Revisiting $\mu$SR Studies of Ion Dynamics in the Light of Extended Kubo-Toyabe Model

The dynamical Kubo-Toyabe (dKT) function is extended to describe the spin relaxation under the coexisting dynamical and static internal magnetic fields. A detailed re-evaluation of the previous ${\mu }^{\pm }$SR data in Na${}_x$CoO${}_2$ using this function disfavors the conventional interpretation based on sodium-ion diffusion and instead supports the ${\mu }^{+}$ self-diffusion scenario. This also resolves the long-standing inconsistencies in the dKT-function-based $\mu$SR studies on ion diffusion from the viewpoint of classical over-barrier-jump mechanism.