Combined rotational diffusion of a superparamagnetic particle and its magnetic moment: Solution of the kinetic equation

Abstract

This paper develops a theory of rotational diffusion of a uniaxial superparamagnetic particle, suspended in a fluid. A kinetic equation for the joint distribution function of orientations of anisotropy axis and magnetic moment of the particle is examined and a consistent method of its solution is proposed. The approach involves the introduction of a kinetic operator that generates the time evolution of the distribution function, and the usage of quantum-mechanical formalism, particularly, elements of the representation theory and the theory of additions of angular momenta. A convenient representation, in which matrix of the kinetic operator is close to diagonal, is specified. In this representation the evolution equation takes a form of linear recurrence–differential relations for statistical moments of the distribution function. Their numerical integration can be performed by one of the standard methods, giving the average (observed) magnetization of the system for any instant of time. The presented scheme assumes that frequency of an applied magnetic field is well below ferromagnetic resonance range (that is typical for majority of experiments) but it does not impose any restrictions on the field amplitude, material parameters of the particle, viscosity of fluid or temperature. It can serve as a theoretical basis for a consistent description of relaxation spectrum, dynamic magnetic susceptibility and nonlinear magnetic response of a dilute magnetic fluid with considering the interplay between mechanical and magnetic degrees of freedom of suspended nanoparticles. Also, it can be used for cross-checking of approximate models.