IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 10, OCTOBER 2005 3121 Micromagnetic Simulation of the Imaginary Part of the Transverse Susceptibility Dorin Cimpoesu , Alexandru Stancu , Senior Member, IEEE, Ioan Dumitru , and Leonard Spinu , Member, IEEE Faculty of Physics, “Alexandru Ioan Cuza” University, 700506 Iasi, Romania Advanced Materials Research Institute (AMRI), University of New Orleans, LA 70148 USA Advanced Materials Research Institute (AMRI) and Department of Physics, University of New Orleans, LA 70148 USA A micromagnetic model based on magnetization vector dynamics described by the stochastic Landau–Lifshitz–Gilbert (SLLG) equa- tion for the transverse susceptibility (TS) of particulate media is presented. The effect of the amplitude of excitatory ac field and the effect of thermal relaxation is taken into account. Index Terms—Magnetic particles, relaxation processes, stochastic processes, transverse susceptibility. I. INTRODUCTION T HE reversible transverse susceptibility (TS) was first analyzed in 1957 by Aharoni et al. [1] for a single-domain Stoner–Wohlfarth particle. In the TS experiment, one applies to the sample a system of two magnetic fields: one dc field and a small amplitude ac field, perpendicular to former. For nonin- teracting uniaxial single-domain fine-particle systems, the field dependence of transverse susceptibility presents characteristic peaks, located at the anisotropy and switching fields. The anisotropy peaks are determined by the particles with the easy axis oriented perpendicular to dc field. Thus, the TS represents an important technique for determination of the anisotropy field and easy-axis texture. Subsequently, a theoretical approach of the TS, which accounts for the higher terms in the uniaxial anisotropy energy expression was derived [2]. A generalized micromagnetic model based on the Landau–Lifshitz–Gilbert (LLG) equation, which is able to take into account any type of anisotropy, distribution of easy-axes orientation, particles positions and volumes, and different strength of magnetostatic interactions between particles was developed in [3]. All these models are frequency independent because it is assumed that, in each moment, the magnetization reaches an equilibrium state independent of the ac field rate. The effect of the frequency of the excitatory ac field on the TS experiment and the effect of the energy dissipation through the imaginary part of the TS are taken into account in a model based on the LLG equation [4]. Thermal relaxation is taken into account using a two-level model such as in [5] and [6], and it is shown that the temper- ature can change the position of the coercivity peak while the anisotropy peaks maintain their position, though their shape is altered. In [7], it is shown that during a TS experiment, the increasing dc field shifts the particle relaxation time toward lower values, making possible the occurrence of the thermal relaxation on the time scale of the ac field period. For a system of identical particles having the easy axis perpendicular to the dc field, the thermal relaxation gives rise to a secondary peak in the real part of the TS and to a peak in the imaginary part of it. Digital Object Identifier 10.1109/TMAG.2005.854889 In this paper, we present a generalized model of the TS that takes into account both the effect of the frequency and the effect of the thermal relaxation. II. THE MODEL The micromagnetic model we have used is based on the sto- chastic Landau–Lifshitz–Gilbert (SLLG) equation [8]. To ac- count for the effects of the interaction of the magnetization with the microscopic degree of freedom, a fluctuating stochastic field is added to the effective field. The SLLG equation is numerically integrated using the method of Heun [9]. The magnetic proper- ties follow from averages over many numerical realizations of the dynamic process. We have considered in simulations a system of identical, spherical, noninteracting, aligned particles with volume , uni- axial anisotropy constant , and saturation magnetization , with the easy axis in the – plane making an angle with the axis. The dc field acts along on the axis, and the ac field along the axis. For low values of the frequency of the ac field, it may be assumed that in the absence of the thermal fluctuations, the magnetization lies in a minimum of the free energy at any moment. It is easy to show that, in this case, the magnetization is in the – plane and let be the angle spanned by the magnetization and axis. In numerical integration of the SLLG equation, the round-off errors increase if the amplitude of the ac field is decreased. As the amplitude of the ac field decreases, the modification in the state of the magnetic moment in comparison with that calculated when only the dc field is applied becomes smaller and more difficult to calculate with accuracy. Therefore, in the simulation, we have used an ac field amplitude that was sufficiently high to avoid the increase of the round-off errors. The TS is calculated as where is the particle’s anisotropy field, and . There- fore, the normalized susceptibility does not depend on the volume or anisotropy. 0018-9464/$20.00 © 2005 IEEE