Calculation of the Elastic Shear Constants of Magnesium and Magnesium Alloys

Leigh's calculation of the elastic shear constants of aluminum are extended to include crystals of lower symmetry, and the results are used to interpret the constants of pure magnesium and dilute magnesium alloys. The requirement that pure magnesium be in equilibrium with respect to shears that change the $\frac{c}{a}$ ratio of this hexagonal crystal is consistent only with the assumption that electron overlap has already occurred across the Brillouin zone faces perpendicular to the $c$ axis, a result which is at variance with the usual assumptions for this metal. Assuming $c$ overlap and one other type of electron overlap in the pure metal, it is found that the total number of overlap electrons (or, equivalently, the number of holes) is 1.70×${10}^{21}$ per ${\mathrm{cm}}^3$, i.e. 2% of the valence electron density. The overlap which appears at an electron/atom ratio of 2.01 is assumed to be an overlap in the equatorial plane of Brillouin zone; this overlap produces an abrupt change in one of the shear constants as the electron/atom ratio is increased beyond 2.01, a behavior of the type predicted by Leigh for aluminum-zinc alloys. The abrupt change in the constant is characteristic of $T=0\mathrm{}°$K, however; at room temperature, the predicted effect is much more gradual and is found to agree with the experimental results.