Magic number of electron concentration in the icosahedral cluster of AlxMg40X60−x (X=Zn, Cu, Ag, and Pd) 1/1-cubic approximants

A X-ray Rietveld analysis is carried out for the AlxMg40X60−x (X=Zn, Cu, Ag, and Pd) 1/1-cubic approximants. The electronic structure of the icosahedral cluster was calculated by means of the DVXα method with the refined atomic positions. The results obtained from the Rietveld refinement revealed that, regardless of the alloy systems, the number of the valence electrons per atom in the 12 atoms icosahedron, e/aico, always shows a unique dependence on the total number of the valence electron in the lattice, e/alat. This suggests that the cluster is stabilized at a certain number of the valence electrons. The number of the valence electrons per atom in this cluster e/aico is found to be 2.14–2.18, which is lower than that of the overall e/alat=2.2–2.35. Notably, at this particular number e/a=2.15, the Frank–Kasper-type icosahedral quasicrystal is the most stabilized and a thermodynamically stable quasicrystal is formed in the Al–Mg–Pd and Al–Mg–Zn alloy systems. We found from the DVXα calculation that an energy gap is formed at the Fermi level at this critical e/aico value.