Both ${\mathrm{Cu}}_5{\mathrm{Zn}}_8$ and ${\mathrm{Cu}}_9{\mathrm{Al}}_4$ $\gamma$ brasses containing 52 atoms in its cubic cell have been recognized as a typical Hume-Rothery alloy phase, since these structurally complex compounds are stabilized at the same electron concentration of 1.61 electrons per atom in spite of different amounts of solute elements Zn and Al involved. Both compounds are found to possess the pseudogap at the Fermi level. Its origin has been studied by performing the full-potential linearized augmented plane-wave (FLAPW) band calculations to lay a firm foundation on the empirical Hume-Rothery electron concentration rule in a straightforward way. This was accomplished through the study of the Fourier coefficient in the plane-wave expansion of the FLAPW wave function in the intermediate region outside the muffin-tin potential for states near the Fermi level. It is concluded that electrons contributing to forming the pseudogap resonate with crystal planes associated with {330} and {411} zones and give rise to standing waves for both ${\mathrm{Cu}}_5{\mathrm{Zn}}_8$ and ${\mathrm{Cu}}_9{\mathrm{Al}}_4$ $\gamma$ brasses. This clearly explains why these two compounds having different solute concentrations are stabilized at the same $e∕a$ value of 1.61.

• Received 19 August 2004

DOI:https://doi.org/10.1103/PhysRevB.71.165103