Optimized Analysis of the AC Magnetic Susceptibility Data in Several Spin-Glass Systems using the Vogel-Fulcher and Power Laws

In spin-glasses (SG), the relaxation time $\tau$ ($= 1/2{\pi}f$) vs. $T_f$ data at the peak position $T_f$ in the temperature variation of the ac magnetic susceptibilities at different frequencies f is often fit to the Vogel-Fulcher Law (VFL): $\tau=\tau_0\exp[E_a/k_b(T_f-T_0)]$ and to the Power Law (PL): $\tau = \tau_0^*[(T_f-T_{SG}/T_{SG}]^{-z\nu}$. Both these laws have three fitting parameters each, leaving a degree of uncertainty since the magnitudes of the evaluated parameters $\tau_0$, $E_a/k_B$, $\tau_{0^*}$ and $z\nu$ depend strongly on the choice of $T_0$ and $T_{SG}$. Here we report an optimized procedure for the analysis of $\tau$ vs. $T_f$ data on several SG systems for which we could extract such data from published sources. In this optimized method, the data of $\tau$ vs. $T_f$ are fit by varying $T_0$ in the linear plots of $\ln \tau$ vs $1/ (T_f - T_0)$ for the VFL and by varying $T_{SG}$ in the linear plot of $\ln \tau$ vs. $\ln (T_f - T_{SG})/ T_{SG}$ for the PL till optimum fits are obtained. The analysis of the associated magnitudes of $\tau_0$, $E_a/k_B$, $\tau_{0^*}$ and $z\nu$ for these optimum values of $T_0$ and $T_{SG}$ shows that magnitudes of $\tau_{0^*}$, $\tau_0$ and $z\nu$ fail to provide a clear distinction between canonical and cluster SG. However, new results emerge showing $E_a/(k_BT_0) < 1$ in canonical SG whereas $E_a/(k_BT_0) >1$ for cluster SG systems and the optimized $T_0 <$ optimized $T_{SG}$ in all cases. Although some interpretation of these new results is presented, a more rigorous theoretical justification of the boundary near $E_a/(k_BT_0) \sim 1$ is desired along with testing of these criteria in other SG systems.