Quartz, cuts

In the approximation Eq. 4, the transformation factor l/(4 /c) does not contain wave velocity or coupling coefficient and is therefore independent of the quartz cut neither does it include the frequency. The transformation factor is real and constant if the electrode radius-thickness ratio is constant. Therefore the real part of Zl directly determines the real part of Zmi. whereas the imaginary part of Zl defines the imaginary part of Zmi- In a narrow frequency range one can represent Re(Zm) as Rl and Im(ZL) as cuLl. In general, Rl = Rl(oo) and II = Tl( ) remain frequency-dependent due to Zl = Zl(c<>) (not linear). 

Temperature dependence of the piezoelectric du and du coefficients is displayed in Fig. 7.4. The piezoelectric coefficient of X-cut and XY-cut (it is d 2 = —du for quartz) for transversal effect is changing significantly (decreasing) with increasing temperature. Second independent piezoelectric coefficient du contributes in an opposite sense (see Fig. 7.4). It makes possible to find special quartz cut orientation with compensated temperature dependence in transversal mode. Piezoelectric coefficient is independent on the temperature for such cut practically in certain limited temperature range. 

Calculation of the desired crystal cut orientation is simple. Let us suppose quartz cut (XYa)f (working in transversal mode) with the lengthy-axis rotated by an angle I about the crystallographic x-axis. The rotated components of the piezoelectric tensor can be calculated using Eqs. (5.10) and (5.15)... 

Figure 59. Wave properties associated with several different types of quartz cuts 260) see also Figure 48...

Parrish, W., and Gordon, S.G. Precise angular control of quartz-cutting by X-rays. 

QMB sensors are placed in cylindrical cavities obtained by wet/dry etching procedures. Fig. 2 shows the quartz-substrate system together with the quartz metallization contacts (electrodes 1 and 2). Three peripheral points of the bottom surface (S2) of each quartz rest on the substrate (among them only Pj and P2 are shown in the figure). Point P2 electrically connects the bottom quartz electrode. In this arrangement a suitable quartz-cut orientation still allows the quartz structure to oscillate. Contacts Ci and C2 are then connected to the amplifiers which are integrated on the same substrate. 

Other Industrial Applications. High pressures are used industrially for many other specialized appHcations. Apart from mechanical uses in which hydrauhc pressure is used to supply power or to generate Hquid jets for mining minerals or cutting metal sheets and fabrics, most of these other operations are batch processes. Eor example, metallurgical appHcations include isostatic compaction, hot isostatic compaction (HIP), and the hydrostatic extmsion of metals. Other appHcations such as the hydrothermal synthesis of quartz (see Silica, synthetic quartz crystals), or the synthesis of industrial diamonds involve changing the phase of a substance under pressure. In the case of the synthesis of diamonds, conditions of 6 GPa (870,000 psi) and 1500°C are used (see Carbon, diamond, synthetic). 

Of all the piezoelectric crystals that are available for use as shock-wave transducers, the two that have received the most attention are x-cut quartz and lithium-niobate crystals (Graham and Reed, 1978). They are the most accurately characterized stress-wave transducers available for stresses up to 4 GPa and 1.8 GPa, respectively, and they are widely used within their stress ranges. They are relatively simple, accurate gauges which require a minimum of data analysis to arrive at the observed pressure history. They are used in a thick gauge mode, in which the shock wave coming through the specimen is... 

The contribution to the stress from electromechanical coupling is readily estimated from the constitutive relation [Eq. (4.2)]. Under conditions of uniaxial strain and field, and for an open circuit, we find that the elastic stiffness is increased by the multiplying factor (1 -i- K ) where the square of the electromechanical coupling factor for uniaxial strain, is a measure of the stiffening effect of the electric field. Values of for various materials are for x-cut quartz, 0.0008, for z-cut lithium niobate, 0.055 for y-cut lithium niobate, 0.074 for barium titanate ceramic, 0.5 and for PZT-5H ceramic, 0.75. These examples show that electromechanical coupling effects can be expected to vary from barely detectable to quite substantial. 

Typical current pulses observed for x-cut quartz, z-cut lithium niobate, and y-cut lithium niobate are shown in Fig. 4.3. Following a sharp rise in current to an initial value (the initial rise time is due to tilt, misalignment of the impacting surfaces), the wave shapes show either modest increases in current during the wave transit time for quartz and z-cut lithium niobate... 

Fig. 4.3. Typical normalized piezoelectric current-versus-time responses are compared for x-cut quartz, z-cut lithium niobate, and y-cut lithium niobate. The y-cut response is distorted in time due to propagation of both longitudinal and shear components. In the other crystals, the increases of current in time can be described with finite strain, dielectric constant change, and electromechanical coupling as predicted by theory (after Davison and Graham [79D01]).

Fig. 4.4. The piezoelectric charge produced by elastic strain in x-cut quartz and z-cut lithium niobate is well represented by a quadratic relationship without a need for fourth-order contributions.

The measured relationships between piezoelectric polarization and strain for x-cut quartz and z-cut lithium niobate are found to be well fit by a quadratic relation as shown in Fig. 4.4. In both materials a significant nonlinear piezoelectric effect is indicated. The effect in lithium niobate is particularly notable because the measurements are limited to much smaller strains than those to which quartz can be subjected. The quadratic polynomial fits are used to determine the second- and third-order piezoelectric constants and are summarized in Table 4.1. Elastic constants determined in these investigations were shown in Chap. 2. 

In the case of x-cut quartz there is excellent agreement between second-order constants determined in the shock-compression studies and ultrasonic... 

Fig. 4.5. The degree of approximation for the increase of current in time for uncoupled and weakly coupled solutions for impact-loaded, x-cut quartz and z-cut lithium niobate is shown by comparison to the numerically predicted, fully coupled case. In the figure, the initial current is set to the value of 1.0 at the measured value (after Davison and Graham [79D01]).

The ratio of third- to second-order piezoelectric constants has also been determined for x-cut quartz with the acceleration pulse loading method [77G05]. Two experiments yielded values for Cm/Cu of 15.0 and 16.6 compared to the ratio of 15.3 [72G03] determined from the fit to the 25 shock loading experiments. 

The determination of piezoelectric constants from current pulses is based on interpretation of wave shapes in the weak-coupling approximation. It is of interest to use the wave shapes to evaluate the degree of approximation involved in the various models of piezoelectric response. Such an evaluation is shown in Fig. 4.5, in which normalized current-time wave forms calculated from various models are shown for x-cut quartz and z-cut lithium niobate. In both cases the differences between the fully coupled and weakly coupled solutions are observed to be about 1%, which is within the accuracy limits of the calculations. Hence, for both quartz and lithium niobate, weakly coupled solutions appear adequate for interpretation of observed current-time waveforms. On the other hand, the adequacy of the uncoupled solution is significantly different for the two materials. For x-cut quartz the maximum error of about 1%-1.5% for the nonlinear-uncoupled solution is suitable for all but the most precise interpretation. For z-cut lithium niobate the maximum error of about 8% for the nonlinear-uncoupled solution is greater than that considered acceptable for most cases. The linear-uncoupled solution is seriously in error in each case as it neglects both strain and coupling. 

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