Superparamagnetic dynamics and blocking transition in Fe$_3$O$_4$ nanoparticles probed by vibrating sample magnetometry and muon spin relaxation.

The magnetic properties of Fe$_3$O$_4$ nanoparticle assemblies have been investigated in detail through a combination of vibrating sample magnetometry and muon spin relaxation ($\mu$SR) techniques. Two samples with average particle sizes of 5 nm and 20 nm, respectively, were studied. For both samples, the magnetometry and $\mu$SR results exhibit clear signatures of the superparmagnetic state at high temperature and the magnetically blocked state at low temperature. The $\mu$SR data demonstrate that the transition from the superparamagnetic to the blocked state occurs gradually throughout the sample volume over a broad temperature range due to the finite particle size distribution of each sample. The transition occurs between approximately 3 K and 45 K for the 5 nm sample and 150 K and 300 K for the 20 nm sample. The magnetometry and $\mu$SR data are further analyzed to yield estimates of microscopic magnetic parameters including the nanoparticle spin-flip activation energy $E_A$, magnetic anisotropy $K$, and intrinsic nanoparticle spin reversal attempt time $\tau_0$. These results highlight the complementary information about magnetic nanoparticles that can be obtained by bulk magnetic probes such as magnetometry and local magnetic probes such as $\mu$SR.