Radio-Frequency Vibrations of Rectangular AT-Cut Quartz Plates

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Abstract
A combined theoretical and experimental study is presented of the coupled thickness shear and flexural vibration modes of rectangular AT-cut quartz plates. The theoretical approach (taking into account piezoelectric effects) involves a variational technique which allows for a significant refinement of the mathematical analysis over that heretofore presented in the literature. A reliable experimental technique was used to obtain the functional dependence of frequency vs crystal dimensions. Comparison of experimental crystal resonant frequencies and polarization distributions with those derived from our theory reveals excellent agreement. A discussion is included describing the nature and significance of charts relating resonant frequencies with the ratio length/thickness of crystal dimensions.