Interpretation of the Hume-Rothery rule in quasicrystals and their approximants

Abstract We have performed the LMTO-ASA band calculations for the γ-phase Cu5Zn8 compound, the Frank–Kasper type Al30Mg40Zn30 and Al52.5Li32.5Cu15 1/1–1/1–1/1 approximants. In γ-brass, where d-states exist in the middle of the valence band and the sp-d hybridization effect is significant near the Fermi level EF, we show that it is critically important that the {3 3 0} and {4 1 1} zones satisfy the Hume-Rothery matching rule and that they are strongly coupled with the sp-d hybridization to produce a deep pseudogap across EF. In the nearly-free-electron-like Al–Mg–Zn approximant, it is shown that the {5 4 3}, {7 1 0} and {5 5 0} zone planes are essentially responsible for a pseudogap at EF. In the Al–Li–Cu approximant, the deep pseudogap at EF is proven to arise mainly from the Al-3p/Al-3p hybridization effect, but that the {6 3 1} zone effect is still significantly important.