Crystallographic point symmetry groups

Creation operator, 505 representation of, 507 Critical value, 338 Crystallographic point groups irreducible representations, 726 Crystallographic symmetry groups construction of mixed groups, 728 Crystal, eigenstates of, 725 Crystal symmetry, changes in, 758 Crystals... 

All the possible combinations of these symmetry elements result in 32 crystallographic point-group symmetries or crystal classes their symbols are listed in Table 3.3. Notice that in putting together the symbols to denote the symmetries of any crystal classes the convention is to give the symmetry of the principal axis first for instance 4 or 4, for tetragonal classes. If there is a plane of symmetry perpendicular to the principal axis, the two symbols are associated as in 4 m or Aim (4 over m), then the symbols for the secondary axes, if any, follow, and then any other symmetry planes. In a symbol such as Almmm, the second and third m refer to planes parallel to the four-fold axis. 

The environment of an ion in a solid or complex ion corresponds to symmetry transformations under which the environment is unchanged. These symmetry transformations constitute a group. In a crystalline lattice these symmetry transformations are the crystallographic point groups. In three-dimensional space there are 32 point groups. 

The site symmetry of each atom must be one of the 32 crystallographic point groups shown in Fig. 10.7, since these are the only point groups compatible with three-dimensional space groups. 

For the monoclinic system it is essential to have one twofold axis, either 2(C2) or 2(m), and it is permitted, of course, to have both. When both are present the point group is that of the lattice, 2lm Cy). There are no intermediate symmetries. By proceeding in this way, we can arrive at the results shown in column 4 of Table 11.4, where each of the 32 crystallographic point groups (i.e., crystal classes) has been assigned to its appropriate crystal system. 

For crystals, the point group must be compatible with translational symmetry, and this requirement limits n to 2,3,4, or 6. (This restriction applies to both proper and improper axes.) Thus the crystallographic point groups are restricted to ten proper point groups and a total of... 

The symmetry elements, proper rotation, improper rotation, inversion, and reflection are required for assigning a crystal to one of the 32 crystal systems or crystallographic point groups. Two more symmetry elements involving translation are needed for crystal structures—the screw axis, and the glide plane. The screw axis involves a combination of a proper rotation and a confined translation along the axis of rotation. The glide plane involves a combination of a proper reflection and a confined translation within the mirror plane. For a unit cell... 

The 32 crystallographic point groups result from combinations of symmetry based on a fixed point. These symmetry elements can be combined with the two translational symmetry elements the screw... 

A single crystal, considered as a finite object, may possess a certain combination of point symmetry elements in different directions, and the symmetry operations derived from them constitute a group in the mathematical sense. The self-consistent set of symmetry elements possessed by a crystal is known as a crystal class (or crystallographic point group). Hessel showed in 1830 that there are thirty-two self-consistent combinations of symmetry elements n and n (n = 1,2,3,4, and 6), namely the thirty-two crystal classes, applicable to the description of the external forms of crystalline compounds. This important... 

The thirty-two crystal classes (crystallographic point groups) described in Section 9.1.4 can also be classified into the same seven crystal systems, depending on the most convenient coordinate system used to indicate the location and orientation of their characteristic symmetry elements, as shown in Table 9.2.1. 

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. 

The 32 crystallographic point groups, first mentioned in Table 7.1, are now described in Table 7.8 (ordered by principal symmetry axes and also by the crystal system to which they belong). The 230 space groups of Schonflies and Fedorov were generated systematically by combining the 14 Bravais lattices with the intra-unit cell symmetry operations for the 32 crystallographic point... 

Table 7.8 The 32 Crystallographic Point Groups, Listed by Main Symmetry Axes or Plane, Using Both the Schoenflies Notation (S, e.g., C2v) and the Hermann-Mauguin or International Notation (HM, e.g., mm2)3...

In order to extend these ideas to three dimensions it is sufficient to combine the 32 crystallographic point groups with the possible translational symmetries in unit cells of the different crystal systems. 

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