An empirical model for deriving information on ion dynamics from $\mu$SR data via the conventional Kubo-Toyabe function analysis

We present an empirical model that analytically relates the parameters of the extended Kubo-Toyabe (KT) function $G_z^{\mathrm{E}\mathrm{A}}(t;\mathrm{\Delta },{\nu }_{\mathrm{i}},Q)$, which describes muon spin relaxation ($\mu$SR) induced by jump diffusion of ions using the Edwards-Anderson parameter $Q$, to those of the conventional KT function $G_z^{\mathrm{K}\mathrm{T}}(t;{\mathrm{\Delta }}_{\mathrm{K}\mathrm{T}},{\nu }_{\mathrm{K}\mathrm{T}})$ that serves as an approximation of $G_z^{\mathrm{E}\mathrm{A}}(t)$. The proposed formulae successfully reproduce the anomalous temperature dependences of the linewidth ${\mathrm{\Delta }}_{\mathrm{K}\mathrm{T}}(T)$ and the fluctuation rate ${\nu }_{\mathrm{K}\mathrm{T}}(T)$ which are artifacts observed upon analyzing $\mu$SR data on ion diffusion using $G_z^{\mathrm{K}\mathrm{T}}(t)$. Applying this model to ${\mathrm{\Delta }}_{\mathrm{K}\mathrm{T}}$ and ${\nu }_{\mathrm{K}\mathrm{T}}$ published in the literature of battery materials yields activation energy $E_{\mathrm{i}}$ and pre-exponential factor ${\nu }_{\mathrm{i}0}$ in the Arrhenius equation for ion diffusion, ${\nu }_{\mathrm{i}}(T)={\nu }_{\mathrm{i}0}{e}^{-E_{\mathrm{i}}\mathrm{/}kT}$, that are in excellent agreement with those deduced using other techniques, transforming peculiar artifacts into a powerful diagnostic tool.