$μ$SR evidence of a marked exchange interaction effect on the local spin dynamics of Tb-based molecular nanomagnets

We report on the spin dynamics of two Terbium-based molecular nanomagnets, Tb-SQ and Tb-Trp, investigated by means of longitudinal muon spin relaxation ($\mu$SR) measurements as a function of applied field, flanked by AC susceptibility characterization. In the two molecules Tb(III) magnetic ion has an isotructural coordination sphere, but in the former the Tb(III) is coordinated by an organic paramagnetic ligand (SQ), while the latter is coordinated by a diamagnetic one (Trp). Thus Tb-SQ presents an exchange interaction between the Tb(III) ion and a radical while Tb-Trp does not. Both the samples exhibit a muon spin-lattice relaxation rate $\lambda_1(T, B_L)$ peak in the temperature range 10-25 K at all applied longitudinal magnetic fields $B_L = 50, 150, 300$ mT. In Tb-SQ, $\lambda_1(T, B_L)$ displays a BPP-like behavior led by three different correlations times: the first, dominating for $T\geq15K$, follows a thermally activated law $\tau_c =\tau_0 \exp(\sigma_A / k_B T)$ with energy barrier $\sigma_A/k_B$, while the second and third ones, dominating respectively for $8<15$ K and $T<8$ K, follow a power-law-like behavior $\tau_c = c_0 T^{-\alpha}$ with two different values of $c_0$ and $\alpha$. On the other hand, the temperature and field behavior of $\lambda_1(T, B_L)$ in Tb-Trp strongly deviates from a BPP law, displaying a strongly anomalous character. Our results indicate that, in the absence of an exchange interaction and maintaining all the other relevant interactions constants, the local spin dynamics of single ion magnets strongly differ from the one observed in the presence of such interaction. The combination of $\mu$SR and AC susceptibility allows us to disentangle the different Orbach, Raman and direct mechanisms which are the key ingredients that control the spin dynamics in Tb-SQ, and evidence the potentiality of $\mu$SR in elucidating complex spin dynamics.