Low-temperature quantum diffusion of light particles: The domains of the various mechanisms in the static-energy-asymmetry-temperature plane

Abstract

The motion of a light particle in a solid coupled to conduction electrons and/or phonons is investigated within the framework of a two-state model, which may be taken as representing the particle ground states in two neighbouring potential wells. With regard to the coupling of the particle to phonons, an important distinction arises between (i) special two-phonon processes, termed diphonon processes, which result from nonlinear particle-lattice coupling and which may give rise to quasielastic phase-destroying scattering and (ii) essentially inelastic processes which may arise from linear as well as from nonlinear particle-lattice coupling.

An important parameter affecting particle motion is the static energy shift e between the particle ground states in the two wells. Under favourable conditions it may be experimentally controlled within certain limits. The two-state model allows us to estimate reliably the boundary in the ϵ-T plane separating the regimes of incoherent hopping from that of coherent bandlike motion, and to discuss its dependence on the particle mass and the coupling parameters. Within the regime of incoherent motion a number of subregimes may exist. The situation becomes particularly interesting if a condensation of the electron system takes place, as in the case of a Bardeen-Copper-Schriefer superconductor. Then domains have to be distinguished in which the hopping rates are dominated by either quasiparticle coupling, diphonon processes, one-phonon processes, Cooper-pair breaking, coupling to normal-conducting electrons, or multiphonon-assisted processes.

In the series of light particles carrying one positive elementary charge, which includes the positive muon (μ⁺) and the nuclei of the hydrogen isotopes, the range of validity of the two-state description extends to rather high temperatures for the μ⁺ but only to considerably lower temperatures for the heavier particles. Nevertheless, for protons the limiting temperature can be almost as high as the Debye temperature. Towards low temperatures the validity of the two-well description of unrestricted particle motion in a crystal is limited by the onset of coherent motion.