The crystal classes

A plane of symmetry divides any figure into mirror images. The cube has nine planes of symmetry, as shown in Fig. 27.15. 

Finally, the cube has a center of symmetry. Possession of a center of symmetry, a center of inversion, means that if any point on the cube is connected to the center by a line, that line produced an equal distance beyond the center will intersect the cube at an equivalent point. More succinctly, a center of symmetry requires that diametrically opposite points in a figure be equivalent. These elements together with rotation-inversion are the symmetry elements f or crystals. The elements of symmetry f ound in crystals are (a) center of symmetry (b) planes of symmetry (c) 2-, 3-, 4-, and 6-fold axes of symmetry and (d) 2- and 4-fold axes of rotation-inversion. Of course, every crystal does not have all these elements of symmetry. In fact, there are only 32 possible combinations of these elements of symmetry. These possible combinations divide crystals into 32 crystal classes. The class to which a crystal belongs can be determined by the external symmetry of the crystal. The number of crystal classes corresponding to each crystal system are triclinic, 2 monoclinic, 3 orthorhombic, 3 rhombohedral, 5 cubic, 5 hexagonal, 7 tetragonal, 7. 

The most common method of determining the symmetry class is by examining the symmetry of the x-ray diffraction pattern of a small single crystal specimen. However, 

The / fold axis of rotation-inversion rotation through 3607a followed by inversion through the center. 

The division of crystals into crystal systems and crystal classes is based on the symmetry of the crystal as a finite object, or the symmetry of a single unit cell. In a unit cell all of the corners are equivalent points, since by translation along the axes the entire pattern can be 

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Table 3 is arranged by crystal class (14). The crystal class of a given pigment is determined almost solely by the ratio of the ionic sizes of the cation and the anion and thek respective valences. Hence, for any given stoichiometry and ionic size ratio, only one or two stmctures will be possible. In some classes (spkiel, zkcon), a wide range of colors is possible within the confines of that class. Pigments within a given class usually have excellent chemical and... 

State the crystal classes and crystal systems to which the following space groups belong ... 

Crystals can only be piezoelectric when they are non-centrosymmetric. In addition, they may not belong to the crystal class 4 32. The effect is thus restricted to 20 out of the 32 crystal classes. 

Lattice equivalent ( translationengleich , abbreviation t). M contains all the translations of G, the crystal class of M is of lower symmetry than that of G. 

Unit cell parameters were obtained from a study conducted on a single crystal of the material. The crystal class was monoclinic within the P2, or P2,/m space groups. The unit cell was characterized by the following lattice parameters a = 9.686(2) A, 6 = 8.792(4) A, c = 10.085(6) A, p = 92.33(4)°. 

Since this chapter is aimed at presenting much of the information leading to the discovery of the new high Tc oxide superconductors, we shall present here an overview of the oxide systems which have been studied for superconducting properties prior to 1985. In the next section of this chapter, we will give a more detailed and descriptive narration of the work performed on oxide systems, presented in terms of the crystal classes which have yielded the most important oxide superconductors. 

Ferroelectrics. Among the 32 crystal classes, 11 possess a centre of symmetry and are centrosymmetric and therefore do not possess polar properties. Of the 21 noncentrosymmetric classes, 20 of them exhibit electric polarity when subjected to a stress and are called piezoelectric one of the noncentrosymmetric classes (cubic 432) has other symmetry elements which combine to exclude piezoelectric character. Piezoelectric crystals obey a linear relationship P,- = gijFj between polarization P and force F, where is the piezoelectric coefficient. An inverse piezoelectric effect leads to mechanical deformation or strain under the influence of an electric field. Ten of the 20 piezoelectric classes possess a unique polar axis. In nonconducting crystals, a change in polarization can be observed by a change in temperature, and they are referred to as pyroelectric crystals. If the polarity of a pyroelectric crystal can be reversed by the application on an electric field, we call such a crystal a ferroelectric. A knowledge of the crystal class is therefore sufficient to establish the piezoelectric or the pyroelectric nature of a solid, but reversible polarization is a necessary condition for ferroelectricity. While all ferroelectric materials are also piezoelectric, the converse is not true for example, quartz is piezoelectric, but not ferroelectric. 

The various names used formerly for the crystal classes are to be... 

Pot a fuller discussion of the phenomenon, and a list of the crystal classes which (according to current theories) may exhibit it, see Wooster (1938) and International Tables (1952). Rotation of the plane of polarization, though difficult to observe and measure except in large isotropic and uniaxial crystals, and therefore of rather limited application in identification or structural investigations, is a phenomenon of great interest that has played an important part in the historical development of chemistry and crystallography. 

The statistical methods are valuable because they detect symmetry elements which are not revealed by a consideration of absent reflections, or by Laue symmetry5. In principle, it is possible to distinguish between all the crystal classes (point-groups) by statistical methods in fact, as Rogers (1950) has shown, it is possible by X-ray diffraction methods alone (using absent reflections as well as statistical methods) to distinguish between nearly all the space-groups (see p. 269). 

In Table 11.4 we list the crystal classes along with the minimum symmetry necessary for each, and the maximum (i.e., lattice) symmetry possible for each. 

Within each of the six crystal systems, there arc specific crystal classes. Each class displays distinctive symmetry elements. There are 32 possible classes distributed among the six crystal systems. One of the crystal classes within each system possesses all of the symmetry elements that re characteristics of its space lattice cell. These are ealled the holohedral class of that system. Other classes within each system possess somewhat fewer symmetry elements and are called tncmhedral classes. 

Knowledge of the diffraction symmetry of a crystal is useful for its classification. If the Laue group is observed to be 4/mmm, the crystal system is tetragonal, the crystal class must be chosen from 422,4mm, 42m, and 4/mmm, and the space group is one of those associated with these four crystallographic point groups. 

Figure 1.1 Top Quartz crystal exhibiting the true symmetry of the crystal class to which quartz belongs. Bottom The forms comprising such a quartz crystal. From left to right, the hexagonal prism, trigonal dipyramid, rhombohedron, and trigonal trapezohedron.

In many of the molecular packing studies, the crystal classes are taken from the experimental X-ray diffraction determinations. The optimal packing is then determined for the assumed crystal class. In other cases, the crystal classes have also been established in the optimization calculations. 

The manner in which a body responds to small external mechanical forces in the elastic regime is determined by the elastic constants (Table 10.2). The number of elastic constants for a given crystal is dependent of the crystal class to which it belongs. 

Upon inspection of Table 10.3, it can be seen that there are twelve nonzero-valued elastic-stiffness coefficients. Some of these are related by the crystal class and some by transpose symmetry, with the result that there are only six independent coefficients Cn = C22 C12 C13 = C23 C33 C44 = C55 Cee- All other components are zero-valued. Hence, the matrix with all the nonzero independent coefficients designated as such is straightforwardly written as ... 

The many-electron wave function in a crystal forms a basis for some irreducible representation of the space group. This means that the wave function, with a wave vector k, is left invariant under the symmetry elements of the crystal class (e.g. translations, rotations, reflections) or transformed into a new wave function with the same wave vector k. 

ZnO is a hexagonal matmal (symmetiy class nm) with elastic cmistants as given in Table 2.2 and piezoelectric constants as given in Table 2.3. Applying the matrices q>-propriate to the crystal class results in the following equations ... 

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