Force-Frequency Coefficient of Singly Rotated Vibrating Quartz Crystals

Abstract

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 <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</inf> . An empirical chart, showing the coefficient K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</inf> 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 <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</inf> | ≅ 30 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−15</sup> (msN <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> ) for angles θ = +17°, +26.5° and +64°. A crystal cut (yxl) − 18.5° shows a constant coefficient K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</inf> = +5.3 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−15</sup> (msN <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> ) for all angles of ψ. Coefficients K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</inf> (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.