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Updated physics.rst in the methods manual
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Adam Nelson
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2 changed files with 109 additions and 18 deletions
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@ -51,6 +51,7 @@ implement a method of reducing the number of energy grid searches in order to
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speed up the calculation.
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Logarithmic Mapping
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+++++++++++++++++++
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To speed up energy grid searches, OpenMC uses logarithmic mapping technique
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[Brown]_ to limit the range of energies that must be searched for each
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@ -60,6 +61,7 @@ the nuclide energy grids. By default, OpenMC uses 8000 equal-lethargy segments
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as recommended by Brown.
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Other Methods
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+++++++++++++
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A good survey of other energy grid techniques, including unionized energy grids,
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can be found in a paper by Leppanen_.
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@ -68,7 +70,7 @@ can be found in a paper by Leppanen_.
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Multi-Group Data
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----------------
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The data governing the interaction of neutrons with various nuclei or materials
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The data governing the interaction of particles with various nuclei or materials
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are represented using a multi-group library format specific to the OpenMC code.
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The format is described in the MGXS library specification_
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The data itself can be prepared via multiple paths including: generation via
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@ -88,23 +90,23 @@ tallies are requested by the user, then the library must also contain the
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fission cross section (:math:`\sigma_{f,g}`) or the fission energy release
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cross section (:math:`\kappa\sigma_{f,g}`).
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After a scattering collision, the outgoing neutron experiences a change in both
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energy and angle. The probability of a neutron resulting in a given outgoing
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After a scattering collision, the outgoing particle experiences a change in both
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energy and angle. The probability of a particle resulting in a given outgoing
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energy group (`g'`) given a certain incoming energy group (`g`) is provided
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by the scattering matrix cross sections themselves. The angular information,
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however, can be expressed either via Legendre expansion of the neutron's
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however, can be expressed either via Legendre expansion of the particle's
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change-in-angle (:math:`\mu`), a tabular representation of the probability of
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a neutron experiencing a given :math:`\mu`, or a histogram representation of the
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probability of a neutron experiencing a given :math:`\mu`. The formats used to
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a particle experiencing a given :math:`\mu`, or a histogram representation of the
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probability of a particle experiencing a given :math:`\mu`. The formats used to
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represent these are described in the library format specification_.
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Unlike the continuous-energy mode, the multi-group mode does not explicitly
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track neutrons produced from scattering multiplication (i.e., :math:`(n,xn)`)
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track particles produced from scattering multiplication (i.e., :math:`(n,xn)`)
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reactions. These are instead accounted for by adjusting the weight of the
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neutron after the collision such that the correct total weight is maintained.
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particle after the collision such that the correct total weight is maintained.
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The information for how to adjust this weight is optionally provided by the
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`multiplicity` data which exists as a group-wise matrix. This data represents
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the average number of neutrons emitted from a scattering reaction, given a
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the average number of particles emitted from a scattering reaction, given a
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scattering reaction has occurred:
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.. math::
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@ -113,16 +115,16 @@ scattering reaction has occurred:
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\sigma_{s,g \arrow g'}}
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This data is provided as a group-wise matrix since the probability of producing
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multiple neutrons in a scattering reaction depends on both the incoming energy,
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multiple particles in a scattering reaction depends on both the incoming energy,
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`g`, and the sampled outgoing energy, `g'`.
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If this scattering multiplication information is not provided in the library
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then no weight adjustment will be performed. This is equivalent to neglecting
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any additional neutrons produced in scattering multiplication reactions.
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any additional particles produced in scattering multiplication reactions.
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However, this assumption will result in a loss of accuracy since the total
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neutron population would not be conserved. This reduction in accuracy due to
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the loss in neutron conservation can be mitigated by reducing the absorption
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cross section as needed to maintain neutron conservation. This adjustment can
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particle population would not be conserved. This reduction in accuracy due to
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the loss in particle conservation can be mitigated by reducing the absorption
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cross section as needed to maintain particle conservation. This adjustment can
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be done when generating the library, or by OpenMC. To have OpenMC perform the
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adjustment, the total cross section (:math:`\sigma_{t,g}`) must be provided.
