Longitudinal Field $\mu $SR Study of Spin Dynamics and Onset of Magnetic Correlations in LiHo$_{x}$Y$_{1-x}$F$_{4}$

Nanomagnets - non-interacting spins typically imbedded in a solid host material - are intensely studied for applications ranging from high density data storage to quantum computing. We are studying the magnetic ion Ho$^{3+}$ spin dynamics in LiHo$_{x}$Y$_{1-x}$F$_{4}$ in the dilute limit (x $<<$ 1) using the muon spin relaxation in applied longitudinal magnetic fields as a function of temperature and magnetic field strength. At low concentrations (x $\le $ 0.01) and for all magnetic field values studied, the spin lattice relaxation rate, 1/T$_{1}$, shows a characteristic peak, usually associated with relaxation by spin fluctuations, near T = 10 K; this peak is consistent with the single-ion energy level diagram of Ho$^{3+}$. At larger x this peak is lost in low fields; stronger fields ($>$ 600 G) restore the peak but show qualitatively different low temperature behavior. This suggests that both magnetic disorder and Ho-Ho interactions play an increasingly important role at larger x values.