Neutron transport (also known as neutronics) is the study of the motions and interactions of neutrons with materials. Nuclear scientists and engineers often need to know where neutrons are in an apparatus, in what direction they are going, and how quickly they are moving. It is commonly used to determine the behavior of nuclear reactor cores and experimental or industrial neutron beams. Neutron transport is a type of radiative transport.

Neutron transport has roots in the Boltzmann equation, which was used in the 1800s to study the kinetic theory of gases. It did not receive large-scale development until the invention of chain-reacting nuclear reactors in the 1940s. As neutron distributions came under detailed scrutiny, elegant approximations and analytic solutions were found in simple geometries. However, as computational power has increased, numerical approaches to neutron transport have become prevalent. Today, with massively parallel computers, neutron transport is still under very active development in academia and research institutions throughout the world. It remains a computationally challenging problem since it depends on time and the 3 dimensions of space, and the variables of energy span several orders of magnitude (from fractions of meV to several MeV). Modern solutions use either discrete ordinates or Monte Carlo methods, or even a hybrid of both.

The neutron transport equation is a balance statement that conserves neutrons. Each term represents a gain or a loss of a neutron, and the balance, in essence, claims that neutrons gained equals neutrons lost. It is formulated as follows:

Where the equation for the precursors of delayed neutrons is as follows:

${\displaystyle {\frac {\partial C_{i}}{\partial t}}({\mathbf {r}},t)dt={\tilde {\beta }}_{i}({\mathbf {r}})\int _{0}^{\infty }dE\nu _{p}({\mathbf {r}},E)\Sigma _{f}({\mathbf {r}},E,t)\phi ({\mathbf {r}},E,t)-\lambda _{i}({\mathbf {r}})C_{i}({\mathbf {r}},t),}$

${\displaystyle \quad i=1,...,N}$

All notations are as follows:

The transport equation can be applied to a given part of phase space (time t, energy E, location ${\displaystyle \mathbf {r} ,}$ and direction of travel ${\displaystyle \mathbf {\hat {\Omega }} .}$) The first term represents the time rate of change of neutrons in the system. The second terms describes the movement of neutrons into or out of the volume of space of interest. The third term accounts for all neutrons that have a collision in that phase space. The first term on the right hand side is the production of neutrons in this phase space due to fission, while the second term on the right hand side is the production of neutrons in this phase space due to delayed neutron precursors (i.e., unstable nuclei which undergo neutron decay). The third term on the right hand side is in-scattering, these are neutrons that enter this area of phase space as a result of scattering interactions in another. The fourth term on the right is a generic source. The equation is usually solved to find ${\displaystyle \phi (\mathbf {r} ,E),}$ since that will allow for the calculation of reaction rates, which are of primary interest in shielding and dosimetry studies.