7.2 Quantum Diffusion in Insulators

Diffusion of interstitial muonium in insulators was studied by making use of the effect of the motion of the muonium atom on the spin polarization relaxation functions in both transverse and longitudinal magnetic fields.

Quantum diffusion of neutral interstitials in insulators is dominated by the interaction of the particle with excitations of the lattice. These excitations are manifested in the displacements of the lattice atoms from their equilibrium positions, which results in the loss of translational symmetry (for the bare interstitial), reducing the tunnelling rate. For muons in metals this interaction is neglible compared to the Coulomb interaction with conduction electrons. Quantum diffusion of neutral interstitials in insulators gives us a means of studying this interaction.

Scattering of phonons results in a reduction of the effective tunnelling bandwidth. At temperatures where the incoherent channel has not been completely quenched the key characteristic of 2-phonon quantum diffusion (in an otherwise perfect lattice) is an increasing hop rate as the temperature decreases. Below about $T=\Theta_{\rm D}/10$ theory predicts that the hop rate will follow a T^-7 power law. At a higher temperature this strong temperature dependence drops off as the phonon spectrum becomes fully excited. Structure in the phonon density of states is not very important to the overall temperature dependence of the hop rate.

At sufficiently low temperatures the site-to-site static level shifts that are always present in a real crystal will inevitably become important. The small difference between energy levels must be made up by differences in phonon energies, and in this situation the 2-phonon hop rate increases with temperature as T⁷, below about $T=\Theta_{\rm D}/10$.This interplay between the phase damping rate $\Omega_2$ and typical static shifts $\xi$ was demonstrated by muonium diffusion in solid nitrogen. Between about 4 K and 20 K the muonium hop rate follows a T^6.7(1) power law, slightly weaker than the predicted T⁷ law for the case in which small static shifts are present, well below the Debye temperature. At higher temperatures, between 32 k and 46 K the hop rate was found to decrease with rising temperture, qualitatively consistent with theory for the case that $\Omega_2$ completely dominates the static shifts. However, the measured temperature dependence was found to be much stronger than theory predicts at such a large fraction of the Debye temperature, assuming that it is phonons that are being scattered.

It is thought that at higher temperatures activated tunnelling diffusion of polarons should be sensitive to the fluctuations in the height and width of the potential barrier separating interstitial sites. The most favourable conditions for detecting this would therefore be at temperatures where short-wavelength phonons are excited. These phonons should cause the greatest relative motion of adjacent atoms which define the shape of the barrier.

Muonium diffusion in solid xenon was studied in an effort to detect thia effect, however the data did not reveal a temperature dependence in the apparent activation energy. Furthermore, the results are consistent with diffusion from an excited state, so that the temperature dependence we measure reflects the thermal population of this excited state. A simple calculation based on Xe-Xe and Xe-H pair potentials shows that the separation of levels in the Xe cage, deformed to allow room for the muonium interstitial at the octahedral site, would be about the same as the measured activation energy.

It would appear that current theory is adequate for modelling phonon-mediated quantum diffusion at temperatures where solids are Debye-like and only the long-wavelength phonons are excited. Exactly which excitations are important at higher tenperatures has not yet been resolved. Perhaps the next stage in experimental work is to measure directly the contribution of various modes, not by measuring temperature dependences, but by measuring the diffusion rate while exciting only a narrow part of the excitation spectrum.