Rotated Y-cut Quartz Plate Research Articles

The properties of thickness-twist (TT) wave modes in a rotated Y-cut quartz plate with a functionally graded material top layer

We propose the use of thickness-twist (TT) wave modes of an AT-cut quartz crystal plate resonator for measurement of material parameters, such as stiffness, density and material gradient, of a functionally graded material (FGM) layer on its surface, whose material property varies exponentially in thickness direction. A theoretical analysis of dispersion relations for TT waves is presented using Mindlin’s plate theory, with displacement mode shapes plotted, and the existence of face-shear (FS) wave modes discussed. Through numerical examples, the effects of material parameters (stiffness, density and material gradient) on dispersion curves, cutoff frequencies and mode shapes are thoroughly examined, which can act as a theoretical reference for measurements of unknown properties of FGM layer.

Shear-horizontal waves in a rotated Y-cut quartz plate on an elastic half space

We show the existence of certain Love-type surface waves propagating in an anisotropic elastic plate of rotated Y-cut quartz bonded to an isotropic elastic half space. A transcendental frequency equation that determines the dispersion relations of the waves is obtained, which reduces to the classical frequency equation for Love waves in isotropic elasticity when the anisotropy quartz plate is replaced by an isotropic elastic plate. The frequency equation is solved using MATLAB. Multiple branches of dispersion curves are found and plotted. Displacement profiles are also plotted, showing the penetration depth of the waves into the half space.

Shear-horizontal waves in a rotated Y-cut quartz plate with an isotropic elastic layer of finite thickness

We study shear-horizontal (SH) waves in a rotated Y-cut quartz plate carrying an isotropic elastic layer of finite thickness. The three-dimensional theories of anisotropic elasticity and isotropic elasticity are used for the quartz plate and the elastic layer, respectively. A transcendental frequency equation that determines the dispersion relations of the waves is obtained. The dispersion relations are obtained and plotted by solving the frequency equation using MATLAB. Approximate dispersion relations are also obtained analytically for two special cases. One is for long waves whose wavelength is much larger than the plate thickness. The other is for the case of a very thin elastic layer. The effects of the elastic layer on the dispersion relations are examined. The results obtained are fundamental and useful to acoustic wave sensors for measuring the mechanical and geometric properties of the elastic layer.

Frequency shifts in plate crystal resonators induced by electric, magnetic, or mechanical fields in surface films

We study frequency shifts in plate crystal resonators with surface films. The films are multiphysical, including the effects of inertia, stiffness, intrinsic stress, piezoelectric coupling, and piezomagnetic coupling. Mindlin's two-dimensional equations for a crystal plate with two elastic surface films are generalized to include the multiphysical effects of the films. They are used to study thickness-shear vibrations of a rotated Y-cut quartz plate with initial fields resulting from the mechanical, electric, and magnetic fields in the surface films. Frequency shifts caused by the initial fields are calculated and examined. Results show that plate crystal resonators with multiphysical surface films may be used for electric/magnetic field sensing.

Shear-horizontal waves in a rotated Y-cut quartz plate in contact with a viscous fluid

We study shear-horizontal (SH) waves in a crystal plate of rotated Y-cut quartz in contact with a semi-infinite viscous fluid. The crystal plate and the fluid are governed by the equations of anisotropic elasticity and the theory of Newtonian fluids. A transcendental equation that determines the dispersion relations of the waves is obtained. Approximate analytical solutions to the equation are presented for the case of low viscosity fluids and the case of long waves whose wavelength is much larger than the plate thickness. The effects of the fluid viscosity and density on the dispersion relations of the waves are examined. The results obtained are fundamental and useful to the understanding and design of acoustic wave fluid sensors for measuring fluid viscosity or density.

