Quartz crystal resonator noise and fluctuation-dissipation theorem considerations

Recent experimental results on the short term stability of quartz crystal oscillators have reached or maybe exceeded the theoretical estimates for the level of the intrinsic 1/f noise in a quartz crystal resonator. Hence we reconsider these theoretical estimates, trying to get more accurate values and also discuss their validity. A possible explanation for the discrepancies between Planat's model and experiment is proposed, whereas the validity of the numerical applications published using Handel's model for quartz resonators is questioned. Since the most numerically reasonable candidate may be derived from Handel's theory, we consider the geometrical dependence of the limit set by this theory and show that the level of intrinsic 1/f noise should decrease as the inverse of the third power of the thickness of the resonator. In a second part of this paper, we use the fluctuation dissipation theorem to numerically evaluate the importance of the contribution of thickness fluctuations on the level of noise, at low frequencies.