Doubly rotated contoured quartz resonators

Doubly rotated contoured quartz resonators are used in the design of temperature-compensated stable clocks and dual-mode sensors for simultaneous measurements of pressure and temperature. The design of these devices is facilitated by models that can predict frequency spectra associated with the three thickness modes and temperature and stress-induced frequency changes as a function of crystalline orientation. The Stevens-Tiersten technique for the analysis of the C-mode of a doubly rotated contoured quartz resonator is extended to include the other two thickness modes. Computational results for harmonic and anharmonic overtones of all three thickness modes of such resonators help in optimizing the radius of curvature of the contour and electrode shape for suppression of unwanted modes and prevention of activity dips. The temperature and stress-induced changes in thickness-mode resonator frequencies are calculated from a perturbation technique for small dynamic fields superposed on a static bias. The static bias refers to either a temperature or stress-induced static deformation of the resonator plate. Phenomenological models are also used for calculating the temperature and stress-induced changes in resonant frequencies as a function of crystalline orientation. Results for the SBTC-cut quartz plate with a spherical convex contour of 260 mm indicate that normal trapping occurs for the third (n=3) and fifth (n=5) harmonic of the A-mode, the fundamental (n=1) and third (n=3) harmonic of the B-mode, and the fundamental (n=1) and fifth (n=5) harmonic of the C-mode     

Doubly rotated quartz resonators with a low amplitude-frequencyeffect: the LD-cut

This article deals with a crystal quartz cut with virtually no amplitude-frequency effect (also called "isochronism defect" or "anisochronism"). The cut, called LD-cut (Low isochronism Defect), is set in a BVA-type structure for later use in an Ultra-Stable Oscillator (USO). Various design parameters are presented, along with the properties of the resonator. The resonance frequencies of various modes, the temperature dependence, the motional parameters and the phase noise in relation to the power supplied to the resonator are mainly studied. These properties are compared to those of the SC-cut resonators     

Isochronism defect for various doubly rotated cut quartz resonators

It has been shown in earlier works that the amplitude-frequency effect [also called isochronism defect (ID) or anisochronism] could be a limitation factor on ultrastable oscillators. Theoretical studies based on the nonlinear theory of piezoelectricity have already been developed to explain the amplitude-frequency effect. So, it is possible to estimate the dependence of the ID versus various parameters of the resonator design (overtone rank, radius of curvature, electrodes diameter, etc.). However, because of the lack of available fourth-order elastic coefficients, it is not possible to predict the ID of any resonant frequency of a given trapped energy resonator. To tentatively find orientations of plates exhibiting a quasi- ID, we have realized electroded resonators with different orientations and curvatures. We present results that verify, particularly, the R/sup -1/2/ dependence of the amplitude-frequency effect versus radius of curvature. Moreover, we show that the ID can be positive or negative, that it can vary from one orientation to other one of about one order of magnitude, and that there exists a thermal compensated mode for which the amplitude-frequency effect is .     

Solidly mounted ZnO shear mode film bulk acoustic resonators for sensing applications in liquids

Solidly mounted film bulk acoustic resonators (FBAR) operating at 850 MHz in the shear vibration mode have been fabricated. C-axis inclined zinc oxide (ZnO) thin films realized by modified reactive magnetron sputtering were used: Coupling factors k2 of 1.7% and Q-factors of 312 were determined in air. Q-factors of 192 were measured in water, making these devices attractive for sensing applications in liquids, e.g., biosensing.     

Viscoelastic properties of low-viscosity liquids studied with thickness-shear mode resonators

The network analysis method was applied to AT cut quartz blanks (f(0) = 10 MHz), which were loaded with liquids of low and medium viscosity (water, methanol, ethanol, 1-propanol, 1-butanol, glycerol solutions). The shift of the resonance frequency Δf could be separated into a term due to rigidly coupled mass Δf(rig) and a term due to viscous damping Δf - Δf(rig). From the difference Δf - Δf(rig) and the broadening of the resonance curve, the complex shear modulus G = G' + iωη(L) was calculated. The viscosity coefficients η(L) are in good agreement with literature data. As G' > 0, it can be concluded that the examined fluids also reveal elasticity at shear frequencies in the MHz range. For the low-viscosity liquids, elastic contributions resulting from collective interactions of molecules are measurable but small and neglectable in most applications. The medium viscous liquid glycerol (98%) begins to exhibit considerable elasticity, resulting from the relaxation of separate molecules.     

