An analytical treatment of time- and space-dependent asymptotic behaviour of slowing down neutron

Asymptotic aspects of space- and time-dependent slowing down of a pulsed neutron is analytically studied, based on the time-dependent diffusion equation. The calculations are mainly concerned with the flux distribution ψ(r, u, t) and the spatially dependent most probable slowing down time t_max(r, u). The analytic solution for ψ(r, u, t) indicates that its spatial dependence is determined only by the time-dependent neutron age τ(u, t), while t_max(r, u) is shown to be dominantly influenced by the stationary neutron age τ_s(u).

Through these analyses, the interesting relations are obtained: (a) When $\text{t = t}{}_{\mathrm{s}}\text{≡ «t}\text{a}\text{̊}{}_0\text{{1 − ξ/(ξ + 4)(ξ + 2)}}$, τ(u, t), coincides with τ_s,(u), which implies that the spatial dependence of ψ(r, u, t) at that time is analogous to that of the stationary neutron distribution, where «tå₀ is the first time moment. (b) The most probable slowing down time t_max(r, u) at $\text{r = √«r}{}^2\text{a}\text{̊}\text{)}$ (standard notation) becomes equal to the most probable slowing down time t_m of the spatially independent distribution ψ(u, t).

The cross sections employed here are assumed to be constant, but different cross-sections for the source neutrons are allowed.