# 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 100*neV*. ^ Neutrons with such low energies have long wavelengths; λ ~ 100* nm*. 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 the presence of the gravitational field and look at the transport of neutrons through rough waveguides in the absence of gravity. The effects of gravity are be explored in the last part. To streamline the transition I use the common notations suitable for both types of problems. ^ More specifically, I am studying the diffusion of ultra-cold neutrons in the context of the experiments done at ILL in Grenoble in the frame of the multinational GRANIT collaboration. The parameters used in numerical calculations are the ones most common to ILL experiments. I will be calculating the diffusion coefficient and the mean-free path (MFP) under the conditions of the quantum size effect. Specifically I look at the dependence of the diffusion coefficient and the MFP on the correlation radius of surface inhomogeneities. * R*. In the second and third parts of the thesis I include the study of the neutron diffusion accompanied by slow continuous disappearing of neutrons as a result of penetration into the channel walls. This includes calculating the number of neutrons *N*(*t* = τ* _{ ex},h,R*), where τ

*is the experimental value of the time of flight in GRANIT experiments and*

_{ex}*h*is the channel width. I look not only at the square well geometry, but will also include the effects of the Earth gravitational field. The results show that while the neutrons in the square well potential disappear almost immediately, the small perturbation near the bottom of the well caused by the presence of the Earth's gravitational field drastically changes the results and is solely responsible for the observed exit neutron count in GRANIT experiments. The shape of the curves describing the exit neutron count on the width of the waveguide is extremely robust. Out brute force calculations also confirms that the earlier biased diffusion approximation is quite accurate.^

#### Subject Area

Physics

#### Recommended Citation

Sarah Elizabeth Brent,
"Diffusion of Ultra-cold Neutrons in Randomly Rough Channels"
(2018).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI10786980.

https://digitalcommons.uri.edu/dissertations/AAI10786980