Organic vapor sensing with ionic liquids entrapped in alumina nanopores on quartz crystal resonators

We report on a concept for vapor sensing with the quartz crystal microbalance where the vapor phase is absorbed into small droplets of an ionic liquid. The liquid is contained in the pores of a nanoporous alumina layer, created on the front electrode of the quartz crystal by anodization. Ionic liquids are attractive for vapor sensing because--being liquids--they equilibrate very fast, while at the same time having negligible vapor pressure. Containing the ionic liquids inside cylindrical cavities of a solid matrix removes all problems related to the liquid's softness as well as the possibility of dewetting and flow. The absence of viscoelastic effects is evidenced by the fact that the bandwidth of the resonance remains unchanged during the uptake of solvent vapor. The Henry constants for a number of solvents have been measured.     

Analysis of thickness modes of contoured doubly rotated, quartz resonators

It is the intent of this work to provide a working resource for calculating all three mode families of a doubly rotated, contoured quartz resonator. It is shown that the theoretical development of Stevens and Tiersten [1986] can be used for this purpose. Their approach uses a transformation of the mechanical displacement vector to the eigenvector triad of the pure thickness solution. The solution methodology here reorganizes the transformation matrix Q in their formulation to calculate the other two mode families. Calculations compare well with experimental results for the three mode families of an SC-cut crystal and an FC-cut crystal and with published calculations for the SBTC-cut mode family with major displacement along the as blank axis. The key constants for the SC-cut are presented for workers to use in the future. In addition, the equations of motion and boundary conditions are derived for the two additional mode families using assumptions parallel to those used by Stevens and Tiersten. Calculations with these equations are presented for completeness to support the present conclusions by showing the equivalence of either simply reorganizing the Q matrix or using separate equations for each of the three mode families     

Stress-induced frequency shifts in rotated Y-cut langasite resonators with electrodes considered

Stress-induced frequency shifts of the thickness-shear vibration of rotated Y-cut langasite resonators are studied with the electrode considered. The relationship between mass loading and force-frequency coefficients is explored for several kinds of electrode materials. Comparisons between Y-cut langasite and AT-cut quartz resonators are also presented.     

Determination of the shear impedance of viscoelastic liquids using cylindrical piezoceramic resonators

In this paper, a new method for determining the rheological parameters of viscoelastic liquids is presented. To this end, we used the perturbation method applied to shear vibrations of cylindrical piezoceramic resonators. The resonator was viscoelastically loaded on the outer cylindrical surface. Due to this loading, the resonant frequency and quality factor of the resonator changed. According to the perturbation method, the change in the complex resonant frequency /spl Delta/~/spl omega/ = /spl Delta/w/sup re/ + j/spl Delta//spl omega//sup im/ is directly proportional to the specific acoustic impedance for cylindrical waves Zc of a viscoelastic liquid surrounding the resonator, i.e., /spl Delta/~w /spl sim/ jZ/sub c/, where j = (-1)/sup 1/2/. Hence, the measurement of the real and imaginary parts of the complex resonant frequency determines the real part, R/sub c/, and imaginary part, X/sub c/, of the complex acoustic impedance for cylindrical waves Z/sub c/ of an investigated liquid. Further-more, the specific impedance Z/sub L/ for plane waves was related to the specific impedance Z/sub c/ for cylindrical waves. Using theoretical formulas established and the results of the experiments performed, the shear storage modulus /spl mu/ and the viscosity /spl eta/ for various liquids (e.g., epoxy resins) were determined. Moreover, the authors derived for cylindrical resonators a formula that relates the shift in resonant frequency to the viscosity of the liquid. This formula is analogous to the Kanazawa-Gordon formula that was derived for planar resonators and Newtonian liquids.     

Optimal electrode shape and size of a few singly rotated quartz and langasite resonators

Based on Mindlin's early work, we calculate and plot optimal electrode shape and size of Y-cut quartz, At-cut quartz, and Y-cut langasite plate thickness-shear resonators. The electrodes obtained are optimal in that they satisfy Bechmann's number in every direction. The results are useful in the design optimization of these resonators.     

Measured properties of doubly rotated dilithium tetraborate (Li(2)B(4)BO(7)) resonators and transducers

Dilithium tetraborate plates, cut to the nominal angles of cut of (YXwl)40 degrees /33 degrees (TA-cut) and (YXwl)19 degrees /56 degrees (TC-cut), have been supplied by a crystal grower and have been processed into plate resonators. The mode spectra and electrical circuit parameters of the resonators have been examined for the fundamental, third, and fifth harmonics for all three thickness modes of the resonators. The observed values of frequency constants and piezoelectric coupling are in good agreement with the predicted values. The frequency-temperature behavior of the zero-temperature coefficient cuts has also been measured.     

