Thermal Fluctuations of a Single-Domain Particle

TLDR

Thermal agitation causes the direction of a uniformly magnetized fine ferromagnetic particle to fluctuate, a phenomenon central to superparamagnetism and magnetic aftereffect. The authors model the particle’s thermal dynamics with Gilbert’s equation augmented by a random field, deriving a Fokker‑Planck equation for orientation probabilities; under axial symmetry the problem reduces to a minimization, enabling approximate solutions and Kramers‑type barrier‑crossing analysis, with limits of the discrete‑orientation approximation discussed.

Abstract

A sufficiently fine ferromagnetic particle has a uniform vector magnetization whose magnitude is essentially constant, but whose direction fluctuates because of thermal agitation. The fluctuations are important in superparamagnetism and in magnetic aftereffect. The problem is approached here by methods familiar in the theory of stochastic processes. The "Langevin equation" of the problem is assumed to be Gilbert's equation of motion augmented by a "random-field" term. Consideration of a statistical ensemble of such particles leads to a "Fokker-Planck" partial differential equation, which describes the evolution of the probability density of orientations. The random-field concept, though convenient, can be avoided by use of the fluctuation-dissipation theorem. The Fokker-Planck equation, in general, is complicated by the presence of gyroscopic terms. These drop out in the case of axial symmetry: then the problem of finding nonequilibrium solutions can be restated as a minimization problem, susceptible to approximate treatments. The case of energy barriers large in comparison with $\mathrm{kT}$ is treated both by approximate minimization and by an adaptation of Kramers' treatment of the escape of particles over barriers. The limits of validity of the discrete-orientation approximation are discussed.