Thermal-Neutron Decay in Small Systems

The pulsed-neutron experiment discrete time-decay constants are examined in slab and spherical geometries using a one-term degenerate isotropic scattering kernel. The integral form of the space-, energy-, and time-dependent neutron-transport equation is considered in the proof of four theorems that determine the nature of the decay constants as a function of system size. The theorems are verified by actual calculation of the decay constants for the simpler of the two degenerate-kernel models considered. The spatial eigenfunctions that become flatter as system size is decreased are also computed. The one-velocity problem is solved as a special case. Pulsed-neutron experiment size-dependent extrapolation distances are defined and calculated in such a way as to bring exp (iB · r) theory decay constant results into agreement with those obtained by a more rigorous treatment of the spatial dependence, even for vanishingly small systems. Again, the monoenergetic problem is included as a special case. The variable extrapolation distances approach the Milne problem value as system size is increased. The variation of the extrapolation distance with system dimension is discussed in terms of opposing effects of the thermalization and transport phenomena. Estimates of leakage angular distributions and energy spectra in slabs are calculated from single iterations (performed analytically) on spatial functions synthesized from asymptotic solutions using the size-dependent extrapolation distances. The nature of the singularity in the angular distributions within extremely small systems is investigated. Finally, physical explanations for the changes in the leakage angular distributions and energy spectra (which are diffusion cooled) with slab dimensions are proffered.