Analysis of piezoelectric bulk-acoustic-wave resonators as detectors in viscous conductive liquids

An analytical solution for the resonance condition of a piezoelectric quartz resonator with one surface in contact with a viscous conductive liquid is presented. The characteristic equation that describes the resonance condition and accounts for all interactions including acoustoelectric interactions with ions and dipoles in the solution is obtained in terms of the crystal and liquid parameters. A simple expression for the change in the resonance frequency is obtained. For viscous nonconductive solutions, the frequency change is reduced to a relationship in terms of the liquid density and viscosity. For dilute conductive liquid, the change in frequency is derived in terms of the solution conductivity and dielectric constant. The boundary conditions for the problem are defined with and without the electrical effects of electrodes. Experiments were conducted with various viscous and conductive chemical liquids using a fabricated miniature liquid flow cell containing an AT-cut quartz crystal resonator. The results, which show good agreement with the theory, on the use of quartz crystal resonators as conductivity and/or viscosity sensors are reported.     

Cavity-ring-down principle for fiber-optic resonators: experimental realization of bending loss and evanescent-field sensing

A novel measurement principle for fiber-optic sensing is presented. Use of a cavity-ring-down scheme enables measurements of minute optical losses in high-finesse fiber-optic cavities. The loss may be induced by evanescent-field absorption, fiber bending, fiber degradation, Bragg gratings, or any other effect that might change the fiber transmission or cavity reflector properties. The principle is proved to be rather insensitive to ambient perturbations such as temperature changes. A high-sensitivity measurement of loss due to bending is presented as a proof-of-principle. With a cavity finesse of 627 a sensitivity for induced loss of 108 ppm (4.68 x 10(-4) dB) is achieved. Preliminary measurements of evanescent-field absorption are also discussed.     

Simulation and optimization of polymer-coated microsphere resonators in chemical vapor sensing

This study presents a chemical vapor sensor based on polymer-coated microsphere resonators. A theoretical simulation of the sensor response is performed, and optimization of the polymer layer thickness is investigated. Results show that the sensor exhibits a good linearity and a low detection limit of the refractive index change. Especially at the thermostable thickness of the polymer layer, the refractive index detection limit of the wavelength around 780 nm can be as low as ~2×10?? refractive index unit for a spectral resolution of 10 fm, without any temperature control. Because of the good sensing performance and simple manipulation, the proposed sensor is a very promising platform for chemical vapor detections.     

Theoretical and experimental mass-sensitivity analysis of polymer-coated SAW and STW resonators for gas sensing applications

Polymer-coated surface transverse waves (STW) resonators have recently been successfully studied for organic gas sensing applications. The first results indicate increased absolute and even relative sensitivity as compared to similar resonators with surface acoustic waves (SAW). However, the gain in sensitivity is accompanied by the adverse effect of an increased attenuation and the advantage frame is difficult to establish quantitatively. In this paper, a new set of experimental samples with Parylene C-coated quartz substrates are studied. The samples are matched in frequency and wavelength. The results are compared and the obtained features explained using available theoretical algorithms for analyzing layered SAW and Love configurations, and a recently developed STW algorithm. The approximate limits of advantageous applicability of the STW resonator gas sensors are discussed.     

The application of two- and three-layer dielectric resonators tothe investigation of liquids in the microwave region

Measurement methods for electrical permittivity and loss tangent of dielectric liquids are presented. The methods utilize dielectric resonators with either a two- or three-layer structure. The techniques are analyzed from the point of view of measurement optimization. The results of measurements of liquid samples are presented     

Wireless sensing using oscillator circuits locked to remote high-Q SAW resonators

This paper introduces a method of wireless read out of high Q surface acoustic wave (SAW) resonator sensors. The resonator is excited by a short RF pulse and decays after switching off the interrogating signal. In the measurement system, a gated phase locked loop (GPLL) locks to the resonance frequency of the SAW resonator within a few bursts. Then the frequency of the GPLL oscillator is synchronized to the resonance of the sensor and can be measured easily. The concept is intended to yield an alternative to interrogators with expensive signal processing. Considering the inherent limitations, the proposed system presents a low cost solution for temperature, force, torque, etc. measurements. We describe the sensors, the signals, and the implemented system. Results of temperature measurements using quartz resonators are presented, and merits and disadvantages are discussed.     

Application of electrostatic pull-in instability on sensing adsorbate stiffness in nanomechanical resonators

Recent theoretical and experimental research showed that the response of micro/nanocantilevers to detect materials is not always simply related to extra mass. Stiffness of adsorbates and surface stress-induced changes in the stiffness, arising from adsorption, can produce frequency shifts that are several times greater in magnitude than those induced by mass loading. Consequently, the calculated adsorbed mass does not fully represent the real adsorbed mass, making the measurements qualitative. Therefore, a proper method for measuring the stiffness of adsorbed layers has to be combined with resonance frequency measurement to quantitatively analyze changes in both the mass and the stiffness. This letter presents the theory for application of electrostatic pull-in instability for measuring the stiffness of adsorbates at the surface of cantilever resonators.