X-ray diffraction topography analysis of TS-TT vibrations in contoured AT-cut quartz resonators

The coupled thickness-shear and thickness-twist (TS-TT) modes of vibration in AT-cut quartz resonators are studied through comparison of theoretically predicted frequencies and mode's shape with experimental data, obtained by Lang's technique. X-ray diffraction patterns of various crystallographic planes are analyzed, and the kinetic energy distribution mechanism of wave motions within a wide frequency range is investigated. According to the analysis of trapped-energy resonators, an X-ray technique for predicting the frequency, spectrum of contoured resonators with unknown geometry of the crystal plate, is presented. This is a consequence to the existing important relationships, relating to the disposition of the anharmonic frequencies for resonators, operating with trapped energy of the acoustical vibrations. The frequency spectrum analysis of resonators with different geometric contours shows that the X-ray diffraction topography predictions are in good agreement with experimental data.