Thoughts on Space Groups

In identifying symmetry elements present in reciprocal space, we are seeking to establish symmetry relationships between intensities in various parts of the three-dimensional diffraction pattern. In doing so, it is necessary to remember that a symmetry relationship observed for a single plane of the diffraction pattern, because of Freidel s law, may not pertain to the entire pattern, and this can only be ascertained by examining additional planes through reciprocal space. 

For macromolecular crystals, the symmetry of the diffraction pattern (the Laue symmetry) must be generated by Friedel s law, plus the rotational components of symmetry axes present in the crystal. Once the rotational elements have been identified, it is necessary to deduce whether they are pure rotational operators or some sort of screw axes. For dyads, 

FIGURE 6.21 In (a), the MO diffraction plane of R3 canavalin exhibits 6mm symmetry, but because of Friedel s law it could arise as a consequence of either a true sixfold axis or a threefold axis plus the Friedel center of symmetry. In (b), the M2 image, which is along the same direction but does not contain Friedel related reflections, exhibits only threefold symmetry. This demonstrates that the crystal does in fact belong to the trigonal system and not the hexagonal system. 

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