The Hume–Rothery electron concentration rule for a series of gamma-brasses studied by full-potential linearized augmented plane wave (FLAPW)

Abstract

A theoretical interpretation of the Hume–Rothery electron concentration rule has been attempted by performing full-potential linearized augmented plane wave (FLAPW) band calculations for a series of gamma-brasses: Cu₅Zn₈, Cu₉Al₄ and TM₂Zn₁₁ (TM = Ni, Pd, Co and Fe). The Fourier spectrum of the FLAPW wavefunction was evaluated for states at point N on the set of {110} zone planes of the reduced bcc Brillouin zone at energy eigenvalues sustaining the pseudogap. It turned out that electrons near the Fermi level resonate with periodic potentials due to {330} and {411} lattice planes to open the pseudogap for all gamma-brasses studied. The effective electron per atom ratio (e/a) was evaluated by averaging the wave vector 2(k _i  + G) having the largest Fourier component of the FLAPW wavefunction at energy E over selected electronic states k _i in the Brillouin zone, with subsequent construction of the dispersion relation E versus in the extended zone scheme. The e/a value proved to be close to e/a = 21/13 = 1.62, the value empirically assigned for the gamma-brass in the Hume–Rothery electron concentration rule, for Cu₅Zn₈, Cu₉Al₄, Ni₂Zn₁₁ and Pd₂Zn₁₁, but to be as small as 1.43 and 1.31 for Co₂Zn₁₁ and Fe₂Zn₁₁, respectively.