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With this information, OpenMC will then adjust the absorption cross section as
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@ -135,12 +137,12 @@ follows:
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The above method is the same as is typically done with most deterministic methods.
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Note that this method is less accurate than using the scattering multiplication
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weight adjustment since simply reducing the absorption cross section does not
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include any information about the outgoing energy of the neutrons produced in
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include any information about the outgoing energy of the particles produced in
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these reactions.
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All of the data discussed in this section can be provided to the code
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independent of the neutron's direction of motion (i.e., isotropic), or the data
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can be provided as a tabular distribution of the polar and azimuthal neutron
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independent of the particle's direction of motion (i.e., isotropic), or the data
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can be provided as a tabular distribution of the polar and azimuthal particle
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direction angles. The isotropic representation is the most commonly used,
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however inaccuracies are to be expected especially near material interfaces
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where a material has a very large cross sections relative to the other material
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@ -4,6 +4,12 @@
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Physics
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=======
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There are limited differences between physics treatments used in the
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continuous-energy and multi-group modes. If distinctions are necessary, each
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of the following sections will provide an explanation of the differences.
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Otherwise, replacing any references of the particle's energy (`E`) with
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references to the particle's energy group (`g) will suffice.
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-----------------------------------
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Sampling Distance to Next Collision
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-----------------------------------
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@ -79,6 +85,10 @@ originating from :math:`(n,\gamma)` and other reactions.
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Elastic Scattering
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------------------
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Note that the multi-group mode makes no distinction between elastic or
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inelastic scattering reactions. The spceific multi-group scattering
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implementation is discussed in the multi-group-scatter_ section.
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Elastic scattering refers to the process by which a neutron scatters off a
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nucleus and does not leave it in an excited. It is referred to as "elastic"
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because in the center-of-mass system, the neutron does not actually lose
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@ -170,6 +180,10 @@ final direction in the lab system.
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Inelastic Scattering
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--------------------
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Note that the multi-group mode makes no distinction between elastic or
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inelastic scattering reactions. The spceific multi-group scattering
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implementation is discussed in the multi-group-scatter_ section.
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The major algorithms for inelastic scattering were described in previous
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sections. First, a scattering cosine is sampled using the algorithms in
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:ref:`sample-angle`. Then an outgoing energy is sampled using the algorithms in
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@ -186,12 +200,67 @@ secondary photons from nuclear de-excitation are tracked in OpenMC.
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:math:`(n,xn)` Reactions
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------------------------
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Note that the multi-group mode makes no distinction between elastic or
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inelastic scattering reactions. The specific multi-group scattering
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implementation is discussed in the multi-group-scatter_ section.
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These types of reactions are just treated as inelastic scattering and as such
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are subject to the same procedure as described in :ref:`inelastic-scatter`. For
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reactions with integral multiplicity, e.g., :math:`(n,2n)`, an appropriate
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number of secondary neutrons are created. For reactions that have a multiplicity
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given as a function of the incoming neutron energy (which occasionally occurs
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for MT=5), the weight of the outgoing neutron is multiplied by the multiplcity.
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for MT=5), the weight of the outgoing neutron is multiplied by the multiplicity.
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.. _multi-group-scatter:
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----------------------
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Multi-Group Scattering
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----------------------
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In multi-group mode, a scattering collision requires that the outgoing energy
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group of the simulated particle be selected from a probability distribution,
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then the change-in-angle selected from a probability distribution according to
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the outgoing energy group, and finally the particle's weight adjusted again
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according to the outgoing energy group.
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The first step in selecting an outgoing energy group for a particle in a given
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incoming energy group is to select a random number (:math:`\xi`) between 0 and
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1. This number is then compared to the cumulative distribution function
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produced from the outgoing group (`g'`) data for the given incoming group (`g`):
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.. math::
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CDF = \sum_{g'=0}^{h}\Sigma_{s,g \arrow g'}
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If the scattering data is represented as a Legendre expansion, then the
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value of :math:`\Sigma_{s,g \arrow g'}` above is simply the 0th order. If the
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data is provided as tabular or histogram data, then the value of
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:math:`\Sigma_{s,g \arrow g'}` is the sum of all bins of data for a given `g`
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and `g'` pair.
