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Calculation of incoherent quantum 1/f frequency fluctuations in low-Q BAW and SAW quartz resonators

Quantum 1/f frequency fluctuations arise both in bulk acoustic wave and surface acoustic wave quartz resonators from the presence of the quantum 1/f effect in the process rates which remove phonons from the main oscillator mode. Therefore, these fluctuations come from the quantum 1/f noise present in the dissipation rate. The present paper includes the incoherence effects caused by the finite "size" of the phonons, measured by their finite intrinsic coherence length. The result of this calculation is a non-monotonic dependence of the 1/f frequency noise on resonator volume, marked first by a linear increase, and followed by a inverse proportionality with volume, as noticed by T.E. Parker at NIST. The I/V dependence is similar to the behavior noticed for current carriers in semiconductors and metals, where the coherence length refers to electrons and is much smaller.