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.