Phase stability in binary alloys

The hypothesis that Hume-Rothery's rules for phase stability in binary alloys are a consequence of the divergent slope of the wave-number-dependent dielectric function at 2k_F is investigated. The relative energies of the fcc, bcc, and hcp structures of the alloy systems CuAl, LiMg, and CuZn are calculated with a realistic local pseudopotential model. Striking evidence of the screening anomaly is obtained in all cases. The calculations suggest alloy phase boundaries in the three systems studied which are for the most part in excellent agreement with experiment. The only exceptions are pure Cu and pure Zn, for which an incorrect structure is predicted, the former by a remarkably small 3 × 10^-5 ryd per electron. It is suggested that both discrepancies might be rectified if the forces between d electrons on neighbouring sites were properly accounted for. In particular, it is argued that the attractive forces arising from the overlap of neighbouring d electrons and from their Van der Waals interactions provide a substantial impetus for a distorted structure in Zn. A thermodynamic instability arising from the divergent slope of the dielectric function is considered. Rough calculations are presented which indicate that when higher order terms in the electron-ion interaction are properly considered this instability will disappear, without, however, the phase boundaries being significantly altered. The principal conclusion, which is consistent with previous suggestions, is therefore that the Hume-Rothery rule arise not so much from the screening divergence itself as from the rapid variation of the dielectric function in the immediate vicinity of 2k_F.