Dynamical Determinitation of the Elastic Constants and Their Temperature Coefficients for Quartz

A mathematical theory is presented giving the relations between the elastic constants and the frequencies of piezoelectric vibration for an infinite plate of homogeneous but aeolotropic material. Since no such theory is known for finite plates it is, strictly speaking, impossible to determine the elastic constants of the material from measurements on such plates. However, the viewpoint of perturbation theory indicates that certain frequencies of vibration of finite plates asymptotically approach the frequencies of vibration of infinite plates of the same material, orientation and thickness as the harmonic order increases. The experimental results confirm that this is an excellent method of eliminating edge effects and accurate data that are seemingly quite independent of mother crystal and proportions of the plate are obtained. However, in the evaluation of the elastic constants and their temperature coefficients from these data it is found that there is a small but not easily explained discrepancy between these results and the classical elastic theory for substances of the symmetry class $D_3$ generally accepted for quartz. Detailed observations have been made in the temperature interval from 0° to 80°C but additional results enable the calculation of the values of certain of the elastic constants at higher temperatures up to the alpha-beta transition point.