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We investigate transition rates between different spin configurations for $S\ge 1$ spins weakly coupled to a $d$-dimensional phonon bath. This study ismotivated by understanding observed magnetization relaxation as a function oftemperature in diverse magnetic systems such as arrays of magneticnanoparticles and magnetic molecules. We assume that the magnetization of thespin system relaxes through consecutive emission or absorption of a singlephonon. From a weak, linear spin-phonon coupling Hamiltonian, we derivetransition rates that would be used to examine dynamic properties of the systemin kinetic Monte Carlo simulations. Although the derived phonon-assistedtransition rates satisfy detailed balance, in the case of two and threedimensional phonon baths, transitions between degenerate states are notallowed. Thus, if there are no alternative paths along which the spin systemcan relax, the relaxation time diverges. Otherwise, the system finds otherpaths, which leads to an increase in the relaxation time and energy barrier.However, when higher-order phonon processes are included in the transitionrates, it is found that the system can reach the states which were inaccessibledue to the forbidden transitions. As a result, the system recovers some of thedynamic properties obtained using the Glauber transition rate.

Transition rates for a S \ge 1 model coupled to a phonon bath