Lagrangian temperature coefficients of the piezoelectric stress constants and dielectric permittivity of quartz

Piezoelectric, Lagrangian equations for the frequency-temperature behavior of quartz are presented. From the solutions of the third order temperature perturbations of these Lagrangian equations for the thickness resonances of infinite quartz plates with air-gap electrodes, regression equations for determining the temperature derivatives of elastic constants, piezoelectric constants and dielectric permittivities am developed. By using these regression equations, and the measured data on temperature coefficients of frequency by Bechmann, Ballato and Lukaszek [1962] for doubly rotated cuts, the first, second, and third temperature derivatives of elastic constants, piezoelectric constants and dielectric permittivities are obtained. The second and third temperature derivatives of the piezoelectric constants and dielectric permittivities were not available in the literature and are published here for the first time. These temperature derivatives will provide a more accurate map of the temperature stable cuts for bulk wave and surface acoustic wave quartz resonators.