Quantum diffusion of muons and muonium atoms in solids Vyacheslav G. Storchak Rutherford Appleton Laboratory, Chilton, Oxfordshire OX11 OQX, United Kingdom; TRIUMF, 4004 Westbrook Mall, Vancouver, British Columbia, Canada V6T 2A3; and Kurchatov Institute, Kurchatov Sq. 1, Moscow 123182, Russia Nikolai V. Prokof’ev Kurchatov Institute, Kurchatov Sq. 1, Moscow 123182, Russia The diffusion of muons and muonium through solids has been studied over many years using the technique of spin relaxation. At low temperatures, the motion is due to tunneling between lattice sites, and the competition between tunneling rates and decoherence rates is important in determining the dynamics. Coherent propagation is seen in superconductors and insulators at low temperature where dissipation is small. At higher temperatures the motion undergoes a crossover from bandlike propagation to incoherent hopping between neighboring sites. This review covers both theory and experiment, emphasizing the mechanisms for dissipation, the role of barrier fluctuations, and effects of crystal disorder on the transport. The review of experimental data includes an analysis of barrier penetration bandwidths for muon and muonium diffusion in a variety of metals and insulators. [S0034-6861(98)00503-0] CONTENTS I. Introduction 929 II. General Remarks on Quantum Diffusion 930 III. Some Theoretical Aspects of Quantum Diffusion 931 A. Hamiltonian and coherent band motion in a perfect crystal 931 B. Coherent (dissipationless) motion in a disordered crystal 933 C. Coupling to the environment. Interaction Hamiltonians 935 D. Incoherent tunneling 936 E. Kinetic equation for spin depolarization in the hopping regime 938 1. Long-range trapping 939 2. Inhomogeneous T 2 relaxation 939 3. Inhomogeneous T 1 relaxation 940 IV. Experimental Techniques 940 A. Transverse-, zero-, and longitudinal-field muon spin relaxation 940 B. Transverse-, zero-, and longitudinal-field muonium spin relaxation 942 V. Quantum Diffusion of  + in Metals 945 A. Transition probabilities and diffusion rates in metals and superconductors 945 B. Experimental results on  + quantum diffusion in metals 946 1. Copper 947 2. Aluminum 950 3. Other metals (V, Nb, Bi) 953 a. Vanadium 953 b. Niobium 954 c. Bismuth 956 VI. Quantum Diffusion of Muonium Atoms in Insulators 957 A. Transition probabilities, trapping, and diffusion rates in insulators 958 B. Experimental results on muonium quantum diffusion in ionic insulators, compound semiconductors, and cryocrystals 958 1. Two-phonon muonium diffusion 958 a. Ionic insulators 959 b. Compound semiconductors 960 c. Solid nitrogen 962 2. Muonium band tunneling 963 a. Ionic insulators, GaAs and CuCl 963 b. Solid neon and solid nitrogen 965 3. One-phonon muonium quantum diffusion 966 a. Ionic insulators and GaAs 966 b. Solid nitrogen 967 c. Solid xenon and solid krypton 967 4. Muonium quantum diffusion in imperfect crystals 970 a. Inhomogeneous quantum diffusion in solid nitrogen 970 b. Inhomogeneous Mu diffusion in Na-doped KCl 972 c. Trapping phenomena in insulators 973 VII. Conclusions 973 Acknowledgments 974 References 974 I. INTRODUCTION The diffusion of muons (  + ) and muonium (the bound state of  + +e - , called Mu) in solids is domi- nated by quantum tunneling between lattice sites at low temperature. This is a very interesting subject from a theoretical point of view, with the many degrees of free- dom of the solid environment playing an essential role in the coherence and barrier penetration rates. Muons are uniquely suited for particle transport tunneling studies because of their intermediate mass, 200 times greater than the electron but several orders of magnitude less than the lattice nuclei. The atoms are too heavy to have perceptible tunneling except in rare cases, and electrons are so light they they are usually delocalized in the lat- tice. The dynamics of barrier penetration in a periodic lat- tice is called quantum diffusion. There are several im- 929 Reviews of Modern Physics, Vol. 70, No. 3, July 1998 0034-6861/98/70(3)/929(50)/$25.00 © 1998 The American Physical Society