A macroscopic equation of motion for the magnetization of a ferromagnet at elevated temperatures should contain both transverse and longitudinal relaxation terms and interpolate between LandauLifshitz equation at low temperatures and the Bloch equation at high temperatures. It is shown that for the classical model where spin-bath interactions are described by stochastic Langevin fields and spin-spin interactions are treated within the mean-field approximation (MFA), such a "LandauLifshitz-Bloch" (LLB) equation can be derived exactly from the Fokker-Planck equation, if the external conditions change slowly enough. For weakly anisotropic ferromagnets within the MFA the LLB equation can be written in a macroscopic form based on the free-energy functional interpolating between the Landau free energy near TC and the "micromagnetic" free energy, which neglects changes of the magnetization magnitude |M|, at low temperatures. [S0163-1829(97)

The dynamical theory of thermally activated resonant magnetization tunneling in uniaxially anisotropic magnetic molecules such as Mn 12 Ac (Sϭ10) is developed. The observed slow dynamics of the system is described by master equations for the populations of spin levels. The latter are obtained by the adiabatic elimination of fast degrees of freedom from the density matrix equation with the help of the perturbation theory developed earlier for tunneling level splitting ͓D. A. Garanin, J. Phys. A 24, L61 ͑1991͔͒. There exists a temperature range ͑thermally activated tunneling͒ where the escape rate follows the Arrhenius law, but has a nonmonotonic dependence on the bias field due to tunneling at the top of the barrier. At lower temperatures this regime crosses over to the non-Arrhenius law ͑thermally assisted tunneling͒. The transition between the two regimes can be first or second order, depending on the transverse field, which can be tested in experiments. In both regimes the resonant maxima of the rate occur when spin levels in the two potential wells match at certain field values. In the thermally activated regime at low dissipation each resonance has a multitower self-similar structure with progressively narrowing peaks mounting on top of each other.

We calculate the contribution of the Néel surface anisotropy to the effective anisotropy of magnetic nanoparticles of spherical shape cut out of a simple cubic lattice. The effective anisotropy arises because deviations of atomic magnetizations from collinearity and thus the energy depends on the orientation of the global magnetization. The result is second order in the Néel surface anisotropy, scales with the particle's volume and has cubic symmetry with preferred directions [±1, ±1, ±1].PACS numbers: 61.46.+w, 75.70.Rf With the decreasing size of magnetic particles, surface effects are believed to become more and more pronounced. A simple argument based on the estimation of the fraction of surface atoms shows that for a particle of spherical shape and diameter D (in units of the lattice spacing), this fraction is an appreciable number of order 6/D. Regarding the fundamental property of magnetic particles, the magnetic anisotropy, the role of surface atoms is augmented by the fact that these atoms in many cases experience surface anisotropy (SA) that by far exceeds the bulk anisotropy. As was suggested by Néel [1] and microscopically shown in Ref.[2], the leading contribution to the anisotropy is due to pairs of atoms and can be written aswhere m i is the reduced magnetization (spin polarization) of the ith atom, e ij are unit vectors directed from the ith atom to its neighbors, and L ij is the pairanisotropy coupling that depends on the distance between atoms. Eq. (1) describes in a unique form both the bulk anisotropy including the effect of elastic strains and the effect of missing neighbors at the surface that leads to the SA. In particular, for an unstrained simple cubic (sc) lattice the bulk anisotropy in Eq. (1) The 1/D surface contribution to K V,eff is in accord with the picture of all magnetic atoms tightly bound by the exchange interaction whereas only the surface atoms feel the surface anisotropy. This is definitely true for magnetic films where a huge surface contribution to the effective anisotropy has been observed. The same is the case for cobalt nanoclusters of the form of truncated octahedrons [5] where contributions from different faces, edges, and apexes compete resulting in a nonzero, although significantly reduced, surface contribution to K V,eff . However, for symmetric particle shapes such as cubes or spheres, the symmetry leads to vanishing of this (first-order) contribution. In this case one has to take into account deviations from the collinearity of atomic spins that result from the competition of the SA and the exchange interaction J. The resulting structures (for the simplified radial SA model) can be found in Refs. [6,7,8] (see also Fig. 1 for the NSA). In the case L > ∼ J deviations from collinearity are very strong, and it is difficult if not impossible to characterize the particle by a global magnetization suitable for the definition of the effective anisotropy. On the other hand, in the typical case L ≪ J the magnetic structure is nearly collinear with small deviations tha...