Shear-horizontal waves in a rotated Y-cut quartz plate with an imperfectly bonded mass layer

We study shear-horizontal (SH) waves in an unbounded plate of rotated Y-cut quartz carrying a thin mass layer imperfectly or nonrigidly bonded to the surface of the quartz plate. The imperfect interface is described by the socalled shear-lag model that allows the displacement to be discontinuous across the interface. A transcendental frequency equation that determines the dispersion relations of the waves is obtained. Exact and approximate solutions to the frequency equation are presented. The effects of the mass layer and the imperfect interface on the dispersion relations are examined. A quantitative criterion is given which distinguishes whether the combined effect of the mass layer and the imperfect interface raises or lowers the wave frequencies.

Thickness-shear vibration of rotated Y -cut quartz plates with unattached electrodes and asymmetric air gaps

Thickness-shear vibration modes of rotated Y-cut quartz plates with unattached electrodes and asymmetric air gaps are analyzed. An exact solution is obtained from the equations of piezoelectricity. Numerical results calculated from the solution show that the resonant frequencies are sensitive to the dimensions of the air gaps. This offers new design parameters for better resonator performance.

Thickness-shear vibrations of rotated Y-cut quartz plates with imperfectly bonded surface mass layers

A solution is obtained from the three-dimensional equations of linear piezoelectricity for the pure thickness-shear vibration of rotated Y-cut quartz or langasite plates with imperfectly bonded surface mass layers. The solution includes a few results in the literature as special cases. It is shown that the mass layers lower the resonant frequencies when they are relatively perfectly bonded to the crystal surface, and that loosely bonded mass layers may raise the frequencies. The results are useful in the analysis of frequency stability of quartz resonators and acoustic wave sensors.

Thickness-shear vibration of rotated Y-cut quartz plates with relatively thick electrodes of unequal thickness

An exact solution is obtained from the three-dimensional equations of linear piezoelectricity for thickness-shear vibrations of rotated Y-cut quartz plates with relatively thick electrodes of unequal thickness at its major faces. Effects of the shear stiffness of the electrodes on resonant frequencies are examined.

A corrected modal representation of thickness vibrations in quartz plates and its influence on the transversely varying case.

A modal representation of the thickness vibrations of rotated Y-cut quartz plates, which was used in the treatment of driven transversely varying thickness modes, is shown to be defective in certain respects. The differential equations and edge conditions for transversely varying thickness modes have been used in the accurate treatment of trapped energy resonators, monolithic crystal filters, and contoured quartz resonators, even though those defects were present. In this work those defects in the thickness solution are corrected along with the influence on the differential equations and edge conditions in the transversely varying case. The corrected modal representation shows that, because in practical applications to the above mentioned devices, the driving frequency is always near a thickness resonant frequency, essentially the same results will be obtained with the corrected representation as were obtained with the defective one, which explains why the results obtained with the defective equations were so accurate.

Twinning of a Quartz Plate at Low Temperature Using a Laser Beam

We have previously reported an artificial twinning (x-axis inversion) technique that involves the thermal treatment method and the laser beam scanning method for a rotated Y-cut (RY-cut) quartz plate. The heating temperature for twinning was about 520–540°C using the thermal treatment method. Furthermore, in the laser beam scanning method, a high preheating temperature of about 480°C was necessary to obtain a well-controlled twinning pattern. We present here twinning at low temperature (350°C ∼ room temperature) using the laser beam scanning method. This method can be used the potential to form an arbitrary fine twinning pattern at room temperature. High reproducibility of a fine twinning pattern was obtained at a preheating temperature of 350°C. A relatively good twinning pattern was also obtained by the laser beam scanning method without preheating. It was estimated that the shear strain caused by laser irradiation was of the order of 10-4, for the case there the twinning occurred. The large temperature gradient produced in the plate by laser beam irradiation drastically reduced the treatment temperature.

