Magnetic relaxation time for an ensemble of nanoparticles with randomly aligned easy axes: A simple expression
Artek R. Chalifour, Jonathon C. Davidson, Nicholas R. Anderson, Thomas Crawford, Karen L. Livesey
Abstract
A critical parameter in characterizing the properties of single-domain nanoparticles is their magnetic relaxation time. It must be known, for example, to estimate the anisotropy from magnetization versus temperature measurements. The time it takes for the magnetization to relax also determines the behavior of particles in various oscillating applied fields, which is critically important for their application in magnetic particle imaging and hyperthermia treatment. However, an analytic expression for this relaxation time has been generally missing. Brown's [Phys. Rev. 130, 1677 (1963)] famous result is only valid for the easy anisotropy axes of each particle in the ensemble aligned along the external field direction and overestimates the relaxation time. Despite this overestimation, this expression is most commonly used to extract magnetic nanoparticle parameters such as anisotropy energy from magnetometry data. Here, we use Brown's formalism to derive a different, simple, approximate relaxation time expression that is valid for randomly aligned easy axes. Using parameters appropriate for magnetite, we compare our results to other results in the literature and with stochastic Landau-Lifshitz-Gilbert simulations to show that our result is more accurate across a range of applied field strengths and temperatures. We note that several analytic expressions for the relaxation time do a reasonable job, as long as one uses a full calculation for the attempt time, rather than the commonly used estimate ${\ensuremath{\tau}}_{0}\ensuremath{\sim}1$ ns.