3-28
The maximum Q of a resonator can be expressed as:
where f is the frequency in Hz, and t is an empirically determined “motional time constant” in seconds, which varies with the angles of cut and the mode of vibration.  For example, t = 1 x 10-14s for the AT-cut's c-mode (Qmax = 3.2 million at 5 MHz), t = 9.9 x 10-15s for the SC-cut's c-mode, and t = 4.9 x 10-15s for the BT-cut's b-mode.
Other factors which affect the Q of a resonator include:
l   Overtone l   Blank geometry (contour,
l   Surface finish          dimensional ratios)
l   Material impurities and defects l   Drive level
l   Mounting stresses l   Gases inside the enclosure
l   Bonding stresses                                       (pressure, type of gas)
l   Temperature l   Interfering modes
l   Electrode geometry and type l   Ionizing radiation
Factors that Determine Resonator Q
   The motional time constant  = R1C1 (= 10-14 s for an AT-cut) is related to the internal friction, i.e., to the attenuation of an acoustic wave as the wave travels in a resonator.  It is independent of frequency.
   See also “Ions in Quartz…” and “Internal Friction in Quartz” at the end of Chapter 5.


A. Ballato, "Piezoelectric Resonators," in B. Parzen, Design of Crystal and Other Harmonic Oscillators, pp. 66-122, John Wiley and Sons, Inc., 1983.

V. B. Braginsky, V. P. Mitrofanov & V. I. Panov, Systems with Small Dissipation, The University of Chicago Press, 1985.