Isochronism defect for various doubly rotated cut quartz resonators

It has been shown in earlier works that the amplitude-frequency effect [also called isochronism defect (ID) or anisochronism] could be a limitation factor on ultrastable oscillators. Theoretical studies based on the nonlinear theory of piezoelectricity have already been developed to explain the amplitude-frequency effect. So, it is possible to estimate the dependence of the ID versus various parameters of the resonator design (overtone rank, radius of curvature, electrodes diameter, etc.). However, because of the lack of available fourth-order elastic coefficients, it is not possible to predict the ID of any resonant frequency of a given trapped energy resonator. To tentatively find orientations of plates exhibiting a quasi-null ID, we have realized electroded resonators with different orientations and curvatures. We present results that verify, particularly, the R(-1/2) dependence of the amplitude-frequency effect versus radius of curvature. Moreover, we show that the ID can be positive or negative, that it can vary from one orientation to other one of about one order of magnitude, and that there exists a thermal compensated mode for which the amplitude-frequency effect is null.