Centering operations, “systematic absences

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. 

The same idea pertains when there are pure translational operators within the crystal, as when centering operations (see Chapter 3) relate equivalent sets of asymmetric units by translations of half of the unit cell lengths (e.g., C centering 0,0,0 and, , 0 etc.). Centering operations also produce subperiodicities within the unit cell, and as above, these subperiodicites in real space produce systematic absences in diffraction space. 

FIGURE 6.10 In (a) are the hOl diffraction intensities from a crystal of fructose 1,6 bisphosphatase from rabbit liver having space group 1222. In (b) is the hkO plane of reflections from C2221 canavalin. In both images the checkerboard patterns produced by systematic absences resulting from the I and C centering operations are evident. 

In Table 7.1, where real and reciprocal cell dimensions, or other distances are related, an orthogonal system is assumed for the sake of simplicity. For nonorthogonal systems, the relationships are somewhat more complicated and contain trigonometric terms (as we saw in Chapter 3), since the unit cell angles must be taken into account. Rotational symmetry is preserved in going from real to reciprocal space, and translation operations create systematic absences of certain reflections in the diffraction pattern that makes them easily recognized. As already noted, because of Friedel s law a center of symmetry is always present in diffraction space even if it is absent in the crystal. This along with the absence of... 

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