Low temperature limitation on the quality factor of quartz resonators

The quality factor (Q) for different resonators driven at several overtones has been determined between 1.5 and 300 K. These measurements give an improved interpretation of the Q-factor limitation. A significant consequence is a better understanding of the relationship between Q and random frequency fluctuations. The curves of 1/Q=F(T) show the usual features between 20 and 300 K. These include the sodium ion peak at 55 K as well as a peak at 20 K. However, the region of principal interest lies between 1.5 and 20 K. At very low temperatures, a plateau is always observed. If for a given resonator, the 1/Q value of this plateau is subtracted from the experimental values, the residual 1/Q is almost a linear function of T/sup 4/. This variation obeys the Landau-Rumer theory of acoustic wave absorption caused by phonon-phonon interaction in the crystal. Thus the authors conclude that between 6 and 20 K the main limitation in Q is due to the crystal itself, but the plateau is not caused by intrinsic crystal properties. Data for different overtones enable the elimination, at least for the seventh overtone, of damping effects of the crystal supports. Measurements have also been carried out on crystals with crystal surfaces with different roughness characteristics. These roughness characteristics have been measured using optical interferometry. With a well polished, good crystal the authors measured, for the seventh overtone at 3 K a loaded Q of 25 million corresponding to an unloaded Q of 33 million. Better control of surface roughness and crystal quality could lead to a Q of about 100 million in the liquid helium region. This value would be for an 11 MHz, seventh overtone crystal having an diameter of 15 mm.<
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