Axial Ratios in Hexagonal Crystals

A theoretical study has been made of the deviations of the axial ratios of hexagonal metal crystals from the ideal close-packing value, both for pure metals and alloys. The deviation is expressed in terms of the "effective stress" which would tend to change the axial ratio in the close-packed configuration, and the observed elastic constants. Using a change of scale procedure, a formula for the "effective stress" is derived in terms of integrals over the electronic wave function in the close-packed configuration. When the Hartree-Fock approximation is used to evaluate the expression for the "effective stress," three contributions are found: (1) a "kinetic stress," (2) an "electrostatic stress," and (3) an "exchange stress." Estimates of each term for pure beryllium indicate that (1), which resembles the effects estimated by Jones and Goodenough, is most important; (2) is negligible; and (3) may be appreciable. It is found that the change in axial ratio with alloying is due to a different mechanism than that proposed by Jones. However, an argument is presented which leads to qualitatively the same conclusions as his concerning the connection between the change in axial ratio and the band structure.