The excitation spectra of magnetic bubble lattices

The excitation spectra of a hexagonal lattice of magnetic bubbles are calculated using the methods of lattice dynamics. The bubbles are allowed one internal zero-mode radial degree of freedom and two translational degrees of freedom of the center of mass resulting in three branches of free oscillation. The equations of motion, whose Fourier transform yields a third order secular determinant corresponding to two acoustical and one optical branch, are obtained from a Lagrangian with a Rayleigh dissipative function. The inter-bubble potential is approximated by that of a dipole-dipole interaction and restricted to nearest neighbors. The dispersion curves are found for the symmetry directions kxand ky. For a dense lattice with no coupling between the radial and translational degrees of freedom the optical and acoustical branches cross. The inclusion of coupling has drastic effects on the dispersion relationships. The acoustical branches consist of admixtures of translational and radial oscillations. Because of damping a critical wavenumber exists below which no real part of the frequency exists. Changes in the acoustical branches can be brought about by changing the external magnetic field.