The analysis of unwanted-mode vibration patterns in AT-cut quartz oscillator crystals, revealed by X-ray diffraction topography. I. Interpretation of the X-ray diffraction topographs

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Abstract
It is shown that, by suitable design of the gold electrodes, it is possible to excite unwanted modes of plane-wave character and symmetry appropriate to the AT cut. The amplitude at each point has a spatial distribution which has been interpreted in terms of two superimposed components, one simply periodic in the X direction, the other more slowly varying. It is proved theoretically and experimentally that the unwanted mode vibrates at twice the frequency of the fundamental mode. The characteristics of both components of the unwanted mode are summarized.