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A combination of electronic-structure methodologies from density functional theory (DFT) through a tight-binding (TB) model to analytic bond-order potentials (BOPs) has been used to investigate structural trends within TCP phases, which we recently discussed using an empirical structure map [Acta Materialia 59, 749 (2011)]. First, DFT is used to calculate the structural energy differences across the elemental 4d and 5d transition metal series and the heats of formation of the binary alloys Mo-Re, Mo-Ru, Nb-Re, and Nb-Ru, where we show that the valence electron concentration stabilizes A15, σ, and χ phases but destabilizes μ and Laves phases. Second, a one-parameter canonical d-band TB model in combination with the structural energy difference theorem is found to reproduce the observed elemental DFT structural trends. The structural energy difference theorem is also used to rationalize the influence of the relative size differences on the stability of μ and Laves phases in binary systems. Third, analytic BOP theory using the TB bond integrals as input is shown to converge to the TB structural energy difference curves as the number of moments in the BOP expansion is increased. In order to provide a simple interpretation of these structural energy difference curves in terms of analytic response functions and the differences in the moments of the density of states (DOS), an expression is used for the difference in the band energy that is correct to first order in the Fermi energy differences. We find that the fourth-moment contribution separates the A15, σ, and χ phases from the μ and Laves phases in agreement with the empirical structure map due to difference in the bimodality of the corresponding DOS caused mainly by distortions in their coordination polyhedra from ideal Frank-Kasper polyhedra. Finally, it is shown that at least six moments are needed to predict the structural trend A15→σ→χ.