Journal of Magnetism and Magnetic Materials 242–245 (2002) 1021–1023 Intrinsic magnetic resonance in nanoparticles: Landau damping in the collisionless regime Yu.L. Raikher*, V.I. Stepanov Institute of Continuous Media Mechanics, Ural Division of RAS, 1 Korolyov St., 614013, Perm, Russia Abstract Inauniaxiallyanisotropicparticlethefrequencyofintrinsicferromagneticresonancedependsontheangledeviation of the magnetic moment from its equilibrium position. At low spin–lattice relaxation and in the presence of superparamagnetism, the regime of relaxation in an assembly of such particles closely resembles the Landau damping regime, which is well known in the kinetic theory of plasma. In the high-temperature limit in the case of polydisperse systems, particle volume and surface anisotropy affect the absorption lines differently. r 2002 Published by Elsevier Science B.V. Keywords: Nanoparticles; Ferromagnetic resonance; LinewidthFfrequency dependent; Superparamagnetism Superparamagnetism is a well-known thermofluctua- tional effect responsible for observable loss of equili- brium magnetization in nanosize objects. Less known is the fact that the rotary diffusion of the magnetic moments of single-domain particles also strongly affects the precession mode. The essential r # ole of thermal noise in the formation of the ferromagnetic resonance (FMR) spectra of fine-particles was reported in 1974 [1], and thusithastakenovertwodecadestofullyunderstandits effect on the dynamic susceptibility. Let a magnetic moment l be deviated at zero temperature from its equilibrium position. Then, according to the Landau– Lipshitz equation dl=dt ¼gðl  H eff Þ gaðl ðl  H eff ÞÞ ð1Þ (with g and a being the gyromagnetic ratio and the damping parameter, respectively) vector l begins to precesswithanangularvelocity BgjH eff j circumscribing a conical surface with the symmetry axis along the effective field H eff : With time the cone shrinks due to spin–lattice relaxation, and l > ; the projection on the plane normal to H eff ; decays as a deterministic vector. Now let us introduce a heat bath, i.e., fluctuations that disorient the magnetic moment at random. In this way one ‘‘switches on’’ the orientational diffusion that accelerates the decay of the macroscopic (observable) quantity /l > S; theensembleaverageof l > : InRef.[1], the appropriate theory was developed for the case of intrinsic FMR, where H eff consists solely of the internal field of magnetic anisotropy H a ¼ 2K=I ; with K being the volume density of the magnetic anisotropy energy and I the particle magnetization. Assuming the pertur- bations l > to be infinitesimal, the conclusion obtained in Ref. [1] was that with increasing temperature the characterofthemagneticmomentmotionchangesfrom a resonance to an overdamped one. In the high- temperaturelimitthedecayrate(andthus,thelinewidth Do) is proportional to akT ; i.e., the product of the damping parameter and temperature. In other words, under heating, the absorption line of intrinsic FMR experiences unlimited homogeneous widening. The predictions of Ref. [1] looked so reasonable that for a long period no revision seemed necessary. However, on closer inspection, the claimed proportion- ality DopakT that yields Do ¼ 0at a-0 whatever the temperature, does not seem physically justified. A correct way to analyze the case of a-0 was outlined in Ref. [2] and applied to intrinsic FMR in Ref. [3]. It turned out that at a ¼ 0 the linear-response (Kubo) theory gives an exact solution for the dynamic suscept- ibility. The result is not a zero-width line, as it follows from the approximation [1] in the limit a-0: In the linear-response treatment, one explicitly takes into *Corresponding author. Fax: +7-3422-336957. E-mail address: raikher@icmm.ru (Y.L. Raikher). 0304-8853/02/$-see front matter r 2002 Published by Elsevier Science B.V. PII:S0304-8853(01)01358-0