Tetragonal system, classes

Fig. 36. Tetragonal system. (See also Fig. 28.) a. Unit cell type. b. Phloroglucinol diethyl ether. Class 4jm. c. Wuifenite, PbMoQ4. Class 4. d. Anatase, TiOs. Class 4/mmm. e. Zircon, ZrSi04. Class 4/wmm.

The tetragonal system, a = b c and a = /3 = y = 90°, has one axis of fourfold symmetry in the [001] direction, two sets of orthogonal axes of twofold syrranetry in the (100) and (110) directions, a set of three orthogonal mirror planes as well as two other mirror planes, (110) and (llO) as shown in Figure 4.10e. The Schoenflies symbol for this class is using the same reasoning as for D2h, except in this case n = 4. The full H-M symbol is abbreviated to A/mmm, where the /mmm indicates a mirror plane normal to the fourfold axis and mirror planes parallel to it. 

The information obtainable from the Laue symmetry is meagre it consists simply in the distinction, between crystal classes, and then only in the more symmetrical systems—cubic, tetragonal, hexagonal, and trigonal (see Table VI). But it is useful in cases in which morphological features do not give clear evidence on this point. 

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. 

With the discovery of superconductivity (Tc = 15.5 K) in the Y-Ni-B-C system [6, 80], a new class of quaternary borocarbide superconductors has emerged. Superconductivity has been observed in several rare earth (Lu, Tm, Er and Ho) nickel borocarbides[80], and with transition metals such as Pd and Pt. The superconducting phase having the composition of YNi2B2C, crystallizes [81] in a tetragonal structure with alternating Y-C and Ni2B2 layers. Band structure calculations [82] indicate that these materials, unlike cuprate superconductors, are three-dimensional metals. 

Careful measurement of mineral specimens allowed crystals to be classified in terms of six crystal families, called anorthic, monoclinic, orthorhombic, tetragonal, hexagonal and isometric. This classification has been expanded slightly by crystallographers into seven crystal systems. The crystal systems are sets of reference axes, which have a direction as well as a magnitude, and hence are vectors1. The crystal families and classes are given in Table 1.1. 

Tetragonal crystal system the holohedric group Dgh, and those sub-groups which do not belong to classes 1, 2, and 3 Dgh Dg, Cgyj, Dai Cgh, Cg, Sg... 

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. 

For example, the crystal class for I4c2 is 4m2, an alternative symbol for 42m. The crystal system is tetragonal and the Bravais lattice is tetragonal I. 

Formula Chemical class Chemical type Pb,(C03)Ci, Anhydrous carbonate with hydroxyl or halogen (AB),(X03>Z, Crystal system Mineral group Space group Tetragonal Phosgenite P4/mbm... 

Lyotropic cubic phases have been the subject of many structural studies [138,157-159]. Their structure is more complicated and less readily visualized than that of other phases. Almost all 3D fluid phases so far observed are of cubic symmetry, although rhombohedral, tetragonal, and orthorhombic phases of inverse topology have been detected in a few systems (based, for example, on SDS or lipids) [160]. There are two distinct classes of cubic phases [110] ... 

Each crystal system contains several classes that exhibit only a partial symmetry for instance, only one-half or one-quarter of the maximum number of faces permitted by the symmetry may have been developed. The holohedral class is that which has the maximum number of similar faces, i.e. possesses the highest degree of symmetry. In the hemihedral class only half this number of faces have been developed, and in the tetrahedral class only one-quarter have been developed. For example, the regular tetrahedron (4 faces) is the hemihedral form of the holohedral octahedron (8 faces) and the wedge-shaped sphenoid is the hemihedral form of the tetragonal bipyramid Figure 1.9). 

This notation defines the symmetries of a crystalline material in terms of point, plane, and space groups and is closely tied to the crystal systems (triclinic, monoclinic, tetragonal, etc.) and crystal class. It is also related to Schoenflies and orbifold notations and offers a definition of the relevant symmetry axes. 

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