Study of eigenvalue formulations in the PN approximation of the neutron transport equation.

Transport processes of particles and physical quantities through matter constitute a physical fundamental occurring in an extremely broad range of applications. The original Boltzmann conservation equation, derived in study through a statistical approach of the transport of molecules in a medium, it is nowadays properly declined for the study of the transport of several different types of particles, from the spreading of cancer cells in human bodies, to the photons transport through matter and neutrons throughout a nuclear reactor core. As regards the case of neutrons, an accurate knowledge of the neutrons population distribution, consequence of the transport processes involving these particles and the nuclei of the background matter, constitutes a crucial basic for any assessments concerning reactor physics analysis, as design and safety analysis computations. Specifically, a peculiar issue is establishing whether the neutron population throughout the reactor core is independent of time, decreases or increases with it, through the different operating stages of the nuclear device. This task is referred to as the criticality problem, commonly intended as the research of the combination of material composition and reactor geometrical configuration, which allows the nuclear device to achieve a stationary energy production. In criticality analysis, the study of the time-dependent behaviour of a neutron population is treated as an eigenvalue problem, representing a fundamental field of interest both for the description of deviations from the reactor stationary configuration, and for the use of the higher eigenvalue modes as a valid tool for the characterization of localized phenomena and the study of nuclear reactor kinetics. The most widely inspected eigenvalue is the effective multiplication factor k, together with the time-eigenvalue α, particularly exploited for subcritical systems analysis. Though, other eigenvalue types, as the effective multiplication factor per collision γ and the most unexplored effective density factor δ, may constitute an interesting field of investigation for physical interpretation of eigenvalue problems. Scope of this work consists in the study of the solution of the Boltzmann equation applied to the transport of neutrons in an infinite-slab geometry reactor core, in the most simplified case of a homogeneous medium, through the application of the spherical harmonics method to a monoenergetic model. Hence, the PN approximation is applied to the different eigenvalue formulations of the neutron transport equation in order to investigate its time-independent solution for simplified models, in attempting to promote a deeper knowledge about the relations occurring in solving with different eigenvalue forms the transport equation, at diverse orders of approximation accuracy. The first three chapters are devoted to the outline of the theoretical physics foundations of the present work. The successive one is dedicated to the description of the numerical algorithm implemented by the author in MATLAB environment, to solve the time-independent neutron transport equation in the specific eigenvalue form through PN approximation. The last part is finally devoted to the examination of the results obtained by the algorithm solutions, with the appropriate comparisons between both different eigenvalue forms and approximation orders of accuracy, to the conclusions report and the proposal of future application fields.