Isochronism defect for various doubly rotated cut quartz resonators

It has been shown in earlier work, that the amplitude-frequency effect (also called isochronism defect or anisochronism) could be a limitation factor on ultra-stable oscillators. Theoretical studies based on the non-linear theory of the piezoelectricity have already been developed to explain the amplitude-frequency effect. So it is possible to estimate the dependence of the isochronism defect versus design parameters of resonators (radius of curvature, electrodes diameter, overtone rank, ...). However, due to the unknown of the fourth order elastic coefficients, it is not possible to predict isochronism defect of any resonant frequency of a given energy trapped resonator. To tentatively find the orientation of a plate which does not exhibit an isochronism defect, we have realised electroded resonators with different orientations and curvatures. We present results which verify, particularly, R^-1/2 dependence of amplitude-frequency effect versus radius of curvature. Moreover, we show that the isochronism defect can be positive or negative and can vary from one orientation to other one of about two orders of magnitude