Higher-Order Temperature Coefficients of the Elastic Stiffinesses and Compliances of Alpha-Quartz

The first-, second-, and third-order temperature coefficients of the elastic stiffnesses and compliances of alpha-quartz have been derived from thickness mode resonances of double-rotated quartz plates employing Christoffel´s theory of wave propagation. The temperature dependence of all possible thickness modes can be calculated from the values of the elastic stiffnesses and their temperature coefficients as derived during this investigation. A curve showing the locus of the first-order zero temperature coefficient of frequency of thickness-shear modes has been calculated and compared with experiments. The second- and third-order temperature coefficients of frequency of the first-order zero quartz cuts are given. Applications to AT, BT, CT, and DT cuts are made by comparing the calculated with the experimental values which characterize the temperature behavior of frequencies and new useful piezoelectric cuts of quartz are indicated.