Rectangular parallellepiped resonance method for piezoelectric crystals and elastic constants of alpha-quartz

A theory is presented on the free osculations of a rectangular parallelepiped of piezoelectric crystal, by extending the theory of the rectangular parallelepiped resonance (RPR) method to determine elastic constants of crystals, as exemplified by an alpha-quartz specimen. The piezoelectric contribution to resonance frequencies was examined numerically on the specimen, and it was revealed that piezoelectricity causes approximately 5 kHz increase around 1 MHz. The resonance frequencies of the specimen were measured and inverted to elastic constants by least squares inversion. The inversion was by both the previous non-piezoelectric or elastic theory and by the present piezoelectric theory. The use of the non-piezoelectric theory resulted in an overestimate of 2σ or 0.6% in c ₁₁ and underestimate of σ or 6% in c ₁₂. These are the constants expected to be most affected by piezoelectricity. Errors are less than σ in the other constants. During measurement, it was found that the force applied to hold the specimen caused deviations from free oscillation and experimental errors of 5 kHz. The correction for this force is of some importance in RPR studies of piezoelectric crystals.