Local icosahedral packing energetics for d-band metals

Local icosahedral order in solid environments is modelled by polytope {3, 3, 5} (a packing of atoms on S³, the surface of the unit sphere in four dimensions, with perfect icosahedral symmetry) and by a 147 atom extension of the Mackay icosahedron. The bonding energies of each are calculated in a one-electron d-band, tight binding model, as a function of d-band filling. The energies per site on the polytope, and on the central atom of the extended Mackay icosahedron, are then compared with those in the f.c.c. structure to reveal chemical trends in the relative stability of local icosahedral packing. The calculated structural energy differences show that even in fairly close-packed metals, local icosahedral packing is preferred only over a limited range of d-band fillings, roughly between 2 and 5 d-electrons per atom, even when frustration effects are artificially turned off by the use of the polytope. It is also observed that the magnitude of these energy differences is of the same order of magnitude as the structural energy differences between f.c.c. and b.c.c. structures and typically exceed the elastic energy differences which arise as a result of atomic size mismatches. These phase stability results are in marked contrast with those calculated in pair potential models.