Exact Time-Dependent Second Spatial Moment of the One-Speed Neutron Transport Model

Consider one-speed model neutron migration in an infinite homogeneous medium. Let a neutron be released from the origin at time zero. A probabilistic argument is used to show that without approximation the neutron’s mean square distance from the origin at time t, given that absorption has not occurred, is , where v is the neutron speed, λ_s is the scattering mean-free-path, and is the mean cosine of the scattering angle. The expression outside the brace is the diffusion theory result. For large t, the exact result tends to the diffusion theory result, while for small t, the exact result tends to (vt)², an extreme nondiffusion result.