Numerical solution of the phase-space dependent backward master equation for the probability distribution of neutron number in a subcritical multiplying sample

Abstract

The ability to model low neutron populations is of great importance in the nuclear safeguards and nonproliferation space. Historically, the modelling of the counting distributions of such low neutron populations has been restricted to either Monte Carlo methods or the calculation of the statistical moments of the distributions. This thesis uses an alternative method of calculating the neutron number probability distributions, focussing on the backward Master equation, allowing the full neutron phase space to be considered. The work detailed herein represents a significant step forward in such backward Master equation methods, which have previously been restricted to the point model. This thesis begins by considering the backward Master equation in the point model, where the mathematical framework is developed for both a single initiating neutron and a spontaneous fission source within a fissile material. The point model formulation allows for limited comparisons to be made between the results obtained in the backward Master equation method and those obtained through semi-analytical solutions and additionally the forward Master equation method. From this foundation, the complexity of the model is increased: firstly through the incorporation of spatial and energy dependence in the diffusion approximation and then using a fully phase space dependent transport setting. The results obtained in this thesis show complex behaviours resulting from the inclusion of the spatial, energy and angular dependence of the neutrons, which the previously used point models are incapable of demonstrating. Furthermore, significant differences are observed in the neutron number probability distributions resulting from the different models considered - highlighting the value of performing the fully phase space dependent calculations.