Symmetry analysis of magnetic structures and spin-waves on the basis of a stabilizer concept

The common scheme for the group theory analysis of a crystal magnetic structure and a spin-wave spectrum is developed. This scheme is based on the representation theory of a crystal space group G (for magnetic structure analysis) and on the co-representation theory of the Shubnikov group M of a magnetically ordered crystal (for spin wave analysis).Unlike the standard calculation, the procedure proposed in this paper is based on the conception of the stabilizer-the symmetry group of single atoms, and this considerably symplified the calculation work, reducing it to the work with the elements of the stabilizer and with the representatives of the expansion of the G or M groups with respect to the group of the stabilizer. It is enough to calculate the components of the basic functions only at one atom, as for the other atoms they may be found by the action of the representatives.For the symmetrization of the spin wave Hamiltonian matrix a new mathematical procedure is used the expansion of group G over the double coset classes. It lets us easily find the symmetric relations between the different matrix elements. The method is illustrated in the example of the complicated magnetic structure of a garnet Dy₃Al₅O₁₂. The advantages of the new method are especially obvious in the case of structures with a great number of magnetic atoms in a primitive cell.