1063-7834/02/4412- $22.00 © 2002 MAIK “Nauka/Interperiodica” 2276 Physics of the Solid State, Vol. 44, No. 12, 2002, pp. 2276–2280. Translated from Fizika Tverdogo Tela, Vol. 44, No. 12, 2002, pp. 2174–2177. Original Russian Text Copyright © 2002 by Kalmykov, Titov. 1. Single-domain ferromagnetic particles are char- acterized by an anisotropic internal potential with sev- eral minima separated by barriers. If the particles are small (~10 nm in diameter), the barriers are relatively low. In this case, the magnetization vector M(t ) can change its orientation by surmounting the barrier with the aid of thermal fluctuations. Thermal instability of the magnetization gives rise to superparamagnetism [1], because each particle behaves as a paramagnetic atom with a magnetic moment of ~10 4 –10 5 Bohr mag- netons. Since the magnetic dipole moment of a particle is large, the Zeeman energy can be comparable to the thermal energy kT even in a moderate external field H 0 . Therefore, when analyzing the magnetization relax- ation in external ac fields, nonlinear effects should be taken into account [2–4]. Until recently, only the theory of the linear response of superparamagnetic particles in weak magnetic fields was relatively well developed. Due to its complexity, the theory of nonlinear response was significantly less developed; this partial develop- ment was achieved predominantly using perturbation methods (see, e.g., [5–8]). A certain advance in the study of nonlinear effects was made in [9–11]; in [9, 10], the kinetics of particles with uniaxial anisotropy in strong ac fields was considered, while in [11], the nonlin- ear response of such particles to a momentary change in an external strong static magnetic field was investigated. The objective of this paper is to extend the results obtained in [11] to the case of particles with cubic anisot- ropy, i.e., to investigate the magnetization kinetics in an external strong static magnetic field when this field is suddenly changed in both magnitude and direction. 2. Let the external static magnetic field applied to the system under study be suddenly changed from H I to H II at time t = 0. We consider the relaxation of the mag- netization vector M(t ) of a system of noninteracting superparamagnetic particles from equilibrium state I (in the field H I ), characterized by a distribution function W I (at t ≤ 0), to equilibrium state II (in the field H II ) with distribution function W II (at t ∞). The dynamics of the projection M r of the magnetization vector M(t ) onto an arbitrary direction specified by a unit vector r = (ν X , ν Y , ν Z ) is described by the normalized relaxation function (1) Angular brackets with indices I and II indicate the aver- age with respect to the equilibrium distribution func- tions W I and W II , respectively, and angular brackets without any index indicate the average over realizations of the random variable M r (t). This problem is essen- tially nonlinear, because the amplitude of the change in the external magnetic field is assumed to be arbitrary. The relaxation processes in a system of superpara- magnetic particles, as a rule, are considered within the diffusion approximation of Brown [2]. For our prob- lem, the nonlinear response of the system in the diffu- sion approximation can be found by solving the Fok- ker–Planck equation for the probability density func- tion W(M, t ) of the magnetization [2]: (2) ft () M r 〈 〉 t () M r 〈 〉 II – M r 〈 〉 I M r 〈 〉 II – ----------------------------------------. = 2 τ N ∂ ∂ t ---- W = ∆ W βα 1 – u ∇V II ∇W × ( ) ∇ W∇V II ( ) + [ ] , + t 0, > MAGNETISM AND FERROELECTRICITY Nonlinear Response of Superparamagnetic Particles with Cubic Anisotropy to a Sudden Change in the Applied Strong Static Magnetic Field Yu. P. Kalmykov* and S. V. Titov** * Centre d’Etudes Fondamentales, Université de Perpignan, Perpignan Cedex, 66860 France e-mail: kalmykov@univ-perp.fr ** Institute of Radio Engineering and Electronics, Russian Academy of Sciences, pl. Vvedenskogo 1, Fryazino, Moscow oblast, 141190 Russia e-mail: svt245@ire216.msk.su Received September 19, 2001; in final form, February 4, 2002 Abstract—The nonlinear response of superparamagnetic particles with cubic anisotropy to a sudden change in an applied strong static magnetic field is analyzed. The relaxation function spectrum and the relaxation time of the magnetization are calculated for typical values of the anisotropy and dissipation parameters. © 2002 MAIK “Nauka/Interperiodica”.