Systematic behavior of the hexagonal axial ratio for the d transition metals

The d transition elements are observed to show a c/a axial ratio which sometimes deviates substantially from the ideal value, root 8/3, when they are assumed to crystallize in the hcp structure, as shown by a first-principles treatment. The reason is investigated by combining a first-principles method together with a more transparent structure-dependent model based on canonical d bands. From this model it is shown that the Madelung energy and, to a smaller extent, the Born-Mayer repulsion stabilize the hcp crystal at the ideal axial ratio, while the canonical one-electron d-eigenvalue sum W-d is the dominant term that favors a deviation of the axial c/a ratio from the ideal one. W-d generally has a distorted asymmetric parabolic c/a dependence, with a maximum in the vicinity of the ideal axial ratio 1.633. The asymmetry varies regularly with the d occupation. It is mainly this asymmetry of the d-band contribution that gives rise to the axial c/a nonideality and to its regular variation with d occupation. Our analysis adds additional insight to the mechanisms which determine the structural properties of the transition metals. [S0163-1829(99)08405-2].