Symmetry of the mixing Hamiltonian in lattices with a basis

To determine substitutional and injection superstructures in alloys having a complex lattice with a basis by the method of concentration waves it is necessary to know the eigenvectors of the Fourier transform matrices V_pq() of the mixing potential of a homogeneous solid solution. A technique has been developed, investigating the variations of the matrix elements V_pq() for symmetry transformations of the spatial lattice group, to solve this problem. The symmetry elements belonging to the wave vector group determine the internal structure of the matrices V_pq() for lattices with a basis. Symmetry elements not appearing in the wave vector group determine the relation between matrices belonging to different rays of the star wave vector. Restrictions on the matrix elements V_pq(), following from symmetry, are found for an h.c.p. lattice. The eigenvectors are calculated for the matrices V_pq(), whose vectors terminate in high-symmetry points, lines, and planes of the Brillouin zone. This made it possible to separate the problem of thermodynamic competition of intermediate phases in an alloy with a hexagonal lattice into symmetric and potential parts, similarly to the way this occurs in Bravais lattices.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 13–25, August, 1993.