Quartz crystals, shape

Fig. 6.4 shows the frequency behavior of a quartz crystal shaped as in Fig. 6.3. The ordinate represents the amplitude of the oscillation or also the current flowing through the crystal as a function of the frequency on the abscissa.

FIGURE 27.20 (a) Top and edge views of quartz crystal with keyhole-shaped vapor-... 

Six years later Sir John Herschel 13> observed that the algebraic sign of the optical rotation of quartz crystals could be correlated with their shape, and the dextro- and levorotatory crystals look like mutual mirror images. This correlation was extended from crystals to molecules by Pasteur 14>, who postulated that the spatial arrangement of atoms in the molecules is responsible for their optical activity, and the dextro-and levorotatory molecules are in a mirror image relation. 

Another problem is an uncertainty involved in the estimation of the double-layer thickness. This thickness is often calculated from the size of the solvent molecule, using macroscopic data (e.g., the molar volume) under a doubtful assumption about the shape of the molecule, which is often taken as spherical. There are some indications, also provided by modern experimental techniques (X-ray spectroscopy, quartz crystal microbalance, QCM), that the density of water near the interface can change drastically (see later discussion). 

Figure 3-13 (a) The shape of a "perfectly" shaped quartz crystal, and (b) the effective "shape of quartz crystal with respect to oxygen diffusion (after coordinate transformation using Equation 3-70d). The length (thickness) to diameter ratio is about 2 1 in (a) and 1 5 in (b). 

Figure 3-26 Quartz crystal growth and diffusion profile in (a) a laboratory-fixed reference frame and (b) an interface-fixed reference frame. At a given time, a given kind of curve is used to outline the crystal shape and plot the concentration profile.

If the unknown crystal can be shaped into a wedge similar to the EFISH cell geometry, a similar sinusoidally varying signal is observed from which tensor elements can be extracted provided a reference sample of known properties is available. Quartz crystals have been characterized extensively (17) and can be obtained in the form of a wedge. They are often used as a reference material for both EFISH measurements and other crystals. 

As early as 1669 the Danish crystallographer Steno made a detailed study of ideal and distorted quartz crystals (Figure 9-5). He traced their outlines on paper and found that the corresponding angles of different sections were always the same regardless of the actual sizes and shapes of the sections. Thus, all quartz crystals, however much distorted from the ideal, could result from the same fundamental mode of growth and, accordingly, corresponded to the same inner structure. 

Figure 9.7. BF micrographs showing the strrun associated with high-pressure clusters of molecular water in as-grown wet synthetic quartz (crystal W2). (a) Region with a water content corresponding to 2(X)H/10 Si. (b) Region with l,600H/10 Si. No strain is associated with the arrowed clusters because they intersect the foil surface the water escapes, and the stress is relaxed. The strmn field of the circled cluster is characteristic of a lens-shaped inclusion whose plane is normal to the foil surface. (From McLaren et al. 1983.)...

Nevertheless, the shape of crystals is really a secondary characteristic, since it depends on, and is a consequence of, the interior arrangement of atoms. Sometimes the external shape of a crystal is rather obviously related to its smallest building block, the unit cell, as in the httle cubical grains of ordinary table salt (NaCl has a cubic lattice) or the six-sided prisms of natural quartz crystals (hexagonal lattice). In many other cases, however, the crystal and its unit cell have quite different shapes gold, for example, has a cubic lattice, but natural gold crystals are octahedral in form, i.e., bounded by eight planes of the form 111. ... 

In addition, crystals show characteristic symmetry properties with respect to their forms and surfaces. Crystals that cannot superimpose themselves are chiral and occur in Nature, for instance, quartz and sodium chlorate. Louis Pasteur noticed that ammonium sodium tartrate came in two asymmetric forms that were mirror images of one another. Pasteur manually separated the left and right crystal shapes from each other to form two piles of crystals in solution, one form rotated light to the left, the other to the right, while an equal mixture of the two forms canceled each other s effect and, does not rotate the polarization of light. 

Pendulums were more accurate, followed by quartz oscillators. In 1880, Pierre and Jacques Curie discovered that sending a current through a quartz crystal could result in a resonance situation with cyclic behavior, making a quartz oscillator that could be used to mark time. Because crystals grow in miniature, quartz crystal watches became the standard in the 1960s. But the periodic nature of resonant quartz excitations depends on the shape of each crystal as well as the ambient temperature and humidity. Every crystal behaves differently, and none can constitute a reputable standard for the accuracy needed by global positioning systems, for example. 

Chemical sensors are small devices for the detection and quantification of gaseous or solvated species. This is an active research area based on the need to obtain increasing amounts of data in chemical and food process streams as well as environmental monitoring. Most sensors consist of an appropriate transduction principle such as the quartz-crystal-microbalancc (QCM) and a chemically sensitive layer that imparts the desired chemical response behaviour. Most often a chemically selective response is desirable. Zeolite molecular sieves offer size- and shape-selective adsorption behaviour that can be combined with appropriate transduction concepts in order to construct chemically selective sensor devices. 

INITIAL INTEGRAL MASS SENSITIVITY Cf OF OSCILLATING QUARTZ CRYSTALS OF DIFFERENT RESONANCE FREQUENCY AND CRYSTAL SHAPE, EXPERIMENTALLY DETERMINED AND CALCULATED BY VARIOUS AUTHORS. 

Figure 27.24 The constancy of interfacial angles, (a) Section of an ideal quartz crystal, (b) and (c) Possible shapes of the crystal section.

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