are treated as scattering and the weight of the
particle is increased by the multiplicity of the reaction. The particle
then continues from step 3. If the reaction is absorption or fission, the
particle dies and if necessary, fission sites are created and stored in the
fission bank.
energy and angle is sampled from the appropriate distribution. In
continuous-energy simulation, reactions of type :math:`(n,xn)` are treated
as scattering and any additional particles which may be created are added
to a secondary particle bank to be tracked later. In a multi-group
simulation, this secondary bank is not used but the particle weight is
increased accordingly. The original particle then continues from step 3.
If the reaction is absorption or fission, the particle dies and if
necessary, fission sites are created and stored in the fission bank.
After all particles have been simulated, there are a few final tasks that must
be performed before the run is finished. This include the following:

93
docs/source/methods/physics.rst
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@ -4,6 +4,12 @@
Physics
=======
There are limited differences between physics treatments used in the
continuous-energy and multi-group modes. If distinctions are necessary, each
of the following sections will provide an explanation of the differences.
Otherwise, replacing any references of the particle's energy (`E`) with
references to the particle's energy group (`g`) will suffice.
-----------------------------------
Sampling Distance to Next Collision
-----------------------------------
@ -79,6 +85,10 @@ originating from :math:`(n,\gamma)` and other reactions.
Elastic Scattering
------------------
Note that the multi-group mode makes no distinction between elastic or
inelastic scattering reactions. The spceific multi-group scattering
implementation is discussed in the :ref:`multi-group-scatter` section.
Elastic scattering refers to the process by which a neutron scatters off a
nucleus and does not leave it in an excited. It is referred to as "elastic"
because in the center-of-mass system, the neutron does not actually lose
@ -170,6 +180,10 @@ final direction in the lab system.
Inelastic Scattering
--------------------
Note that the multi-group mode makes no distinction between elastic or
inelastic scattering reactions. The spceific multi-group scattering
implementation is discussed in the :ref:`multi-group-scatter` section.
The major algorithms for inelastic scattering were described in previous
sections. First, a scattering cosine is sampled using the algorithms in
:ref:`sample-angle`. Then an outgoing energy is sampled using the algorithms in
@ -186,12 +200,69 @@ secondary photons from nuclear de-excitation are tracked in OpenMC.
:math:`(n,xn)` Reactions
------------------------
Note that the multi-group mode makes no distinction between elastic or
inelastic scattering reactions. The specific multi-group scattering
implementation is discussed in the :ref:`multi-group-scatter` section.
These types of reactions are just treated as inelastic scattering and as such
are subject to the same procedure as described in :ref:`inelastic-scatter`. For
reactions with integral multiplicity, e.g., :math:`(n,2n)`, an appropriate
number of secondary neutrons are created. For reactions that have a multiplicity
given as a function of the incoming neutron energy (which occasionally occurs
for MT=5), the weight of the outgoing neutron is multiplied by the multiplcity.
for MT=5), the weight of the outgoing neutron is multiplied by the multiplicity.
.. _multi-group-scatter:
----------------------
Multi-Group Scattering
----------------------
In multi-group mode, a scattering collision requires that the outgoing energy
group of the simulated particle be selected from a probability distribution,
the change-in-angle selected from a probability distribution according to
the outgoing energy group, and finally the particle's weight adjusted again
according to the outgoing energy group.
The first step in selecting an outgoing energy group for a particle in a given
incoming energy group is to select a random number (:math:`\xi`) between 0 and
1. This number is then compared to the cumulative distribution function
produced from the outgoing group (`g'`) data for the given incoming group (`g`):
.. math::
CDF = \sum_{g'=0}^{h}\Sigma_{s,g \rightarrow g'}
If the scattering data is represented as a Legendre expansion, then the
value of :math:`\Sigma_{s,g \rightarrow g'}` above is the 0th order forthe
given group transfer. If the data is provided as tabular or histogram data, then
:math:`\Sigma_{s,g \rightarrow g'}` is the sum of all bins of data for a given
`g` and `g'` pair.
Now that the outgoing energy is known the change-in-angle, :math:`\mu` can be
determined. If the data is provided as a Legendre expansion, this is done by
rejection sampling of the probability distribution represented by the Legendre
series. For efficiency, the selected values of the PDF (:math:`f(\mu)`) are
chosen to be between 0 and the maximum value of :math:`f(\mu)` in the domain of
-1 to 1. Note that this sampling scheme automatically forces negative values of
the :math:`f(\mu)` probability distribution function to be treated as zero
probabilities.
If the angular data is instead provided as a tabular representation, then the
value of :math:`\mu` is selected as described in the :ref:`angle-tabular`
section with a linear-linear interpolation scheme.
If the angular data is provided as a histogram representation, then
the value of :math:`\mu` is selected in a similar fashion to that described for
the selection of the outgoing energy (since the energy group representation is
simply a histogram representation) except the CDF is composed of the angular
bins and not the energy groups. However, since we are interested in a specific
value of :math:`\mu` instead of a group, then an angle selected from a uniform
distribution within from the chosen angular bin.
The final step in the scattering treatment is to adjust the weight of the
neutron to account for any production of neutrons due to :math:`(n,xn)`
reactions. This data is obtained from the multiplicity data provided in the
multi-group cross section library for the material of interest.
The scaled value will default to 1.0 if no value is provided in the library.
.. _fission:
@ -271,10 +342,19 @@ position of the collision site are stored in an array called the fission
bank. In a subsequent generation, these fission bank sites are used as starting
source sites.
The above description is similar for the multi-group mode except the data are
provided as group-wise data instead of in a continuous-energy format. In this
case, the outgoing energy of the fission neutrons are represented as histograms
by way of either the nu-fission matrix or chi vector.
-----------------------------------------
Secondary Angles and Energy Distributions
-----------------------------------------
Note that this section is specific to continuous-energy mode since the
multi-group scattering process has already been described including the
secondary energy and angle sampling.
For any reactions with secondary neutrons, it is necessary to sample secondary
angle and energy distributions. This includes elastic and inelastic scattering,
fission, and :math:`(n,xn)` reactions. In some cases, the angle and energy
@ -890,6 +970,9 @@ space distribution at all, the :math:`(n,2n)` reaction with H-2.
Transforming a Particle's Coordinates
-------------------------------------
Since all the multi-group data exists in the laboratory frame of reference, this
section does not apply to the multi-group mode.
Once the cosine of the scattering angle :math:`\mu` has been sampled either from
a angle distribution or a correlated angle-energy distribution, we are still
left with the task of transforming the particle's coordinates. If the outgoing
@ -941,6 +1024,9 @@ the post-collision direction is calculated as
Effect of Thermal Motion on Cross Sections
------------------------------------------
Since all the multi-group data should be generated with thermal scattering
treatments already, this section does not apply to the multi-group mode.
When a neutron scatters off of a nucleus, it may often be assumed that the
target nucleus is at rest. However, the target nucleus will have motion
associated with its thermal vibration, even at absolute zero (This is due to the
@ -1272,6 +1358,8 @@ described fully in `Walsh et al.`_
|sab| Tables
------------
Note that |sab| tables are only applicable to continuous-energy transport.
For neutrons with thermal energies, generally less than 4 eV, the kinematics of
scattering can be affected by chemical binding and crystalline effects of the
target molecule. If these effects are not accounted for in a simulation, the
@ -1461,6 +1549,9 @@ actual algorithm utilized to sample the outgoing angle is shown in equation
Unresolved Resonance Region Probability Tables
----------------------------------------------
Note that unresolved resonance treatments are only applicable to
continuous-energy transport.
In the unresolved resonance energy range, resonances may be so closely spaced
that it is not possible for experimental measurements to resolve all
resonances. To properly account for self-shielding in this energy range, OpenMC

