5. Outcome of a Simulation: Particle Distribution, Reaction Rates, and Leakage Rates

Deterministic approaches for solving the linear Boltzmann equation yield the global solution in every spatial cell and in every energy group once the iterative processes have been converged. This means that the behavior of particles in space/energy/time is the main output of a transport solve.

Hence, reaction rates for any reaction type can be obtained as a simply post-processing step using the computed flux moments, \(\phi_{\ell,m}\)) . For example, the reaction rate for a certain reaction, over a spatial region of interest (RoI \(\subset\mathcal{D}\)), for a subset of the energy groups \(\mathbb{G} \subset \mathcal{E}\)

\[\text{RR}_\text{type} = \sum_{g \in \mathbb{G}} \int_\text{RoI} d^3r \, \sigma^g_{\text{type}}(\vec{r},t) \phi^g_{0,0}(\vec{r},t)\]

The reaction type can be one of the types used during the simulation (e.g., total) or a type present in the multigroup cross-section library and used only at the post processing stage (e.g., heating).

On the boundary of the domain, OpenSn also computes the half-range angular currents

\[j^{\pm,g}(\vec{r},t) = \int_{\vec{\Omega} \cdot \vec{n}(\vec{r}) \gtrless 0} d\Omega \, |\vec{\Omega} \cdot \vec{n}(\vec{r})| \psi^g(\vec{r},\vec{\Omega},t)\]

With this quantity, one can compute partial leakage rates \(\mathcal{L}^\pm\) on a portion of the domain’s boundary (the boundary of interest, BoI \(\subset\partial\mathcal{D}\)), for a subset of the energy groups \(\mathbb{G} \subset \mathcal{E}\)

\[\mathcal{L}^\pm = \sum_{g \in \mathbb{G}} \int_\text{BoI} d^2r \, j^{\pm,g}(\vec{r},t)\]

These integrals are taken over the physical extent of the model. In reduced Cartesian geometries the ignored dimensions are infinite, so 2D results are per unit length in \(z\) and 1D slab results are per unit area normal to \(z\); in RZ geometry the volume element is \(d^3r = 2\pi r\,dr\,dz\), so results cover the full revolution. See Normalization and Units for the corresponding source conventions.

For net leakage rates across a surface of interest (SoI), that is, in the interior of the domain, we simply use the three first-order flux moments:

\[\begin{split}\begin{bmatrix} \mathcal{L}_x \\ \mathcal{L}_y \\ \mathcal{L}_z \end{bmatrix} = \sum_{g \in \mathbb{G}} \int_\text{SoI} d^2r \, \begin{bmatrix} \phi^g_{1,1}(\vec{r},t) \\ \phi^g_{1,-1}(\vec{r},t) \\ \phi^g_{1,0}(\vec{r},t) \\ \end{bmatrix}\end{split}\]