Formation of Artificial Twinning Quartz Plate with x-axis Inversion Area by Laser Beam Irradiation

We have previously reported an artificial twinning (x-axis inversion) technique by thermal treatment for a rotated Y-cut (RY-cut) quartz plate. However, it is impossible to form an arbitrary x-axis inversion pattern on a quartz plate using this technique. If the x-axis inversion pattern is artificially controlled, it is possible to produce new piezoelectric and/or optical devices. Twinning phenomenon is induced as follows. A metal film (i.e., Cr or NiCr) is deposited on the top surface of a rotated Y-cut quartz plate. Thermal treatment of the plate at about 520–540°C produces x-axis inversion of the quartz under the metal film. The x-axis inversion occurs due to stress that results from the difference in the thermal expansion between the quartz plate and the metal film. This paper presents a new twinning technique using laser scanning that has the potential to form an arbitrary twinning pattern. A quartz plate is placed on a movable stage in a vacuum chamber. On the plate surface, a metal film is deposited and the plate is heated at around 200–500°C. A laser beam is irradiated onto the plate. The metal film on the quartz plate absorbs the light energy and the plate is locally heated. This causes strong mechanical strain and twinning occurs around the irradiated area. As a result, a twinning area with an arbitrary pattern can be produced by scanning the laser spot along the metal film deposited on a RY-cut quartz plate. It was clarified that temperature gradient in the quartz plate is an important factor for twinning. Large temperature gradient produced in the plate by the laser beam irradiation drastically reduces the treatment temperature.

Influence of electrodes arrangement on the quartz resonator equivalent circuit

The strip electrodes deposited on the main surfaces of the rotated Y-cut quartz plate in the Z' direction and overlapping only in the central part of the plate excite a non-negligible electric field parallel to the surface of the plate. The influence of this field on the parameters of the electrical equivalent circuit and frequency-temperature characteristic of the AT-cut quartz plates is examined. The influence of this electric field parallel to the surface depends on the orientation of electrodes in the rectangular coordinate system of quartz.

Piezoelectric Resonance Properties of a β Phase Quartz Plate

We have previously reported an artificial twinning (x-axis inversion) technique for rotated Y-cut quartz plates. The twinning is produced by mechanical stress through a thin metal film deposited on the plate. Such a twinning phenomenon occurs at a considerably lower temperature than the α-β transition temperature, Tc, of 573°C. This paper presents the electromechanical properties of quartz at a temperature region higher than the twinning temperature. Degeneration of the piezoelectric resonances corresponding to the x-axis inversion area and the non-inversion area are observed at a temperature between the twinning temperature and Tc. The elastic stiffness constants c55 and c66, estimated from the degeneration mode resonance frequency, agree with these for the β phase. This suggests that the α-β transition occurs by the thin metal film at a considerably lower temperature than Tc of 573°C. An intense piezoelectric response with a high quality factor above 20,000 is observed from the degeneration temperature to above 660°C.

Bechmann's Number for Harmonic Overtones of Thickness/Twist Vibrations of Rotated-Y-Cut Quartz Plates

Two methods are described for calculating Bechmann's number for harmonic overtones of thickness/twist modes in rotated Y-cut quartz plates: one based on an exact solution of the equations of elasticity and the other based on an asymptotic form of the exact solution. The latter leads to an explicit formula for Bechmann's number and it is shown to give results very close to the former—which requires extensive computations.

Thickness Vibrations of Piezoelectric Plates

The thickness vibrations of an infinite anisotropic plate with electrodes coated on both surfaces are investigated using the linear piezoelectric equations. In general, the three fundamental solutions of the differential equations are found to couple at the traction-free surfaces of the plate. The solution reduces to the well-known uncoupled solution for the purely elastic case when the piezoelectric constants are zero. The transcendental equation which determines the resonant frequencies is derived. The theory is applied to a ferroelectric ceramic in two special cases; one with the plate polarized in the thickness direction, and the other with the plate polarized in its plane. The theory is also applied to a rotated Y-cut quartz plate. In all of these special cases the three solutions of the differential equations uncouple. Two of these solutions are identical with the purely elastic solutions and cannot be excited electrically, whereas the third differs from the purely elastic solution and can be excited electrically. The frequency equation obtained from this piezoelectric solution shows that the resonant frequencies are not integral multiples of the fundamental.