Stochastic theory of multistate diffusion in perfect and defective systems. II. Case studies

The time development of a number of physical systems can be described in terms of the temporal passage of the system through its allowable states. These states may correspond to spatial configurations, energy levels, competing transition mechanisms, or correlated processes. The evolution of certain of these systems can be analyzed by mapping onto a continuous-time random walk on a lattice whose unit cell may contain several internal states. In addition, we study the influence of periodically placed defects, which modify transition rates, on transport properties in several condensed-matter systems. A propagator formalism is reviewed where the matrix propagator (whose dimensions equal the number of internal states in a unit cell) is renormalized owing to the presence of defects. Knowledge of the propagator allows the evaluation of observable quantities such as positional moments, diffusion coefficients, occupation probabilities of states, and line shapes of scattering from diffusing particles. The formalism allows the study of diverse transport systems in a unified manner. We study the multistate diffusion of particles and clusters (specifically dimers) observed via field-ion microscopy on defective surfaces and derive expressions for observables pertinent to the analysis of these experiments. The lattice structure dependence of particle diffusion in systems containing defects which influence their neighbors is demonstrated. Next, the anomalous, non-Arrhenius behavior of the vacancy diffusion constant, observed via tracer-diffusion measurements, is described in terms of competing monovacancy mechanisms: nearest-neighbor and next-nearest-neighbor single jumps, and double jumps in which two atoms jump almost simultaneously, collinearly, as suggested by recent molecular-dynamics studies. We derive an analytic expression for the vacancy diffusion which provides a reinterpretation of fits to experimental data. As a further application of our method we investigate non-Markovian transport due to relaxation effects. Finally, we provide an analytic expression for the scattering law, $S(\overrightarrow{\mathrm{k}},\omega )$, of quasielastic neutron scattering and demonstrate the non-Lorentzian line shape of the scattered intensity for systems containing defects and for correlated motion.