On the origin of the Hume-Rothery rules for phase stability in α and β brasses

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Abstract
The authors investigate the Blandin-Heine-Stroud-Ashcroft suggestion that the Hume-Rothery rules for the phase stability of alpha and beta brasses arise as a consequence of the rapid variation of the free electron density response function in the neighbourhood of 2KF. Using pseudopotential theory the authors show that the most structure-sensitive contribution to the free energy of a random binary alloy is the one-electron contribution arising from the virtual crystal bandstructure of neutral pseudo-atoms. The calculated difference in the one-electron contribution to the free energy between different crystal structures shows rapid variations at concentrations which are characterised by particular electron per atom ratios Z for which 2KF is equal to the first or second reciprocal lattice vector of the alloy.

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