Exchange Charge Contributions to the Failure of the Cauchy Relations in Alkali Halide Crystals

Using a model originally constructed to deal with the theory of dielectric constants of alkali halide crystals, a theory of the elastic constants of these crystals is developed. Exchange charges arising from the requirements of the exclusion principle are approximated as point charges appropriately placed in a Born-Mayer model ionic crystal. The presence of these exchange charges and the manner in which their magnitude and position change in a strained crystal give rise to interionic forces of a many-body character. As a consequence the model predicts deviations from the Cauchy relation $c_{12}=c_{44}.c_{11},c_{12}, \mathrm{and} c_{44}$ are calculated for 16 NaCl structured alkali-halides. In those cases where the method of calculation is expected to be valid the calculated valves indicate that the exchange charge mechanism can account quantitatively for much of the observed deviation of $c_{12}-c_{44}$ from zero for these crystals.