X-cut quartz

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. 

Constants determined from data reported in reference [68G05] and from the piezoelectric studies of X-cut quartz are shown in Table 4.3. The coefficients are found to be constant over the range of strain indicated. 

Fig. 4X When x-cut quartz is subjected to impact loading whose duration is less than wave transit time, an anomalous current pulse can be observed after the stress release. The diagram shows locations at which experiments were conducted and delineates the region of normal and anomalous response (after Graham and Ingram ([72G03]).

The linear piezoelectric tensor relating polarization and stress in x-cut quartz is... 

Fig. 5.2. Current-versus-time records for x-cut quartz impact loaded to stresses of 2.5, 3.9, 4.5, 5.9, 6.5, and 9.0 GPa are shown, illustrating the drastic changes occurring with mechanical yielding and conduction. Time increases from right to left. The current pulses are in the center of each record and are characterized by a brief horizontal trace (zero current before impact) followed by a rapid jump to a current value (after Graham [74G01]).

Interpretation of the experimental study of quartz leads to the conclusions that below 6 GPa and greater than 2.5 GPa x-cut quartz responds as an approximation to the elastic-dielectric model, but that there are very signifi-... 

Observations of current pulses from shock-loaded, x-cut quartz in the vicinity of and above the Hugoniot elastic limit provided rather remarkable confirmation of the nature of the phenomena resulting from mechanical yielding and shock-induced conduction. Lithium niobate provides another opportunity to test the generality of the models. 

These days the most common method employed for the generation and detection of ultrasound utilises the piezoelectric properties of certain crystals one of which is quartz [3]. A simplified diagram of a crystal of quartz is reproduced (Fig. 7.3) which shows three axes defined as x, y and z. If a thin section of this crystal is cut such that the large surfaces are normal to the x-axis (x-cut quartz) then the resulting section will show the following two complementary piezoelectric properties ... 

Figure 3.11 An acoustic interferometer of the type used in the author s laboratory (from Nethery [104]). A X-cut quartz crystal, 100-600 KHz B crystal support mount and aligning screws. Optical flat E is attached to a movable reflector D for generation of ultrasonic standing waves. Invar rod F position is read from precision micrometer slide L.

Figure 1.4. The piezoelectic effect as produced by an x-cut quartz. (Reproduced with permission of Wiiey-VCH, Ref [8].)...

Longitudinal and shear measurements are made separately with different sets of transducers. Quartz transducers are often used. Quartz crystals produce different types of vibrations depending on how they are cut. An X-cut quartz crystal is used for generating longitudinal waves. Y-cut or AC-cut quartz crystals are used to generate shear waves. 

Graham RA (1974) Shock-wave compression of x-cut quartz as determined by electrical response measurements. J Phys Chem Solids 35 355... 

Figure 7. Equipment geometries for studying the wave and acousto-electrical interactions in nematics (1) substrate (y cut, x oriented quartz), (2) glass plate, (3) interdigital transducer, (4) shear transducer (y cut quartz), (5) compression transducer (x cut quartz), (6) nematics, (7) mirror coating, (8) optically transparent electrode, (9) generator, (10) waveguide (substrate), (11) phase meter.

Many apparatuses for measuring ultrasonic absorption have been reported and the state of the art in ultrasonic absorption measurements can be found in Reference 44. Only the swept-frequency resonator and the pulse technique are still in use. The resonator consists of two parallel X-cut quartz crystal plates set at a fixed distance at the top and bottom of a cell filled with the investigated solution (Figure 2.6 top). The input transducer, driven by a swept-frequency... 

In the pulse technique, the two parallel X-cut quartz plates are at a variable distance. e Qng transducer sends pulses of ultrasound of frequency O) through the solution, where they are attenuated before reaching the second trans-... 

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