1 Introduction

This thesis is about two rather diverse topics within the field of muon spin rotation/relaxation/resonance ($\mu{\cal SR}$) spectroscopy: muonium formation in insulators and quantum diffusion of muonium in simple insulating solids. They are linked by the nature of the samples studied - they are van der Waals solids made of atoms or small molecules so weakly interacting they are gases at room temperature, solidifying only at very low temperatures and therefore called cryocrystals.

When energetic positive muons are injected into most insulating materials a fraction of the muons capture an electron from the sample and form a neutral, light hydrogen-like atom, with the muon as its nucleus (chemical symbol Mu.) The chemistry of positive muons is essentially the same as that of protons; those muons that don't form muonium atoms become bound into molecular ions instead. It has long been known that the fraction of each of these chemical species - muonium and the diamagnetic fraction, referring to the total electron spin density at the muon which is zero for a muon in a molecular bond - vary among different insulating materials. [12] This is a relatively old problem in the field of $\mu{\cal SR}$ and of interest to radiation chemists, but is not widely appreciated by many researchers applying $\mu{\cal SR}$ to a growing number of areas in condensed matter physics such as magnetism and superconductivity.

The first part of this thesis is about the use of electric fields in $\mu{\cal SR}$ experiments to extract new information about end-of-track processes leading to muonium formation. In some of these cryocrystals it is possible to influence the muonium and diamagnetic fractions with an external electric field applied either parallel to or anti-parallel to the incoming muons' direction of motion. This is the first observation of an electric field having an effect on muonium formation in a solid. The presence of an effect, and its dependence on orientation of the external electric field, imply that, at least sometimes, the electrons that eventually become bound to muons originate far from the muons. Thus, it seems that electron transport properties of the solid play a role in muonium formation. An overview of this technique and the interpretation of results is given in Chapter 4.

The main topic of this thesis is the use of the muonium that is formed in cryocrystals for the study of quantum diffusion under one- and two-phonon coupling to the vibrational modes of the lattice. Tunnelling diffusion of light interstitials is of great interest because it is inherently quantum mechanical in nature. It provides us with a system in which to study a quantum process subject to a dissipative coupling to a bath of excitations. Muons and muonium are particularly good probes of quantum diffusion since the muon has a mass about 1/9 that of hydrogen; its tunnelling amplitude is larger than that of any ordinary atom in the same potential.

Previously, studies of muon diffision in Cu and Al identified a cross-over from high-temperature, stochastic thermally-activated hopping to a low-temperature regime where the coherent propagation of the muon was disrupted by scattering of electrons and phonons. Minima in the muon hop rates occurred at about 50 K and 5 K in these metals respectively. [42,47] In metals the interaction of the muon with conduction electrons serves to reduce the tunnelling amplitude and suppress the temperature dependence. [50,51]

More recently, diffusion rates of muonium in ionic solids KCl and NaCl have been measured. [66,43] Neutral muonium in insulating materials avoids the interaction with conduction electrons, so in these systems the dominant interaction between the interstitial particle and its environment is with excitations of the lattice. In NaCl and KCl the muonium hop rate also showed minima at about 50 K and 70 K respectively. In the low temperature regime the scattering of phonons (or electrons) serves to diminish the tunnelling bandwidth, so the key characteristic is a hop rate that increases as the temperature drops.

Chapter 5 reviews the theory of phonon-mediated quantum diffusion (due mostly to Kagan and Prokof'ev), with the emphasis on explaining the origin of the temperature dependences one can measure and the influence of the phonon spectrum.

This thesis presents results from experiments in which diffusion rates of muonium in van der Waals solids were measured. Measurements in both low and high temperature regimes are discussed and results are compared with the current theory of quantum diffusion in chapter 6.

The current theory of phonon-mediated quantum diffusion is successful in predicting the observed temperature dependence of the muonium diffusion rate in solid nitrogen at temperatures well below the Debye temperature. The measured temperature dependence of the diffusion rate was found, for the first time, to be very nearly as strong as predicted by theory. Moreover, these results demonstrate qualitatively the influence of static energy level shifts, which inevitably become important at sufficiently low temperatures. Kadono et al. also studied the influence of deliberately induced defects in a sample of KCl doped with varying amounts of Na substitutional impurities. [45] In this inhomogenous system the lattice has both situations present simultaneously. The muonium ensemble was divided into two parts - one trapped by the static shifts and the other, far from impurities, diffusing as in pure KCl.

At temperatures where the phonon scattering rate is sufficient to destroy the coherent channel, tunnelling diffusion proceeds by stocastic, phonon-assisted hopping. Tunnelling at temperatures approaching the Debye temperature should be sensitive to the presence of short-wavelength phonons since these excitations will cause the shape and height of the barrier to tunnelling to fluctuate. Optimal configurations of atoms that define the barrier are therefore expected to enhance the average tunnelling rate. Experiments on muonium diffusion in solid Xe were carried out in an attempt to detect this effect. The results do not show any on-set of this effect as the temperature rises. However, it is possible that the diffusion mechanism in this system is dominated by a classical activation to an excited state, obscuring the temperature dependence of the tunnelling rate.