Cubic point groups described

What rests are the socalled cubic point groups T, Td, Th, O and Oh. The symmetry operators for these groups are conveniently described with reference to a cube, Fig. 1. 

In the trigonal point group 3, the axis of quantization is chosen along the three-fold axis, while the x and y axes may be selected anywhere in the plane perpendicular to the z axis. In the point group 3m, which occurs for many distorted octahedral complexes, there are also vertical mirror planes. The relation to the cubic axes is described by the transformation... 

A centrosymmetric stress cannot produce a noncentrosymmetric polarization in a centrosymmetric crystal. Electric dipoles cannot form in crystals with an inversion center. Hence, only the twenty noncentrosymmetric point groups are associated with piezoelectricity (the noncentrosymmetric cubic class 432 has a combination of other symmetry elements which preclude piezoelectricity). The piezoelectric strain coefficients, dj for these point groups are given in Table 8.7, where, as expected, crystal symmetry dictates the number of independent coefficients. For example, triclinic crystals require the full set of 18 coefficients to describe their piezoelectric properties, but mono-chnic crystals require only 8 or 10, depending on the point group. 

The periodicity of a lattice limits the number of compatible rotation operations to onefold, twofold, threefold, fourfold, and sixfold. This, in turn, limits the number of point groups to thirty-two. Point groups are used to describe individual molecules. Table 14.1 shows the thirty-two point groups in both the Hermann-Mauguin notation and the Schoenflies notation divided into seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

The rhombohedral (or trigonal) crystal system is one of the seven lattice point groups, named after the two-dimensional rhombus. In the rhombohedral system, the crystal is described by vectors of equal length, of which all three are not mutually orthogonal. The rhombohedral system can be thought of as the cubic system stretched diagonal along a... 

The symmetries for the seven basic crystal systems described above assume the full symmetry or holohedry of each of the lattices. When the basis is added, some of these symmetries may be restricted. For example, a face-centered cubic (fee) crystal such as A1 has the full cubic symmetry. However, diamond also has the fee structure with atoms occupying the lattice points as well as every other tetrahedral interstitial point. Its point group is 43m, which implies a rotation-inversion on the fourfold axes. The threefold symmetry is preserved without the threefold rotation-inversion. The twofold symmetry is no longer preserved and the only mirror symmetry is along the 110 planes. 

Green rhombohedral crystals of iron(II) nitrate hexahy-drate, Fe(N03)2-6H20 (ferrous nitrate), with a melting point 60.5 °C, are obtained from solutions made by dissolving iron in dilute nitric acid. With more concentrated acid, oxidation takes place, and monoclinic pale violet iron(III) nitrate non-ahydrate, Fe(N03 )s -91120 (ferric nitrate), melting point 47 °C, may be crystallized. A colorless hexahydrate can also separate as cubic crystals. Various basic iron(in) nitrates have been described. The eight-coordinate iron(in) anion [Fe(N03)4] has an essentially dodecahedral symmetry with four almost symmetrical bidentate nitrate groups. ... 

The structures of the principal allotropic forms of all the elements will be discussed in detail as the chemistry of each element is treated. For illustrative purposes,-we shall mention here only one such structure, the diamond structure, since this is adopted by several other elements and is a point of reference for various other structures. It is shown from two points of view in Fig. 2-8. The structure has a cubic unit cell with the full symmetry of the group Td. However, it can, for some purposes, be viewed as a stacking of puckered infinite layers. It will be noted that the zinc blende structure (Fig. 2-3) can be regarded as a diamond structure in which one-half of the sites are occupied by Zn2 + (or other cation) while the other half are occupied by S2 (or other anion) in an ordered way. In the diamond structure itself all atoms are equivalent, each being surrounded by a perfect tetrahedron of four others. The electronic structure can be simply and fairly accurately described by saying that each atom forms a localized two-electron bond to each of its neighbors. 

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