LXX. Lattice theory of dielectric and piezoelectric constants in crystals

Abstract

Abstract A general perturbation theory has been given by Born for treating long acoustic lattice waves. By comparing the wave equations thus obtained with the equations for elastic waves he obtains the expression for the elastic constants. The formal results however diverge when Coulomb interactions between charged lattice particles are taken into account. The fault in this case can be traced to the breakdown of the expansion procedure employed in the perturbation theory. In this paper it is shown that after the separation of a term by Ewald's theta-function transformation from the lattice restoring force, the perturbation theory can once more be developed. The corresponding wave equation contains an extra contribution due to the term separated which is shown to be the macroscopic electric field associated with the wave. The equation proves to be identical with the rigorous elastic wave equation for a piezoelectric crystal which takes into account the interaction between the elastic stresses and the electric field produced by the piezoelectric polarization. The comparison leads to the expressions for the dielectric and piezoelectric coefficients as well as the elastic constants (which cannot be obtained by the standard static methods when Coulomb interaction is taken into account).