Other crystal systems

In general, any fraction of the sphere volume, symmetrically equivalent to that shaded in 

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In common with other crystal systems where the diffraction tends to fall off rapidly with resolution. 

The list of alternate symbols arises because at an earlier stage in the history of X-ray crystallography, it was the accepted convention to call the unique axis b. The reader can easily see that if this is done, the alternate symbols become correct. Since in the other crystal systems with a unique direction (i.e., tetragonal and hexagonal) the unique direction is called c, the formally correct practice is now to do the same for the monoclinic system. However, the literature is still replete with the old choice of axes and it is necessary to be cognizant of both systems and of their relationship. 

In all other crystal systems we encounter the same general situation, namely, that a few space groups (69, in fact) can be uniquely identified from a knowledge of diffraction symmetry and systematic absences, while the rest form mostly pairs, or small groups that are indistinguishable in this way. Table 11.9 lists for the triclinic, monoclinic, and orthorhombic crystal systems the uniquely determined space groups and the sets with identical systematic absences. 

As a consequence of Friedel s law, the diffraction pattern exhibits the symmetry of a centrosymmetric crystal class. For example, a crystal in class 2, on account of the 1 symmetry imposed on its diffraction pattern, will appear to be in class 2/m. The same result also holds for crystals in class m. Therefore, it is not possible to distinguish the classes 2, m, and 2/m from their diffraction patterns. The same effect occurs in other crystal systems, so that the 32 crystal classes are classified into only 11 distinct Laue groups according to the symmetry of the diffraction pattern, as shown in Table 9.4.1. 

In subsequent experiments, using other crystal systems, such as ferrous sulfate and sodium hydrogen phosphate, it was similarly observed that the first crystallization product to form was the one most closely resembling the structure of the solvent (Nyvlt, 1995). For the case of citric acid, this is the monohydrate, which more closely resembles the aqueous structure. As the temperature of the solution is increased, the structure of the solvent, as well as the solubility of the crystal, changes, resulting in a more thermodynamically stable anhydrous product. This conversion between the kinetic and thermodynamic product occurs at a critical transition temperature, below which the structure of the solution favors the formation of the hydrated product. As the transition temperature is surpassed, the anhydrous product becomes favored. 

In order to specify a crystal direction, a vector is drawn from the origin to some point P. This vector will have projections u on the a axis, d on the b axis, and W on the c axis. The three numbers are divided by the highest common denominator to give the set of smallest integers, u, v, and w. The direction is then denoted in brackets as [uvw]. Sets of equivalent directions are labeled u v w). For cubic systems, the [h k /] direction is always orthogonal to the (hkl) plane of the same indices. With the other crystal systems, this simple relationship does not hold. For example, in the hexagonal lattice, the normal to the (1 0 0) plane is in the [2 1 0] direction, the [100] direction being 120° to the (1 0 0) plane (see Practice Problem 4). 

Other crystal systems -> Tetragonal, orthorhombic, rhombohedral, monoclinic, and triclinic... 

In most of the other crystal systems, the axis with the highest order rotational symmetry is designated the z-axis, which is more commonly called the c-axis. The order of this axis is given first in the point group designation. In the monoclinic systan, the -axis is taken as the two-fold axis or as perpendicular to the mirror plane. 

Crystallographic direction. Glasses and crystals with a cubic structure are optically isotropic. In all other crystal systems n is higher along close packed directions. 

PLZT ceramics belonging to either the tetragonal or rhombohedral crystal systems are classified as optically uniaxial. Which other crystal system or systems are also optically uniaxial ... 

I have demonstrated that it is possible to determine experimentally the dephasing times of optical phonons and their temperature dependence, and that it is equally possible to determine experimentally all components of the third order nonlinear electronic susceptibility. The measurements reported here were taken in the temporal domain. Similar measurements of T2 and X taken in the spectral domain had previously been reported for other crystal systems. 

Finally, let us introduce another simplification. The matrices of the Cy and tensors are symmetric with respect to their main diagonal, which means that the largest number of independent elasticity or compliance moduli is 6 h- (36 - 6)/2 = 21. This is exactly the case for the least symmetrical crystals belonging to the triclinic crystal system. In the opposite extreme case for isotropic media, the number of independent moduli is reduced to two (but never to a single one). All other crystal systems can be arranged by the number of independent elasticity constants in the following series ... 

Each of the other crystal systems has similar restricted symmetries and it can be shown that there is a total of 32 unique sets of point symmetry operations or point groups. The symmetry of every crystalline structure may be described by one of these 32 point groups. Such classification of point symmetries is useful in the search for materials with certain properties. For example, if one is looking for materials with permanent dipole moments, one would look only at systems that are noncentrosymmetric, i.e., systems that do not possess a center of inversion symmetry. The 10 noncentrosymmetric point groups are 1, 2, 3, 4, 6, m, 2mm, 3m, 4mm, and 6mm. 

Construction of a direction specified by four indices is carried out using a procedure similar to the one used for other crystal systems—by the subtraction of vector tail point coordinates from head point coordinates. For the four coordinate axes of Figure 3.10, we use the following designations for head and tail coordinates ... 

One interesting and imique characteristic of cubic crystals is that planes and directions having the same indices are perpendicular to one another however, for other crystal systems there are no simple geometrical relationships between planes and directions having the same indices. 

We determine these indices in a manner analogous to that used for other crystal systems as described previously—that is, taking normalized reciprocals of axial intercepts, as described in the following example problem. 

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