Hysteresis effects in magnetic nanoparticles: A simplified rate-equation approach

Hysteretic properties of an assembly of uniaxial magnetic nanoparticles described as double well systems (DWS) with either collinear or randomly distributed easy axes are studied by means of a rate-equation approach. The resulting picture is sufficiently accurate to be exploited in high frequency applications of nanoparticles such as magnetic hyperthermia. The rate equation scheme allows an exhaustive description of hysteretic effects to be achieved in a rather simple way, with remarkable advantages over treatments based on nonlinear equations of magnetization dynamics and models derived from the Stoner-Wohlfarth theory. Rate equations for the magnetic DWS are then simplified and decoupled by making special assumptions on the escape frequency from the energy wells; it is shown that the simplified rate equation scheme can be applied in extended intervals of frequencies and temperatures, including the ones of interest for present-day practical applications of magnetic nanoparticles. Analytical solutions of the simplified rate equations allow one to explain several hysteretic properties of the system when the magnetic field is applied either parallel or perpendicular to the nanoparticle easy axis. These solutions are suitable to be generalized to the case of an assembly of nanoparticles with randomly distributed easy axes. Minor hysteresis loops of an assembly of DWS exhibit an anomalous behaviour: the magnetization driven by the periodic field initially follows a spiral path in the (H, M) plane; closed, self-similar hysteresis loops corresponding to the system's steady state are achieved only after a sequence of iterations that depends on the loop's vertex field: the smaller the vertex field is, the longer the time needed to reach the steady state. Only major loops (i.e., ones where the magnetization goes from positive to negative saturation) are closed since the very beginning. The anomaly occurs at all angles between magnetic field and easy axis and at all explored frequencies. This effect should be taken in due account in magnetic hyperthermia experiments.