SEM Observations of Mode Shapes of Flexural, Anharmonic, and Thickness-Shear Vibrations in Quartz Resonators

Abstract

Scanning electron microscopy (SEM) observations of mode shapes of flexural, anharmonic and thickness-shear modes of vibrations in rectangular and circular disk quartz resonators have been pre- sented. Flexural vibrations haw been observed propagatng both along and perpendicular to the x-axis of the crystal. In only high-frequency (30-MHz, third overtone) thin unbeveled circular disk resonators, evi- dence of circular flexural waves is found for the first time. Beveling of these resonators makes the flexural waves linear, and they propagate only along the x-axis. The studies suggest the immediate application in time and frequency standardization in using thicker quartz crystal res- onators for obtaining purer thickness vibrations relatively free from flexural components and ewcitng resonators in higher overtones to achieve high frequencies and higher stabilit) index. Crystal resonators (plates or disks) are capable of vibrating in different modes. Primarily, these modes can cause extensional. flexural, or shear (thickness or face) motions. In the actual practice of using bonded resonators, due to finite dimensions and reflection from boundaries, there results a complex elastic coupling of these different modes which gives rise to a host of frequencies that a crys- tal resonator is capable of vibrating at. A variety of experimental techniques (l) have been used for mode identification and also to suppress the undesired modes due to the underlying urge and the associated requirement of using a crystal resonator in the purest possible mode of vibration in frequency-control instrumentation. Among the new sophisticated experimental techniques. scanning electron microscopy (SEM) has been used for about a decade to study vibrations in quartz crystals (2)-(4). In the technique, an electron beam (usually accelerated to about 2 kV) is used to monitor the electrical potential distribution onto the resonator surface, which is due to its piezoelectric effect. The SEM micrographs thus exhibit a particular mode of vibrations of the crystal in operation. At high magnification (2000-5000X), where only a very small area of the surface is examined, the micrographs represent an uniform electric potential and the smearing in the patterns can be used either to measure the displacement or track the direction of vibrations. The SEM has been used in the recent past by Bahadur er d., (5)-(7) and Bahadur and Parshad (3), (4) to monitor the energy trapping and its absence in some resonator disks. Also, some com- plex SEM micrographs were reported occurring both for rectangu- lar plates and circular disks. In some cases of rectangular plates. a regular periodic band structure was attributed (l), 181 to represent appropriate overtones of flexural vibrations having their resonance frequency close to the fundamental thickness-shear vibrations. It was observed that the propagation of flexural vibrations was parallel to the x-axis of the crystal. In the measurements (l). (g) for the SEM examination of the crystal vibrations, a sizeable area of the crystal was unelectroded.