Cube symmetry

The main symmetry elements in SFg can be shown, as in Figure 4.12(b), by considering the sulphur atom at the centre of a cube and a fluorine atom at the centre of each face. The three C4 axes are the three F-S-F directions, the four C3 axes are the body diagonals of the cube, the six C2 axes join the mid-points of diagonally opposite edges, the three df, planes are each halfway between opposite faces, and the six d planes join diagonally opposite edges of the cube. 

The first complex intermetallic compound found to have large clusters of atoms with local icosahedral symmetry was Mg32Al4, which has 162 atoms in a body-centred cubic unit17. The unit cube contains 98 icosahedra, 20 Friauf polyhedra and 44 others. 

We have introduced the term "symmetry" in the solid. Let us now examine exactly what is meant by that term. To illustrate this concept, examine the symmetry elements of our cube, given in the following diagram, given on the next page as 2.1.6. 

These 14 Bravais Lattices are unique in themselves. If we arrange the crystal systems in terms of symmetry, the cube has the highest symmetry and the triclinic lattice, the lowest symmetry, as we showed above. The same hierarchy is maintained in 2.2.4. as in Table 2-1. The symbols used by convention in 2.2.4. to denote the type of lattice present are... 

The unit cell of cubic diamond corresponds to a face-centered packing of carbon atoms. Aside from the four C atoms in the vertices and face centers, four more atoms are present in the centers of four of the eight octants of the unit cell. Since every octant is a cube having four of its eight vertices occupied by C atoms, an exact tetrahedral coordination results for the atom in the center of the octant. The same also applies to all other atoms — they are all symmetry-equivalent. In the center of every C-C bond there is an inversion center. As in alkanes the C-C bonds have a length of 154 pm and the bond angles are 109.47°. 

You say that your nonlinear molecule has the high symmetiy of a regular polyhedron, such as a tetrahedron, cube, octahedron, dodecahedron, icosahedron,... sphere. If it is a sphere, it is monatomic. On the other hand, if it is not monatomic, it has the symmetry of one of the Platonic solids (see the introduction to Chapter 8). 

To illustrate the high molecular symmetry of these compounds, the molecular structure of cage 85 178) is depicted in Fig. 19 32). The skeleton of S can be derived from an Sn4N4 cube enlarged at one edge by the sequence Si—N the analog in hydrocarbon chemistry is the so-called basketane . For further discussions of the synthesis and the structural and physical properties of these compounds the reader is referred to a review article published recently 10). 

The penetrating power of a shaped charge is approximately proportional to the cube of its diameter, but also very dependent on maintenance of exact axial symmetry during construction. It is also proportional to the detonation pressure of the explosive used, so that suitable fillings are cast Pentolite or RDX/TNT. Well-known applications of shaped charges are in the British PIAT and American bazooka. 

Figure 2.9 Schematic structure of two dif- (d) cube. The dpp parameter defines the aver-ferent polyoxometalate complexes with inter- age distance between the two oxygen-based esting coordination symmetry (a) LnW10 and square planes. The djn is the average 0-0 (b) [LnPd"12(AsvPh)8032]5-, and their coor- distance within the oxygen-based square dination polyhedra (c) square antiprism and planes.

Although the effect on the d orbitals produced by a field of octahedral symmetry has been described, we must remember that not all complexes are octahedral or even have six ligands bonded to the metal ion. For example, many complexes have tetrahedral symmetry, so we need to determine the effect of a tetrahedral field on the d orbitals. Figure 17.5 shows a tetrahedral complex that is circumscribed in a cube. Also shown are lobes of the dz- orbital and two lobes (those lying along the x-axis) of the dx> y> orbital. 

Cubed compound, in PVC siding manufacture, 25 685 Cube lattice, 8 114t Cubic boron nitride, 1 8 4 654 grinding wheels, 1 21 hardness in various scales, l 3t physical properties of, 4 653t Cubic close-packed (CCP) structure, of spinel ferrites, 11 60 Cubic ferrites, 11 55-57 Cubic geometry, for metal coordination numbers, 7 574, 575t. See also Cubic structure Cubic symmetry Cubic silsesquioxanes (CSS), 13 539 Cubic structure, of ferroelectric crystals, 11 94-95, 96 Cubic symmetry, 8 114t Cubitron sol-gel abrasives, 1 7 Cucurbituril inclusion compounds,... 

Td, possesses 32 symmetry, and requires a minimum of 12 asymmetric units the cube and octahedron, which belong to the point group Oh, possess 432 symmetry, and require a minimum of 24 asymmetric units and the dodecahedron and icosahedron, which belong to the point group Ih, possess 532 symmetry, and require a minimum of 60 asymmetric units. The number of asymmetric units required to generate each shell doubles if mirror planes are present in these structures. 

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