22
docs/source/methods/tallies.rst
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@ -4,6 +4,10 @@
Tallies
=======
Note that the methods discussed in this section are written specifically for
continuous-energy mode but equivalent apply to the multi-group mode if the
particle's energy is replaced with the particle's group
------------------
Filters and Scores
------------------
@ -32,8 +36,9 @@ OpenMC: flux, total reaction rate, scattering reaction rate, neutron production
from scattering, higher scattering moments, :math:`(n,xn)` reaction rates,
absorption reaction rate, fission reaction rate, neutron production rate from
fission, and surface currents. The following variables can be used as filters:
universe, material, cell, birth cell, surface, mesh, pre-collision energy, and
post-collision energy.
universe, material, cell, birth cell, surface, mesh, pre-collision energy,
post-collision energy, polar angle, azimuthal angle, and the cosine of the
change-in-angle due to a scattering event.
With filters for pre- and post-collision energy and scoring functions for
scattering and fission production, it is possible to use OpenMC to generate
@ -55,9 +60,9 @@ be scored to for each value of the filter variable. If a particle is in cell
:math:`n`, the mapping would identify what tally/bin combinations specify cell
:math:`n` for the cell filter variable. In this manner, it is not necessary to
check the phase space variables against each tally. Note that this technique
only applies to discrete filter variables and cannot be applied to energy
bins. For energy filters, it is necessary to perform a binary search on the
specified energy grid.
only applies to discrete filter variables and cannot be applied to energy,
angle, or change-in-angle bins. For these filters, it is necessary to perform
a binary search on the specified energy grid.
-----------------------------------------
Volume-Integrated Flux and Reaction Rates
@ -196,8 +201,9 @@ One important fact to take into consideration is that the use of a track-length
estimator precludes us from using any filter that requires knowledge of the
particle's state following a collision because by definition, it will not have
had a collision at every event. Thus, for tallies with outgoing-energy filters
(which require the post-collision energy) or for tallies of scattering moments
(which require the scattering cosine), we must use an analog estimator.
(which require the post-collision energy), scattering change-in-angle filters,
or for tallies of scattering moments (which require the scattering cosine of
the change-in-angle), we must use an analog estimator.
.. TODO: Add description of surface current tallies
@ -430,7 +436,7 @@ analytically. For one degree of freedom, the t-distribution becomes a standard
.. math::
:label: cauchy-cdf
c(x) = \frac{1}{\pi} \arctan x + \frac{1}{2}.
c(x) = \frac{1}{\pi} \arctan x + \frac{1}{2}.
Thus, inverting the cumulative distribution function, we find the :math:`x`
percentile of the standard Cauchy distribution to be

2
docs/source/usersguide/beginners.rst
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@ -65,7 +65,7 @@ Now let's look at the pros and cons of Monte Carlo methods:
- **Pro**: Running simulations in parallel is conceptually very simple.
- **Con**: Because they related on repeated random sampling, they are
- **Con**: Because they rely on repeated random sampling, they are
computationally very expensive.
- **Con**: A simulation doesn't automatically give you the global solution