Neutron transport¶
In neutron transport, we are interested in obtaining the distribution of neutrosn in phase space. A logical quantity is then the neutron density in phase space:
However, when we want to calculate how many collisions (with e.g. a detector) occur, this is calculated as:
where \(\sigma_c(\mathbf{r}, \Omega, E)\) is the macroscopic cross section for the collision type of interest and \(v(E)\) is the neutron speed at energy \(E\). The reason is that neutrons with a large speed, have a larger path length in a time interval and thus have a higher probability of colliding.
It is therefore often more convenient to work with the angular neutron flux, defined as:
Assumptions¶
Several assumptions are made about the nature of the neutral particle behaviour.
Particle size is small relative to problem length scales and are classical. They are described by a position \(\mathbf{r}\) and momentum \(\mathbf{p}\).
Between collisions, particles travel in straight lines.
Collisions between neutral particles do not happen at appreciable levels compared to background-neutral collisions and are ignored.
Collisions are instant in time.
Background collisions are isotropic: the probability of a collision is solely determined by the change in angle (before and after).
Fixed materials: they do not change as a result of the neutron flux (linearity).
The timescales of neutron transport are much smaller than the timescales of macroscopic changes in the background medium.
Multiplying neutron reactions are assumed not to lead to criticality.
Neutron transport equation¶
Using these assumptions, it can then be shown, that the behavior of neutrons in a medium is governed by the neutron transport equation (here in steady-state):
where \(\sigma(\mathbf{r}, \Omega)\) is the total macroscopic cross section, \(\sigma_s(\mathbf{r}, \Omega' \to \Omega, E' \to E)\) is the differential scattering cross section from direction \(\Omega'\) and energy \(E'\) to direction \(\Omega\) and energy \(E\), and \(Q(\mathbf{r}, \Omega, E)\) is an external source term.
The first term on the left-hand side represents streaming (similar to \(\mathbf{v}\cdot \nabla f\) in the Boltzmann equation), the second term represents losses due to interactions, the first term on the right-hand side represents scattering from other energies and directions, and the second term on the right-hand side represents external sources (in the sense that they are not produced by any neutron interactions). Intuitively, the transport equation is a balance equation stating that the rate of change of neutrons in a given phase space volume is equal to the neutrons streaming in minus the neutrons streaming out, plus the neutrons created by scattering and external sources minus the neutrons lost due to interactions.
This equation is 6-dimensional (3 spatial dimensions, 2 angular dimensions, 1 energy dimension) and integro-differential. It is often solved using Monte-Carlo methods, where the above equation is sampled.
In deterministic methods, instead, the equation is fully discretised in all 6 dimensions. Since the neutron transport equation is linear, the discetised version will also be a linear system, which will need to be efficiently solved. Furthermore, in the end, we are also interested in obtaining derivatives of a function solution of this equation with respect to some parameters (e.g., how the tritium breeding ratio in fusion blankets changes with respect to material composition or even 3D geometry).
We want to keep this in mind when deriving a formulation of the neutron transport equation, and thus, before discretising, we will review some linear algebra.