Diffusion of Ultra-Cold Neutrons in Randomly Rough Channels

Abstract

This thesis deals with ultra cold neutrons, or, more precisely, with beams of ultra-cold neutrons. ultra-cold neutrons are longwave particles produced in a reactor from which they are coming to experimental cells through narrow channels. The beams are collimated so that the distribution of longitudinal and transverse velocities is narrow. The energies of the neutrons that we consider as ultra cold are somewhere around 100neV. Neutrons with such low energies have long wavelengths; λ ~ 100nm. Neutral particles with such large wavelengths exhibit nearly (locally) specular reflection when reflected by the solid surfaces at almost any angle of incidence. The number of ultra-cold neutrons available for experiment is extremely small. Therefore, a major experimental challenge is not to lose any particles while they travel from the reactor to the lab. Some of the main losses occur in the channel junctions when the neutrons disappear into the gaps between the overlapping channels. We explore the possibility of recovering some of these otherwise "lost" neutrons by making the inside surfaces of the junctions rough: scattering by the surface roughness can send some of the neutrons back out of the gap. This practical goal made us to re-examine diffusion of neutrons through rough channels which is by itself an interesting problem. We assume that the correlation function of random surface roughness is either Gaussian or exponential and investigate the dependence of the mean free path on the correlation radius R of the surface inhomogeneities. My results show that in order to ensure better recovery of the "lost" neutrons the walls of the junction should be made rough with the exponential correlation function of surface roughness with as small a correlation radius as possible. The results also show that the diffusion coefficient and the mean free path of UCN in rough channels exhibit a noticeable minimum at very small values of the correlation radius. This minimum sometimes has a complicated structure. The second goal is the study of UCN in Earth's gravitational field. One of the most interesting features of ultra-cold neutrons is a possible quantization of their vertical motion by the Earth's gravitational field: the kinetic energies are so low that they become comparable to the energy of neutrons in Earth's gravitational field. This results in quantization of neutron motion in the vertical direction. The energy discretization occurs on the scale of several peV. In the first part of my thesis I ignore