Symmetry of the mixing Hamiltonian in lattices with a basis

Abstract

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 Vpq(κ) of the mixing potential of a homogeneous solid solution. A technique has been developed, investigating the variations of the matrix elements Vpq(κ) 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 Vpq(κ) 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 Vpq(κ), following from symmetry, are found for an h.c.p. lattice. The eigenvectors are calculated for the matrices Vpq(κ), 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.