Rotation crystallographic point groups

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... 

Use arabic numerals or combinations of numerals and the italic letter m to designate the 32 crystallographic point groups (Hermann-Mauguin). The number is the degree of the rotation, and m stands for mirror plane. Use an overbar to indicate rotation inversion. 

Figure 2.33. Illustration of crystallographic point group operations. Shown are (a) rotation axis, (b) rotation-inversion axis, and (c) mirror plane.

Note, that since five-fold rotational symmetry cannot be propagated on alattice, there are only 32 Crystallographic point groups, since the icosahedral groups are excluded. 

A solid can belong to one of an infinite number of general three-dimensional point groups. However, if the rotation axes are restricted to those that are compatible with the translation properties of a lattice, a smaller number, the crystallographic point groups, are found. The operators allowed within the crystallographic... 

In order to discuss the rest of the crystallographic point groups, one further has to consider the dihedral rotation groups D4 and D, and the cubic rotation group O. Their character tables, standard basis functions, and a useful choice of group generators are displayed in Tables 6,7, and 8. In this way the material required for symmetry considerations is directly available. 

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. 

I 1.9. Determine the crystallographic point group for each of the following crystals, where the rotational axes and mirror planes are indicated. Use both the Schoenflies and Hermann-Mauguin notations. 

One way to visualize the 32 crystallographic point groups is to use stereographic projections (Figure 2.37). The symbolism used to illustrate rotation axes are as follows ... 

As illustrated by the stereographic projections in Figure 2.37, the 6mm crystallographic point group is observed to have a primary sixfold axis of rotation, with two... 

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