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This paper concerns an investigation of the spin wave excitations in magneticnanoparticles. We provide a detailed derivation of the theoretical method forthe determination of the normal modes of confined magnetic systems based on adiscrete lattice of magnetic moments. The method is based on the damping-freeLandau-Lifshitz equation and general enough to be utilized for the magneticsystem of any dimensionality, magnetic structure, shape, and size. As anexample we explore the influence of the competition between exchange anddipolar interactions on the spectrum of normal modes as well as on thestability of the vortex state in two-dimensional nanorings. We show thelowest-frequency mode to be indicative of the dipolar-to-exchange iterationsratio. We also study behavior of the fundamental mode and present the influenceof both, the discreteness of the lattice and the dipolar-to-exchange iterationsratio, on its hybridization with azimuthal modes. We complete the paper with aselective review of the spin wave excitations in circular dots to compare withthe results obtained for the rings.

Vortices in two-dimensional nanorings studied by means of the dynamical matrix method