Symmetry elements of a cube

Figure 17.4. Labeling of the symmetry elements of a cube as used in the derivation of Ym and of the symmetrized bases. The points 1, 2, 3, and 4 label the three-fold axes. The poles of the two-fold axes are marked a, b, c, d, e, and f. The points 1,2,3,4,5,6,7, and 8 provide a graphical representation of the permutations of [1 1 1] (cf. Table 17.10).

Symmetry elements of a cube. There are three fourfold axes, four threefold axes, and six twofold axes of rotation. There are three 100 mirror planes and six 110 mirror planes (only two are shown). 

Fig. 2-6 Some symmetry elements of a cube, (a) Reflection plane. Ai becomes A2. (b) Rotation axes. 4-fold axis Ai becomes A2 3-fold axis A becomes A 2-fold axis Ai becomes A4,. (c) Inversion center. A becomes A2. (d) Rotation-inversion axis. 4-fold axis A becomes A inversion center A becomes A2.

In the simple structures which we have discussed so far the local environment of the central metal ion has been tetrahedral, octahedral, or cubic. We can describe all of these as being of cubic symmetry since they include some or all of the characteristic symmetry elements of a cube. They possess the maximum symmetry compatible with the coordination number and, with the reservations already noted for eightfold coordination, are just those environments which would follow from the properties of the simple ionic model described in the previous section. There are many... 

PI 2.14 (a) xyz changes sign under the inversion operation (one of the symmetry elements of a cube) hence it... 

EXAMPLE 9.6 List the symmetry elements of a uniform cube centered at the origin with its faces parallel to the coordinate planes. 

There are other symmetry elements such as screw axes that are meaiungful for crystals but not for our macroscopic crystal shapes. Figure 5.3 illustrates some of the symmetry elements for a cube. The most important are the four 3-fold axes along the <111> diagonals. 

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. 

Polyhedral crystals bounded by flat crystal feces usually take characteristic forms controlled by the symmetry elements of the crystal (point) group to which the crystal belongs and the form and size of the unit cell (see Appendix A.5). When a unit cell is of equal or nearly equal size along the three axes, crystals usually take an isometric form, such as a tetrahedron, cube, octahedron, or dodec-... 

Figure 9-23. The diamond structure after Shubnikov and Koptsik [33], (a) A unit cell the edges of the cube are the a, b, and c axes (b) Two face-centered cubic sublattices displaced along the body diagonal of the cube (c) Projection of some symmetry elements of the Fd im space group onto a horizontal plane. The vertical screw axes 4 and 43 are marked by appropriate symbols. Used with permission.

Hie Octahedron and the Cube. These two bodies have the same elements, as shown in Fig. A5-7, where the octahedron is inscribed in a cube, and the centers of the six cube faces form the vertices of the octahedron. Conversely, the centers of the eight faces of the octahedron form the vertices of a cube. Figure A5-7 shows one of each of the types of symmetry element that these two polyhedra possess. The list of symmetry operations is as follows ... 

Evaporation of the water from salt solutions results in solid salt crystals the ions involved form an ion lattice corresponding to the salt structure. If one allows the water to slowly evaporate from the saturated solution, this often results in large and beautiful crystals. Particularly the alum salt, when growing crystals from saturated solutions (E4.1), results in beautifully formed octahedron (see Fig. 4.4). In the process, K+(aq) ions, Al3+(aq) ions and S042- ions join together to form an ion lattice of cubic symmetry. If one adds approximately 10% of urea to a saturated sodium chloride solution the salt crystals do not crystallize in the expected cubic form, but in an octahedron form with identical symmetry elements as the cube. 

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 symmetry elements of the tetrahedral and octahedral groups are oriented with respect to the characteristic directions for a cube (Fig. 2.17) ... 

The tetrahedron symmetry elements can be thought of as a subset of those of a related cube. If we draw a cube and place atoms at half of the eight corners such that on any face they are diagonaiiy opposite to one another, then the atoms will define a tetrahedron, as shown in Figure 3.27a. The paper model of a cube from Appendix 3 is made with one open side so that the tetrahedral model can be inserted into it in the same orientation. This relationship between the cube and the tetrahedron shows how the tetrahedron can be... 

Figure 3.28 The symmetry elements of Oh to which the cube and octahedron belong, (a) Q axes, (b) C4 axes, (c) C2 axes, (d) an axis and the mirror plane that forms part of the operation there are four such axes collinear with the Q axes, (e) The 3ah planes and (f) two of the 6cTd planes these contain the C4 axis along Z the other (Xj planes are in identical pairs containing X or Y.

The orbital complement of the octahedron with sp d hybridization is a seven-vertex polyhedron with sp d (xy, xz, yz) hybridization. However, there are no seven-vertex polyhedra with more than two symmetry elements which can be formed by sp d (xy, xz, yz) hybrids. Nevertheless, the seven-orbital sp d (xy, xz, yz) hybrid is of significance in being the unique sp d manifold to which the f(xyz) orbital is added to form eight hybrid orbitals at the vertices of a cube. In this sense the octahedron is the orbital complement as well as the polyhedral dual of the cube. Note that the maximum electron density of an octahedron but the minimum electron density of the cube are located along the X, y, and z axes in accord with the dual relationship between these polyhedra (Section 2.1). 

There is one other element of symmetry possessed by sodium chloride crystals. For each face, edge, or corner of the cube or octahedron there is an exactly similar face, edge, or comer diametrically opposite the centre of the cube or octahedron (Fig. 19) is therefore called a centre of symmetry. The centre of symmetry possessed by these shapes corresponds with the centre of symmetry in the atomic arrangement the centre of any sodium or chlorine ion is a centre of symmetry, since along any direction from the selected ion the arrangement encountered is exactly repeated in the diametrically opposite direction. 

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