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

The escape rate Γ of the large-spin model described by the Hamiltonian H=-DSz²-HzSz-HₓSₓ is investigated with the help of the mapping onto a particle moving in a double-well potential $U(x)$. The transition-state method yields Γ 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 (dΓ/dT discontinuous at the transition temperature T₀) for hₓ below the phase boundary line hₓ=hxc(hz), where hx,z≡Hx,z/(2SD), and of the second order above this line. In the unbiased case (Hz=0) the result is hxc(0)=1/4, i.e., one fourth of the metastability boundary hₓₘ=1, at which the barrier disappears. In the strongly biased limit δ≡1-hz1, one has hxc(2/3)3/4(√3-√2)δ3/20.2345δ3/2, which is about one half of the boundary value hₓₘ(2δ/3)3/20.5443δ3/2. The latter case is relevant for experiments on small magnetic particles, where the barrier should be lowered to achieve measurable quantum escape rates.