Force-Frequency Coefficient of Singly Rotated Vibrating Quartz Crystals

Frequency perturbations in vibrating quartz crystals, caused by externally applied forces, have been investigated for some time. The parameters affecting the force-frequency sensitivity were recently established and evaluated, making possible the derivation of a normalized coefficient K
f
. An empirical chart, showing the coefficient K
f
for all singly rotated crystals, (yxl)θ, is presented for all directions of force in the azimuth angle ψ. The maximum force-frequency coefficient is found to be |K
f
| ≅ 30 × 10
⁻¹⁵
(msN
⁻¹
) for angles θ = +17°, +26.5° and +64°. A crystal cut (yxl) − 18.5° shows a constant coefficient K
f
= +5.3 × 10
⁻¹⁵
(msN
⁻¹
) for all angles of ψ. Coefficients K
f
(ψ) for BT-, Y- and AT-cut crystals are also shown on separate graphs. The force-frequency behavior of every crystal cut in the singly rotated Y-cut group can be determined from the chart. The chart is a useful tool for the design of frequency standards and precise force-sensing elements. More significantly, the chart provides normalized experimental data to form a basis for a theoretical treatment of the force-frequency effect.