Longitudinal Field $\mu $SR Study of Spin Dynamics and Onset of Magnetic Correlations in LiY$_{1-x}$Ho$_{x}$F$_{4}$ with 0.002$\le \quad x\le $ 0.10

The Ho$^{3+}$ ions in LiY$_{1-x}$Ho$_{x}$F$_{4}$ exhibit a crossover from single ion to spin glass behavior with increasing $x$. We have studied the longitudinal field depolarization rate for samples with 0.002 $\le \quad x \quad \le $ 0.10 over the temperature range 50 mK $\le \quad x \quad \le $ 50 K and for magnetic fields up to 0.1 T. For low concentrations, we find a peak in the temperature-dependent depolarization, as often observed in 1/$T_{1}$ NMR measurements on single molecule magnets (SMM); at high concentrations the depolarization rate increases monotonically with decreasing temperature. These results suggest that the difference in behavior of SMMs systems as seen in NMR and $\mu $SR measurements may be due to differences in the strength of the interactions between the magnetic clusters.