Escape rate of a biaxial nanospin system in a magnetic field: first- and second-order transitions between quantum and classical regimes

We investigate the escape rate of the biaxial nanospin particle with a magnetic field applied along the easy axis. The model studied here is described by the Hamiltonian H = −AS z − BS x − HSz, (A > B > 0). By reducing this Hamiltonian to a particle one, we derive, for the first time, an effective particle potential for this model and find an analytical form of the phase boundary line between firstand second-order transitions, from which a complete phase diagram can be obtained. We also derive an analytical form of the crossover temperature as a function of the applied field at the phase boundary. PACS number : 75.45.+j, 75.50.Tt Typeset using REVTEX 1 Recently the quantum-classical phase transition of the escape rate [1] in nanospin system has been studied intensively. One of the main issues in this subject is to determine whether the transition is first-order or second-order. In this regards, for the uniaxial spin system such as high-spin molecular magnet Mn12Ac [2], two models have been investigated comprehensively : one with a transverse field [3] and the other with an arbitrarily directed field [4], described by the Hamiltonians H = −DS z −HxSx and H = −DS z −HxSx −HzSz respectively. In the first case, by using the method of particle mapping and the Landau theory of phase transition, Chudnovsky and Garanin have shown that the transition order changes from first to second when the field parameter hx ≡ Hx/(2SD) is 0.25. In the case of model with arbitrarily directed field Garanin et al. have obtained the phase boundary line hxc = hx(hz) to show the whole phase diagram in which hxc(0) = 0.25 in the unbiased case and hxc ∼ (1− hz) in the strongly-biased limit. For biaxial spin system such as iron cluster Fe8 [5] Liang et al. considered a model without an applied field, H = K(S z + λS y), (0 < λ < 1) [6]. Using the coherent spin state representation they have shown that the coordinate dependent effective mass leads to the first-order transition and the change between the firstand second-order transitions occurs at the value λ = 0.5 . The biaxial spin model with transverse field, H = K(S z+λS2 y)−HySy, has also been investigated by Lee et al. who demonstrated that various types of combinations of the firstand second-order transitions are possible depending on λ and Hy/KλS [7]. In this paper we study the phase transition of the escape rate of the biaxial spin system with a longitudinal field. We will first derive an effective particle potential by mapping the spin Hamiltonian onto particle one, which is the first derivation for the present model. Then, with the help of the recently developed method for the criterion of the transition order [8,9], we find an analytical form of the phase boundary line from which a complete phase diagram for the order of the phase transition is obtained. Consider a nanospin particle with an applied field H along the easy axis. If the spin particle is a biaxial spin system with XOZ easy plane anisotropy and the easy Z-axis in the XZ-plane the Hamiltonian can be described by 2 H = −AS z −BS x −HSz (1) where the anisotropy constants satisfy A > B > 0. Our model is equivalent to H = K(S z + λS 2 y) − HSx, (λ < 1) if we set A = K,B = (1 − λ)K. In the following, for convenience, we introduce dimensionless anisotropy parameter b ≡ B/A and field parameter α ≡ H/SA, (0 < α < 2) where S is the spin number. For iron cluster Fe8 in Ref.5, S = 10, A = 0.316 K, and B = 0.092 K. We can reduce this spin problem to a particle moving in a potential [10]. The equivalent Schrödinger-like equation is derived as − 1 2m dΨ dx + V (x)Ψ = EΨ, (2)