Group 32 crystallographic point 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... 

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. 

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. 

Table 16.2. Crystal systems, space lattices, and crystallographic point groups.

Find the Bravais lattice and crystallographic point groups that are compatible with a C2 axis. [Hint Use eq. (16.1.17).]... 

Both methods can easily be derived from a very simple model. Consider a unit cell which contains two XY molecules which are equivalent in the group-theoretical sense, i.e. they are transformed into one-another by the operations of the group of the unit cell (this group is the factor group of the space group 10 and is isomorphous with one of the 32 crystallographic point groups).8 ... 

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... 

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