Quantum-classical transition of the escape rate of a uniaxial spin system in an arbitrarily directed field

The escape rate $\Gamma$ of the large-spin model described by the Hamiltonian $\mathcal{H}=-{\mathrm{DS}}_z^2-H_z{S}_z-H_x{S}_x$ is investigated with the help of the mapping onto a particle moving in a double-well potential $U\left(x\right)\mathrm{}$. The transition-state method yields $\Gamma$ in the moderate-damping case as a Boltzmann average of the quantum transition probabilities. We have shown that the transition from the classical to quantum regimes with lowering temperature is of the first order $(d\Gamma /dT$ discontinuous at the transition temperature $T_0)$ for $h_x$ below the phase boundary line $h_x{=h}_{\mathrm{xc}}{(h}_z)$, where $h_{x,z}\equiv H_{x,z}/\left(2SD\right)\mathrm{}$, and of the second order above this line. In the unbiased case ${(H}_z=0)$ the result is $h_{\mathrm{xc}}\mathrm{}\left(0\right)=1/4\mathrm{}$, i.e., one fourth of the metastability boundary $h_{\mathrm{xm}}=1\mathrm{}$, at which the barrier disappears. In the strongly biased limit $\delta \equiv 1-h_z\ll 1$, one has $h_{\mathrm{xc}}\cong {\mathrm{}(2/3)\mathrm{}}^{3/4}\left(\sqrt{3}-\sqrt{2}\right){\delta }^{3/2}\cong 0.2345{\delta }^{3/2}$, which is about one half of the boundary value $h_{\mathrm{xm}}\cong (2\mathrm{}\delta {/3)}^{3/2}\cong 0.5443{\delta }^{3/2}$. The latter case is relevant for experiments on small magnetic particles, where the barrier should be lowered to achieve measurable quantum escape rates.