Effect of Invariance Requirements on the Elastic Strain Energy of Crystals with Application to the Diamond Structure

After demonstrating inconsistencies in some of the better known elasticity calculations, an alternative method of imposing the necessary invariance conditions on the strain energy of a crystal is presented. The new method is equivalent to the Born-Huang procedure but, in addition to providing further insight, also offers one or two operational advantages. For example, it demonstrates that all purely first-neighbor interactions are central only. The method is applied to the calculation of the elasticity of a two-constant model of the diamond type of crystal, and this predicts the relation $2{c}_{44}({c}_{11}+{c}_{12})=({c}_{11}\ensuremath{-}{c}_{12})({c}_{11}+3{c}_{12})$, which is very well satisfied by the experimental data for diamond, silicon, and germanium.