We have found a novel feature of the bistable large-spin model described by the Hamiltonian H 2DS 2 z 2 H x S x . The crossover from thermal to quantum regime for the escape rate can be either first ͑H x , SD͞2͒ or second ͑SD͞2 , H x , 2SD͒ order, that is, sharp or smooth, depending on the strength of the transverse field. This prediction can be tested experimentally in molecular magnets like Mn 12 Ac. [S0031-9007(97)04645-0] PACS numbers: 75.45. + j, 75.50.Tt Transitions between two states in a bistable system can occur either due to the classical thermal activation or via quantum tunneling. A rigorous study of that problem was begun by Kramers [1] and WKB [2][3][4], and a review of the progress that followed can be found in Ref. [5]. At high temperature the transition rate follows the Arrhenius law, G ϳ exp͑2DU͞T͒, with DU being the height of the energy barrier between the two states. In the limit of T ! 0, the transitions are purely quantum, G ϳ exp͑2B͒, with B independent on temperature. Because of the exponential dependence of the thermal rate on T , the temperature T 0 of the crossover from quantum to thermal regime can be estimated as T ͑0͒ 0 DU͞B. For a quasiclassical particle in a potential U͑x͒, Goldanskii [6] noticed the possibility of a more accurate definition, T ͑2͒ 0 "͞t 0 , where t 0 is the period of small oscillations near the bottom of the inverted potential, 2U͑x͒. Below T ͑2͒ 0 , thermally assisted tunneling occurs from the excited levels, that reduces to the tunneling from the ground-state level at T 0. Above T ͑2͒ 0 quantum effects are small and the transitions occur due to the thermal activation to the top of the barrier. Affleck [7] demonstrated that the two regimes smoothly join at T T ͑2͒ 0 . Larkin and Ovchinnikov [8] called this situation the secondorder phase transition from classical to quantum behavior. This means that for G written as G ϳ exp͑2DU͞T eff ͒, the dependence of both T eff and its first derivative on T are continuous at T T ͑2͒ 0 . This situation is not generic, however. The transition between the two regimes can also be of the first order [8,9], i.e., more abrupt, with dT eff ͞dT discontinuous at a certain temperature T ͑1͒ 0 . T ͑2͒ 0 . Chudnovsky derived the criterion allowing one to establish whether first-or second-order transition takes place, based on the shape of the potential U͑x͒. Commonly studied potentials U 2x 2 1 x 4 and U 2x 2 1 x 3 yield the second-order transition. Physically relevant potentials which would exhibit the first-order transitions were not known. In this Letter we show that spin systems readily accessible in the experiment possess both first-and second-order transitions between the classical and quantum behavior of the escape rate. The order of the transition in these systems can be controlled by external magnetic field. Consider a spin system described by the Hamiltonianwhere S ¿ 1. This model is generic for problems of spin tunneling studied by different methods [10 -15]. It is believed to be a good approximation for the molecular magnet...

In conventional micromagnetism magnetic domain configurations are calculated based on a continuum theory for the magnetization which is assumed to be of constant length in time and space. Dynamics is usually described with the Landau-Lifshitz-Gilbert (LLG) equation the stochastic variant of which includes finite temperatures. Using simulation techniques with atomistic resolution we show that this conventional micromagnetic approach fails for higher temperatures since we find two effects which cannot be described in terms of the LLG equation: i) an enhanced damping when approaching the Curie temperature and, ii) a magnetization magnitude that is not constant in time. We show, however, that both of these effects are naturally described by the Landau-Lifshitz-Bloch equation which links the LLG equation with the theory of critical phenomena and turns out to be a more realistic equation for magnetization dynamics at elevated temperatures. who demonstrated complete demagnetization on a timescale of picoseconds. One of the main issues of the high-temperature magnetization dynamics is the rate of the magnetization relaxation due to different processes involving magnon, phonon and electron interactions that contribute to thermal spin disordering.The basis of most of theoretical investigations of thermal magnetization dynamics is a micromagnetic approach which considers the magnetization of a small particle or a discrete magnetic nanoelement as a vector of a fixed length (referred to here as a macro-spin) with the phenomenological Landau-Lifshitz-Gilbert (LLG) equation of motion augmented by a noise term [3]. However, contrary to atomic spins, there is no reason to assume a fixed magnetization length for nanoelements at nonzero temperature. For instance, the latter can decrease in time upon heating by a laser pulse. Hence, from the point of view of modeling of magnetization dynamics, there is a general need for further development of the micromagnetic theory in terms of its ability to deal with elevated temperatures.Within this context we note the failure of micromagnetics in general to deal with the high frequency spinwaves which give rise to the variation of magnetization with temperature. It has been suggested to treat this problem using scaling approaches [4,5]. A similar problem arises in multi-scale modeling (with atomistic and micromagnetic discretizations to treat, for example, interfaces [6]) which can not correctly describe the transfer of high energy spin-waves from atomistic into the micromagnetic region. An alternative approach is the coarse graining model of Dobrovitksi et. al. [7], which has the advantage of being able to link the length-scales but has been developed for simple systems only.Some understanding of the pulsed laser experiments could indeed be obtained in terms of a micromagnetic approach taking into account, in an empirical way, the temperature variation of the intrinsic parameters, particularly the saturation magnetization M s and the anisotropy energy density K. Lyberatos and Guslienko [8] h...

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R. Evans and R. W. ChantrellDepartment of Physics, University of York, Heslington, York YO10 5DD, UKMagnetic nanoparticles with Néel surface anisotropy, different internal structures, surface arrangements and elongation are modelled as many-spin systems. The results suggest that the energy of many-spin nanoparticles cut from cubic lattices can be represented by an effective one-spin potential containing uniaxial and cubic anisotropies. It is shown that the values and signs of the corresponding constants depend strongly on the particle's surface arrangement, internal structure and elongation. Particles cut from a simple cubic lattice have the opposite sign of the effective cubic term, as compared to particles cut from the face-centered cubic lattice. Furthermore, other remarkable phenomena are observed in nanoparticles with relatively strong surface effects: (i) In elongated particles surface effects can change the sign of the uniaxial anisotropy. (ii) The competition between the core and surface anisotropies leads to a new energy that contributes to both the 2 nd − and 4 th −order effective anisotropies. We also evaluate energy barriers ∆E as functions of the strength of the surface anisotropy and the particle size. The results are analyzed with the help of the effective one-spin potential, which allows us to assess the consistency of the widely used formula ∆E/V = K∞ + 6Ks/D, where K∞ is the core anisotropy constant, Ks is a phenomenological constant related to surface anisotropy, and D is the particle's diameter. We show that the energy barriers are consistent with this formula only for elongated particles for which the surface contribution to the effective uniaxial anisotropy scales with the surface and is linear in the constant of the Néel surface anisotropy.

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