Spin-lattice relaxation of magnetic centers in molecular crystals at low temperature

We study the spin-phonon relaxation rate of both Kramers and non-Kramers molecular magnets in strongly diluted samples at low temperature. Using the "rotational" contribution to the spin-phonon Hamiltonian, universal formulae for the relaxation rate are obtained. Intriguingly, these formulae are all entirely expressed via measurable or \emph{ab initio} computable physical quantities. Moreover, they are also independent of the energy gaps to excited states involved in the relaxation process. These obtained expressions for direct and Raman processes offer an easy way to determine the lowest limit of the spin-phonon relaxation of any spin system based on magnetic properties of the ground doublet only. In addition, some intriguing properties of Raman process are also found. Particularly, Raman process in Kramers system is found dependent on the magnetic field's orientation but independent of its magnitude, meanwhile the same process in non-Kramers system is significantly reduced out of resonance, i.e. for an applied external field. Interestingly, Raman process is demonstrated to vary as $T^{9}$ for both systems. Application of the theory to a recently investigated cobalt(II) complex shows that it can provide a reasonably good description for the relaxation. Based on these findings, a strategy in developing efficient single-molecule magnets by enhancing the mechanical rigidity of the molecular unit is proposed.