3. Cross Section Representations

3.1. 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, second-chance fission, etc.), inelastic scattering, \((n,xn)\), \((n,\gamma)\), 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 \(\nu\) 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 \(S(\alpha,\beta,T)\) scattering law data can be used.

3.1.1. 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, \((n,2n)\), 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 a logarithmic mapping technique 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.

3.1.2. 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 0K cross sections in the resolved resonance region can be computed by summing up a contribution from each pole:

\[\sigma(E, T=0\text{K}) = \frac{1}{E} \sum_j \text{Re} \left[ \frac{i r_j}{\sqrt{E} - p_j} \right]\]

Assuming free-gas thermal motion, cross sections in the multipole form can be analytically Doppler broadened to give the form:

\[\sigma(E, T) = \frac{1}{2 E \sqrt{\xi}} \sum_j \text{Re} \left[i r_j \sqrt{\pi} W_i(z) - \frac{r_j}{\sqrt{\pi}} C \left(\frac{p_j}{\sqrt{\xi}}, \frac{u}{2 \sqrt{\xi}}\right)\right]\]
\[W_i(z) = \frac{i}{\pi} \int_{-\infty}^\infty dt \frac{e^{-t^2}}{z - t}\]
\[C \left(\frac{p_j}{\sqrt{\xi}},\frac{u}{2 \sqrt{\xi}}\right) = 2p_j \int_0^\infty du' \frac{e^{-(u + u')^2/4\xi}}{p_j^2 - u'^2}\]
\[z = \frac{\sqrt{E} - p_j}{2 \sqrt{\xi}}\]
\[\xi = \frac{k_B T}{4 A}\]
\[u = \sqrt{E}\]

where \(T\) is the temperature of the resonant scatterer, \(k_B\) is the Boltzmann constant, \(A\) is the mass of the target nucleus. For \(E \gg k_b T/A\), the \(C\) integral is approximately zero, simplifying the cross section to:

\[\sigma(E, T) = \frac{1}{2 E \sqrt{\xi}} \sum_j \text{Re} \left[i r_j \sqrt{\pi} W_i(z)\right]\]

The \(W_i\) integral simplifies down to an analytic form. We define the Faddeeva function, \(W\) as:

\[W(z) = e^{-z^2} \text{Erfc}(-iz)\]

Through this, the integral transforms as follows:

\[\text{Im} (z) > 0 : W_i(z) = W(z)\]
\[\text{Im} (z) < 0 : W_i(z) = -W(z^*)^*\]

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 \(C\) 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.

The data format used by OpenMC to represent windowed multipole data is specified in Windowed Multipole Library Format.

3.1.3. Temperature Treatment

At the beginning of a simulation, OpenMC collects a list of all temperatures that are present in a model. It then uses this list to determine what cross sections to load. The data that is loaded depends on what temperature method has been selected. There are three methods available:

Nearest:Cross sections are loaded only if they are within a specified tolerance of the actual temperatures in the model.
Interpolation:Cross sections are loaded at temperatures that bound the actual temperatures in the model. During transport, cross sections for each material are calculated using statistical linear-linear interpolation between bounding temperature. Suppose cross sections are available at temperatures \(T_1, T_2, ..., T_n\) and a material is assigned a temperature \(T\) where \(T_i < T < T_{i+1}\). Statistical interpolation is applied as follows: a uniformly-distributed random number of the unit interval, \(\xi\), is sampled. If \(\xi < (T - T_i)/(T_{i+1} - T_i)\), then cross sections at temperature \(T_{i+1}\) are used. Otherwise, cross sections at \(T_i\) are used. This procedure is applied for pointwise cross sections in the resolved resonance range, unresolved resonance probability tables, and \(S(\alpha,\beta)\) thermal scattering tables.
Multipole:Resolved resonance cross sections are calculated on-the-fly using techniques/data described in Windowed Multipole Representation. Cross section data is loaded for a single temperature and is used in the unresolved resonance and fast energy ranges.

3.2. 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 MGXS Library Specification. 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 Multi-Group Mode Part I: Introduction 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 (\(\sigma_{a,g}\)) 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 (\(\nu\sigma_{f,g\rightarrow g'}\)), or both the fission spectrum data (\(\chi_{g'}\)) and a fission production cross section (\(\nu\sigma_{f,g}\)), or, . The library must also contain the fission cross section (\(\sigma_{f,g}\)) or the fission energy release cross section (\(\kappa\sigma_{f,g}\)) 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 (\(\mu\)), a tabular representation of the probability distribution function of \(\mu\), or a histogram representation of the same PDF. The formats used to represent these are described in the MGXS Library Specification.

Unlike the continuous-energy mode, the multi-group mode does not explicitly track particles produced from scattering multiplication (i.e., \((n,xn)\)) 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:

\[multiplicity_{g \rightarrow g'} = \frac{\nu_{scatter}\sigma_{s,g \rightarrow g'}}{ \sigma_{s,g \rightarrow g'}}\]

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 (\(\sigma_{t,g}\)) must be provided. With this information, OpenMC will then adjust the absorption cross section as follows:

\[\sigma_{a,g} = \sigma_{t,g} - \sum_{g'}\nu_{scatter}\sigma_{s,g \rightarrow g'}\]

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.