In lattices in which not all ions possess inversion symmetry, internal-strain effects give rise to piezoelectricity and contribute to the elastic response. It is possible to describe these internal strains as static optical-phonon-mode displacements and to analyze the internal-strain contributions to the elastic and piezoelectric constants accordingly. Certain symmetry properties become evident as a result, the most general of which is that only Raman-active modes contribute to internal strain, and only modes simultaneously Raman- and infrared-active produce piezoelectricity. This formalism also provides a basis for a discussion of piezoelectric and elastic anomalies which may accompany incipient instabilities in opticalphonon modes. Finally, it is shown that the anomalous elastic behavior of $\beta$ quartz near the $\alpha -\beta$ transition temperature can be understood by postulating a low-frequency temperature-dependent Raman-active mode of $A_1$ symmetry.

• Received 6 June 1967

DOI:https://doi.org/10.1103/PhysRev.163.924