Stretched exponential relaxation in three dimensional short-range spin glass Cu$_{0.5}$Co$_{0.5}$Cl$_{2}$-FeCl$_{3}$ graphite bi-intercalation compound

Cu$_{0.5}$Co$_{0.5}$Cl$_{2}$-FeCl$_{3}$ graphite bi-intercalation compound is a three-dimensional short-range spin glass with a spin freezing temperature $T_{SG}$ ($= 3.92 \pm 0.11$ K). The time evolution of the zero-field cooled magnetization $M_{ZFC}(t)$ has been measured under various combinations of wait time ($t_{w}$), temperature ($T$), temperature-shift ($\Delta T$), and magnetic field ($H$). The relaxation rate $S_{ZFC}(t)$ [$=(1/H)$d$M_{ZFC}(t)$/d$\ln t$] shows a peak at a peak time $t_{cr}$. The shape of $S_{ZFC}(t)$ in the vicinity of $t_{cr}$ is well described by stretched exponential relaxation (SER). The SER exponent $b$ and the SER relaxation time $\tau_{SER}$ are determined as a function of $t_{w}$, $T$, $H$, and $\Delta T$. The value of $b$ at $T=T_{SG}$ is nearly equal to 0.3. There is a correlation between $\tau_{SER}$ and $1/b$, irrespective of the values of $t_{w}$, $T$, $H$, and $\Delta T$. These features can be well explained in terms of a simple relaxation model for glassy dynamics.