Extended Dynamical Kubo-Toyabe Relaxation for $\mu$SR study of Ion Dynamics: An Introduction

In the analysis of the ion diffusion in metal oxides based on muon spin rotation and relaxation ($\mu$SR), the dynamical Kubo-Toyabe (dKT) function has been routinely used to deduce the jump frequency of ions. This is based on the two beliefs: (1) the fluctuations of the internal magnetic field ${\bm H}(t)$ are determined solely by the relative motion of the muons to the surrounding ions, and (2) the muons are immobile due to bonding to the oxygen. However, these are not necessarily trivial, and we addressed their credibility by developing an extended dKT function corresponding to the realistic situation that only a part of the ions surrounding muon are involved in a single fluctuation of ${\bm H}(t)$ in the ion diffusion, and investigated its behavior in detail. The results show that the new function exhibits qualitatively different behavior from the dKT function, and that it provides a way to determine whether muons or ions are in motion, as well as a means for quantitative analysis based on the assumption of immobile muons. As a typical example, we examine the earlier $\mu^\pm$SR results on Na$_x$CoO$_2$ and demonstrate that the internal field fluctuations observed in $\mu^+$SR are dominated by muon self-diffusion, in contrast to previous interpretations.