Crystallographic point group

Crystallographic point groups are point groups in which translational periodicity is required. The so-called crystallographic restriction states that rotations of a symmetry of 2,3,4, and 6 may occur. 

It is necessary that the sum of the interior angles divided by the number of sides is a divisor of 360°. Therefore, 

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. 

TABLE 14.1 Crystal Systems and Their Point Groups  

Crystal System Point Groups (Hermann-Mauguin) Point Groups (Schoenflies) 

Source From Julian, M. M., Foundations of Crystallography with Computer Applications (Boca Raton, FL CRC Press, 2008), p. 114.  

The orientation of each symmetry element with respect to the three major crystallographic axes is defined by its position in the sequence that forms the symbol of the point group symmetry. The complete list of all 32 point groups is found in Table 1.8. 

The columns labeled First , Second and Third position describe both the symmetry elements found in the appropriate position and their orientation. When the corresponding symmetry element is a rotation axis, it is parallel to the specified crystallographic direction but mirror planes are always perpendicular to the corresponding direction. When the crystal system has a unique axis, e.g. 2 in the monoclinic crystal system, 4 in the 

Crystal First position Second position Third position Point 

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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... 

Invariance principle, 664 Invariance properties of quantum electrodynamics, 664 Inventory problem, 252,281,286 Inverse collisions, 11 direct and, 12 Inverse operator, 688 Investment problem, 286 Irreducible representations of crystallographic point groups, 726 Isoperimetric problems, 305 Iteration for the inverse, 60... 

Table 15 The crystallographic point groups (crystal classes).

Crystal family Symbol Crystal system Crystallographic point groups (crystal classes) Number of space groups Conventional coordinate system Bravais lattices... 

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. 

Table 2.1 presents the non-cubic crystallographic point groups with compared notation. 

Fig. 10.7. The crystallographic point groups arranged according to their order, ms, shown on the left, and linked to show sub- and supergroup relations (adapted from International Tables for Crystallography Vol. A, (1996) Table 10.3.2).

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... 

Table 2.9. The thirty-two crystallographic point groups in both International and Schonflies notation.

Underlines in the International notation for G show which operators are complementary ones. Alternatively, these may be identified from the classes of G H by multiplying each operator by 0 G is the ordinary crystallographic point group from which G was constructed by eq. (14.1.2) H is given first in International notation and then in Schonflies notation, in square brackets. Subscript a denotes the unit vector along [1 1 0]. 

Example 16.1-1 Find the Bravais lattices, crystal systems, and crystallographic point groups that are consistent with a C3z axis normal to a planar hexagonal net. 

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