Greisheimer and Kooreman present a multiphysics Analytical Benchmark that I would like to use OpenMC for the neutronics modeling, within Cardinal. The paper uses deterministic (S_{2}) transport, but it should be possible to use Monte Carlo for this purpose. Since the paper assumes 1D transport with vacuum conditions on the endpoints, OpenMC can handle this by using a vacuum boundary condition on each x boundary and reflective boundary conditions on the y and z boundaries. Additionally, the mesh will be N x 1 x 1, so there is only refinement in the x direction, which will allow for computing \phi(x).

The part that I am looking for feedback on is how to handle the cross sections. This paper assumes a one-group structure for the cross sections, but it also couples some temperature and density feedbacks: namely Doppler Broadening and thermal expansion (in the x direction). Equation (10) in the paper shows that the total macroscopic cross section can be written as \Sigma_{t}(x) = \frac{\Sigma_{t,0}T_{0}}{T(x)}.

Another assumption of the paper is that the temperature and neutron flux have the same shape, or T(x)=f\phi(x). This allows the total macroscopic cross to be written also as \Sigma_{t}(x) = \frac{\Sigma_{t,0}T_{0}}{f\phi(x)}. The paper also explains how to determine \Sigma_{t,0} from other chosen parameters, so the question I have is this: **how can I generate the one-group cross sections as a function of position (or temperature and/or flux)**? It seems like this may require some iteration and some initial guess for the distribution. The paper gives a procedure for the selecting the parameters, and even gives some canonical ones as a way to compare. I am going to use the canonical parameters in this benchmark, so this will determine \Sigma_{t,0} and a few other parameters that are derived from initial choices. For more on the paper and details of the benchmark, see my GitHub repository for this project.