Space group theory for spin waves

The symmetry properties of spin waves in ferro-, antiferro-, and ferrimagnetic crystals can be conveniently discussed in terms of an appropriate space group as are electron energy states in band theory and normal modes in the theory of lattice vibrations. Dimmock and Wheeler have pointed out that the magnetic space group, because of the directional properties of the moments, has fewer elements than the lattice space group, and have discussed the spin-wave properties in terms of this group. While this is strictly correct, the main part of the Hamiltonian usually used in such systems often has much more symmetry than this. For Heisenberg interactions the spins can be rotated independently so that the total group is a product of a lattice space group and a set of spin - space rotations which leave the spin pattern unchanged. This lattice space group is obtained by considering atoms with different spin orientations as inequivalent. The single ion anisotropy restricts the spin rotations. This leads to considerably more degeneracy in the spin-wave spectrum than is predicted by the magnetic space group, the degeneracy being removed in real crystals by small interactions like dipole - dipole. Such groups have been constructed for common systems for use in evaluating the form of spin-wave spectra, selection rules, etc.