Space-groups symmetries glide-reflection

All the above examples applied to point groups. Antisymmetry and color symmetry, of course, may be introduced in space-group symmetries as well as examples illustrate in Figures 8-32, 8-37, and 9-46 (in the discussion of space groups). If we look only at the close-up of the tower in Figure 4-14b, it also has tranlational antisymmetry, specifically anti-glide-reflection symmetry together with similarity symmetry (these symmetries will be discussed in Chapter 8). 

Glide-Reflection. This operation is referred to a symmetry element called a glide plane. We have already employed a glide line (its 2D equivalent) in developing the 2D space groups. 

Crystal symmetries that entail centering translations and/or those symmetry operations that have translational components (screw rotations and glides) cause certain sets of X-ray reflections to be absent from the diffraction pattern. Such absences are called systematic absences. A general explanation of why this happens would take more space and require use of more diffraction theory than is possible here. Thus, after giving only one heuristic demonstration of how a systematic absence can arise, we shall go directly to a discussion of how such absences enable us to take a giant step toward specifying the space group. 

Along with the primitive translations, and glide-reflections when appropriate, there axe other symmetry operations belonging to the space group. In bipartite systems, it is relevant to classify any symmetry operation according to whether it leaves each sublattice invariant or transforms one into each other (see, for instance, Fig. 2). 

Figure 8-1. Planar decoration with two-dimensional space group after Budden [4], (a) The decoration (b) Symmetry elements of the pattern (c) Some of the glide reflection planes and their effects in the pattern.

The internal symmetry of the crystal is revealed in the symmetry of the Bragg reflection intensities, as discussed in Chapter 4. The crystal system is derived by examining the symmetry among various classes of reflections. Key patterns in the diffraction intensities indicate the presence of specific symmetry operations and lead to a determination of the space group. The translational component of the symmetry elements, as in glide planes or screw axes, causes selective and predictable destructive interference to occur. These are the systematic absences that characterize these symmetry elements, described in Chapter 4. 

The additional symmetry elements which are necessary for the 230 space groups to define the symmetry of all crystals (i.e. enantiomorphic and non-enantiomorphic) are glide planes (i.e. mirror reflection + translation) and improper rotation axes (rotation axis + inversion). 

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