Linearized rate-equation approach for double-well systems: Cooling- and temperature-dependent low-field magnetization of magnetic nanoparticles

Low-field magnetization curves for an assembly of polydisperse magnetic nanoparticles described as double-well systems (DWS) with randomly distributed easy axes are generated by means of a rate-equation approach applied to states obtained by cooling with and without a magnetic field. The model has notable advantages over the existing steady-state approaches, allowing one to accurately determine the magnetization frozen in a sample after cooling under field, and to study the effect of the magnitude of the cooling field on the shape of the field-cooled (FC) curve. The DWS scheme keeps validity well above the blocking temperature, making it possible to apply the model over all the region of interest (i.e., when the system is out of equilibrium). A linearized expression for the magnetization curves is obtained and compared with the existing formulas. It is shown that the linearized expression for the zero-field-cooled (ZFC) curve is more satisfactory, while no linearized expression for the FC curve is able to accurately reproduce the behavior derived from the rate equations. A method to obtain the size distribution function is developed; this makes use of the linearized model in the DWS scheme and is based on the analysis of the experimental ZFC curve only. Explicit expressions for the FC/ZFC curves of DWS characterized by an average blocking temperature higher than the starting temperature of the measurement cycle are proposed and shown to naturally explain "anomalous" experimental curves found in the literature.