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Now that the outgoing energy is known the change-in-angle, :math:`\mu` can be
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determined. If the data is provided as a Legendre expansion, this is done by
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rejection sampling of the probability distribution represented by the Legendre
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series. For efficiency, the selected values of the PDF (:math:`f(\mu)`) are
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chosen to be between 0 and the maximum value of :math:`f(\mu)` in the domain of
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-1 to 1.
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If the angular data is instead provided as a tabular representation, then the
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value of :math:`\mu` is selected as described in the later angle-tabular_
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section with a linear-linear interpolation scheme.
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If the angular data is provided as a histogram representation, then
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the value of :math:`\mu` is selected in a similar fashion to that described for
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the selection of the outgoing energy (since the energy group representation is
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simply a histogram representation) except the CDF is composed of the angular
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bins and not the energy groups. However, since we are interested in a specific
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value of :math:`\mu` instead of a group, then an angle selected from a uniform
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distribution within from the chosen histogram bin.
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The final step in the scattering treatment is to adjust the weight of the
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neutron to account for any production of neutrons due to :math:`(n,xn)`
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reactions. This data is obtained from the multiplicity data provided in the
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multi-group cross section library for the material of interest.
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The scaled value will default to 1.0 if no value is provided in the library.
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.. _fission:
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@ -271,10 +340,19 @@ position of the collision site are stored in an array called the fission
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bank. In a subsequent generation, these fission bank sites are used as starting
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source sites.
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The above description is similar for the multi-group mode except the data are
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provided as group-wise data instead of in a continuous-energy format. In this
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case, the outgoing energy of the fission neutrons are represented as histograms
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by way of either the nu-fission matrix or chi vector.
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-----------------------------------------
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Secondary Angles and Energy Distributions
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-----------------------------------------
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Note that this section is specific to continuous-energy mode since the
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multi-group scattering process has already been described including the
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secondary energy and angle sampling.
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For any reactions with secondary neutrons, it is necessary to sample secondary
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angle and energy distributions. This includes elastic and inelastic scattering,
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fission, and :math:`(n,xn)` reactions. In some cases, the angle and energy
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@ -890,6 +968,9 @@ space distribution at all, the :math:`(n,2n)` reaction with H-2.
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Transforming a Particle's Coordinates
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-------------------------------------
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Since all the multi-group data exists in the laboratory frame of reference, this
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section does not apply to the multi-group mode.
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Once the cosine of the scattering angle :math:`\mu` has been sampled either from
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a angle distribution or a correlated angle-energy distribution, we are still
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left with the task of transforming the particle's coordinates. If the outgoing
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@ -941,6 +1022,9 @@ the post-collision direction is calculated as
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Effect of Thermal Motion on Cross Sections
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------------------------------------------
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Since all the multi-group data should be generated with thermal scattering
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treatments already, this section does not apply to the multi-group mode.
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When a neutron scatters off of a nucleus, it may often be assumed that the
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target nucleus is at rest. However, the target nucleus will have motion
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associated with its thermal vibration, even at absolute zero (This is due to the
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@ -1272,6 +1356,8 @@ described fully in `Walsh et al.`_
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|sab| Tables
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------------
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Note that |sab| tables are only applicable to continuous-energy transport.
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For neutrons with thermal energies, generally less than 4 eV, the kinematics of
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scattering can be affected by chemical binding and crystalline effects of the
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target molecule. If these effects are not accounted for in a simulation, the
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@ -1461,6 +1547,9 @@ actual algorithm utilized to sample the outgoing angle is shown in equation
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Unresolved Resonance Region Probability Tables
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----------------------------------------------
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Note that unresolved resonance treatments are only applicable to
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continuous-energy transport.
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In the unresolved resonance energy range, resonances may be so closely spaced
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that it is not possible for experimental measurements to resolve all
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resonances. To properly account for self-shielding in this energy range, OpenMC
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