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.