Magnetization dynamics in Landau–Lifshitz–Gilbert formulation. FMR experiment modeling Mirosław R. Dudek a, * , Nikolaos Guskos b,c , Bogdan Grabiec a , Michał Maryniak c a Institute of Physics, University of Zielona Góra, ul. Szafrana 4a, 65-069 Zielona Góra, Poland b Solid State Physics Section, Department of Physics, University of Athens, Panepistimiopolis, 15 784, Greece c Institute of Physics, Szczecin University of Technology, Al. Piastow 17, 70-310 Szczecin, Poland article info Article history: Available online 30 July 2008 Keywords: Magnetic properties Modeling and simulation abstract A ferromagnetic resonance (FMR) spectrum of carbon coated magnetic nanoparticles in a non-magnetic elastic matrix has been investigated. The experimental absorption data have been compared with anal- ogous data obtained with the help of the stochastic Landau–Lifshitz equation for the magnetic moment of a ferromagnetic single-domain nanoparticle and stochastic equations describing rotational oscillations of the polymer region containing a magnetic nanoparticle. We have shown that if polymer matrix anisotro- pies are blocking the rotational freedom of the easy axes direction of the magnetic nanoparticles, addi- tional resonance peaks can appear in the FMR spectrum which are satellite peaks accompanying the main resonance peak. This is another mechanism for the appearance of additional peaks in the FMR spec- tra besides the spin-waves exchange mode in nanoparticles or inter-particle dipolar interactions. The compound Poisson process has been used to model this effect of additional correlations introduced on the motion of the particle’s magnetic moment. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction The first papers by Neél [1] and Brown [2] on the magnetic re- sponse of single-domain magnetic particles have started a long term discussion on the dynamic properties of materials containing such particles. Recently, this discussion has been even intensified because there have appeared new technologies producing stable magnetic nanoparticles, not degrading in time, with the applica- tions for a new type of high density storage devices or new func- tional materials based on polymers and filled with magnetic nanoparticles. The ferromagnetic resonance experiment (FMR) is the main tool for determining the magnetic dynamical properties of modern magnetic materials. The complex dynamics of the mag- netic moments of magnetic nanoparticles in a non-magnetic ma- trix results in compound FMR spectra in which the main resonance peak corresponding to the uniform resonance mode might be accompanied by a series of peaks originating from such different physical phenomena as a spin-wave exchange mode [3,4] of particles or dipolar inter-particle interactions. The experi- mental measurement data obtained by Owen [5] for a colloidal suspension of Fe 3 O 4 nanoparticles which have been solidified in a constant magnetic field (DC magnetic field) show that the corre- sponding FMR spectra could have an orientational dependence. We will refer to that work in the following part of this paper, as we want to emphasize that both the particle’s magnetic anisotropy and the matrix anisotropy become important factors in an FMR experiment with magnetic nanoparticles in a non-magnetic poly- mer matrix. It is usually, assumed that the nanoparticle moment of inertia has a negligible contribution to the rotational motion equations and it is excluded from consideration. The same line of reasoning leads to neglecting the polymer matrix dynamics in the case when its elastic properties substantially influence the the magnetic nanoparticle’s rotational oscillations. The recent pa- per by Guskos et al. [6] on the magnetic properties of c-Fe 2 O 3 fer- rimagnetic nanoparticles embedded in a multiblock poly(ether- ester) copolymer suggests that the magnetoelastic coupling of oxide nanoparticles with the surrounding polymer becomes important for the polymer dynamics. In particular, the FMR spectra show additional peaks in low temperatures which originate from the orientational anisotropy of frozen polymer blocks. We have used a very simple stochastic model of the FMR exper- iment which is based on the Langevin version of the Landau–Lifshitz–Gilbert [7,8,10,11] equations and the Langevin version of the rotational motion of magnetic nanoparticles being build into the elastic surroundings. The specific relation between resonance peaks and correlations introduced by blocked directions of magnetic particles as well as the number of nanoparticles in the 0022-3093/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2008.06.024 * Corresponding author. E-mail addresses: mdudek@proton.if.uz.zgora.pl (M.R. Dudek), ngouskos@ phys.uoa.gr (N. Guskos), bgrab@proton.if.uz.zgora.pl (B. Grabiec), michalmaryniak@ gmail.com (M. Maryniak). Journal of Non-Crystalline Solids 354 (2008) 4146–4150 Contents lists available at ScienceDirect Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol