Thermal fluctuations of a single-domain particle

Abstract

A statistical ensemble of particles, with moment orientations (, ), can be represented by a surface density W (, , t) of points on the unit sphere. The corresponding surface density J satisfies a continuity equation ∂W∂t=-∇·J. With no thermal agitation, J=WṀM s, where M is the vector magnetization ( M = const = Ms); its rate of change Ṁ is assumed to be given by Gilbert's equation. To include thermal agitation, we may add to J a diffusion term -k′∇W; this gives directly the ''Fokker-Planck'' equation of a previous, more laborious calculation. When ∂∂=0, the equation simplifies and can be replaced by a minimization problem, susceptible to approximate treatment. In the case of a free-energy function with deep minima at =0 and , such treatment leads again to a result derived previously by a method adapted from Kramers and valid when v(Vmax-Vmin)kT is at least several times unity (v=particle volume, Vmax and Vmin=maximum and minimum free energy per unit volume, k=Boltzmann's constant, T=Kelvin temperature). When the minima are not deep, a different treatment is necessary; this leads to a formula valid when v(Vmax-Vmin)kT<<1. © 1963 The American Institute of Physics.