The mass loading effects of adsorbing and desorbing contaminant molecules on the magnitude and characteristics of frequency fluctuations in a thickness-shear resonator are studied. The study is motivated by the observation that the frequency of a thickness-shear resonator is determined predominantly by such mechanical parameters as the thickness of the resonator, elastic stiffnesses, mass loading of the electrodes, and energy trapping. An equation was derived relating the spectral density of frequency fluctuations to: (1) rates of adsorption and desorption of one species of contaminant molecules; (2) mass per unit area of a monolayer of molecules: (3) frequency constant; (4) thickness of resonator; and (5) number of molecular sites on one resonator surface. The induced phase noises were found to be significant in very-high-frequency resonators and are not simple functions of the percentage of area contaminated. The spectral density of frequency fluctuations was inversely proportional to the fourth power of the thickness if other parameters were held constant.
Three-dimensional linear equations of motion for small vibrations superposed on thermal deformations induced by steady, uniform temperature change in quartz are obtained. The material properties of quartz, such as the elastic stiffnesses and thermal expansion coefficients, are assumed temperature dependent and expressible by third-degree polynomials in temperature change. From the solutions of third-order perturbations of these equations for the thickness resonances of infinite quartz plates, six values of the effective third temperature derivatives of elastic stiffnesses C̃(3)pq are calculated by the use of the measured temperature coefficients of frequency by Bechmann, Ballato, and Lukaszek [Proc. IRE 50, 1812 (1962)] for various doubly rotated cuts and the values of the first temperature derivatives C(1)pq and the effective second temperature derivatives C̃(2)pq obtained in a previous study. An infinite system of two-dimensional equations of motion is derived by Mindlin’s method of power-series expansion for crystal plates subject to a steady, uniform temperature change. Four equations, governing the coupled thickness-shear, thickness-twist, thickness-stretch, and flexural vibrations, are extracted from the infinite set and employed to study the frequency-temperature behavior of thickness vibrations of finite SC-cut quartz plates with a pair of free edges. Changes in the thickness-shear resonance frequencies as a function of temperature are predicted and plotted for various values of orientation angles θ and φ, and length-to-thickness ratio a/b. Effects on the frequency-temperature behavior of the plates due to changes in the values of θ, φ, and a/b are observed and discussed.
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