Stochastic model of muon diffusion in the presence of traps: Nonsecular effects on spin depolarization

A stochastic model is presented for calculating the depolarization of the positive muon when the latter undergoes jump diffusion in a solid in the presence of trapping impurities. The theory, restricted to low concentration of traps, is applicable to relaxation studies in both transverse and zero magnetic fields. In the transverse geometry, the dipolar interaction between the μ⁺ and the surrounding nuclear spins can be treated classically, and the analysis is simpler. The results obtained are shown to be identical to those derived before in a ‘two-state’ model, widely used recently in interpreting various experimental studies on trap-limited diffusion of light interstitials (μ⁺,H, etc.) in solids. The zero field technique is more versatile but is also more complicated to analyze as it involves nonsecular terms in the dipolar interaction. Assuming that the local dipolar fields are isotropic with their magnitudes distributed in a Gaussian manner, tractable results can be obtained for the Laplace transform of the zero field relaxation function. The latter needs to be inverted in order to facilitate comparison with experimental data, which are normally recorded in the time space. A scheme to handle this numerical problem is described and results are presented using realistic parameters.