A cluster expansion is used to determine the energy of substitutionally disordered alloys as a function of configuration. The expansion is exact in the sense that the basis functions are complete and orthonormal. The coefficients, effective cluster interactions (ECI's), are computed directly from their definition by means of the method of direct configurational averaging, which is described in detail in the context of a tight-binding linear muffin-tin orbital (TB-LMTO) Hamiltonian. The alloy Hamiltonian is constructed from a combination of the pure-element TB-LMTO Hamiltonians, the hopping integrals between unlike pairs of atoms (simply given by the geometric mean of the pure-element integrals), and the potentials of the alloy, which are computed consistent with the condition that each configurationally averaged atom of the alloy be neutral. This scheme of self-consistency is tested against the results of fully self-consistent LMTO calculations on ordered compounds. The ECI's are computed on the fcc lattice for six alloy systems: Rh-Ti, Rh-V, Pd-Ti, Pd-V, Pt-Ti, and Pt-V. It is shown how the ECI's may be used in conjunction with properties of the energy expansion to exactly solve for the ground-state superstructures of fcc. This ground-state search is contingent upon minimizing the configurational energy subject to a number of geometric constraints. A large number of these constraints are formulated using group-theoretic means on the (13-14)-point clusters of the fcc lattice. The use of this large number of constraints makes possible the inclusion of fourth-nearest-neighbor pair ECI's as well as multiplet ECI's in the ground-state search. Both these types of interactions are shown to be essential towards obtaining a convergent energy expansion. In all six alloy systems, agreement between the theoretically predicted ground states and the experimental evidence of fcc superstructures is excellent: in no case is an unambiguously experimentally determined fcc-based phase missing from the results of the ground-state search.
ASJC Scopus subject areas
- Condensed Matter Physics