Symmetry operation symbolic description

Table Various kinds of symmetry elements and operations Symmetry elements description Symbol Symmetry operation...

It is equivalent to describe the symmetry class of the tetrahedron as 3/2-m or 3/4. The skew line relating two axes means that they are not orthogonal. The symbol 3/2-m denotes a threefold axis, and a twofold axis which are not perpendicular and a symmetry plane which includes these axes. These three symmetry elements are indicated in Figure 2-50. The symmetry class 3/2-m is equivalent to a combination of a threefold axis and a fourfold mirror-rotation axis. In both cases the threefold axes connect one of the vertices of the tetrahedron with the midpoint of the opposite face. The fourfold mirror-rotation axes coincide with the twofold axes. The presence of the fourfold mirror-rotation axis is easily seen if the tetrahedron is rotated by a quarter of rotation about a twofold axis and is then reflected by a symmetry plane perpendicular to this axis. The symmetry operations chosen as basic will then generate the remaining symmetry elements. Thus, the two descriptions are equivalent. 

We begin with the symbolic description of symmetry operations, which is based on the fact that the action of any symmetry operation or any combination of symmetry operations can be described by the coordinates of the resulting object(s). In this section we assume that the initial object has... 

Figure 1.46. Symmetry elements and symbolic description of symmetry operations in the point group symmetry 2/m. Here, the origin of the coordinate system coincides with the center of inversion.

This description formalizes symmetry operations by using the coordinates of the resulting points and, therefore, it is broadly used to represent both symmetry operations and equivalent positions in the International Tables for Crystallography (see Table 1.18). The symbolic description of symmetry operations, however, is not formal enough to enable easy manipulations involving crystallographic symmetry operations. 

The operation of the allowed symmetry elements on the 14 Bravais lattices must leave each lattice point unchanged. The symmetry operators are thus representative of the point symmetry of the lattices. The most important lattice symmetry elements are given in Table 4.3. In all except the simplest case, two point group symbols are listed. The first is called the full Hermann-Mauguin symbol, and contains the most complete description. The second is called the short Hermann-Mauguin symbol, and is a condensed version of the full symbol. The order in which the operators within the symbol are written is given in Table 4.2. 

The unit cell of polyethylene is drawn in Fig. 5.19 in two proj ections. The zig-zag chains are represented such that the carbon atoms are located at the intersections of the backbone bonds and the hydrogen atoms are located at the ends of the heavy lines. All symmetry elements are marked by their symbols as also given in Figs. 5.7-9. With help of the symmetry elements all atom-positions can be fixed in both projections. The symbols are standardized and will be used for the description of all the other crystal structures. Since the structure in the chain direction is best known, most projections will be made parallel to the chain axis which is also the helix axis. Before continuing with the study, it is of value to take the time to check the operation of every symmetry element in Fig. 5.19 with respect to its operation, since the later examples will similarly be displayed and a facility of three-dimensional visualization of crystal stractures is valuable for the understanding of the crystal stractures. 

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