We consider a ferromagnetic linear chain and a simple-cubic lattice of spin-1 ions with isotropic dipolar and quadrupolar pair interactions. We calculate the zero-temperature energy spectra of two-spin—deviation states over the entire range of $\beta$ (the ratio of the quadrupolar to dipolar coupling), $-1<\beta \le 1$, for which the system has a ferromagnetic ground state. We also calculate the spectral density of states for "quadrupolar" excitations, which are single-ion two-spin—deviation excitations. We find that, in addition to the two-spin—wave bound states outside the band, there exists a new "quadrupolar" resonant state within the band, which is seen as a distinctive peak in the spectral density of states for quadrupolar excitations. The linewidth of the peak decreases and the peak moves to a lower position in the band as $\beta$ increases. As $\beta$ approaches 1 the quadrupolar resonant state becomes a "quadrupole wave" excitation, which is an eigenstate of the Hamiltonian and describes the coherent propagation of quadrupolar excitations in the solid. The dispersion relation of the quadrupole wave is the same as that of the spin wave. We also find that, when $\beta$ is sufficiently negative, there is a repulsive interaction between two spin waves, which causes bound states to split off above the band. We show that both the dipolar and quadrupolar pair interactions can in principle be simultaneously determined from the Raman spectra of the isotropic system. Finally, we comment on the generalizations of the present calculations to more realistic cases.

• Received 9 December 1974

DOI:https://doi.org/10.1103/PhysRevB.12.1819