Space Groups and X-Ray Crystallography

Space groups (like point groups) constitute a very pretty branch of mathematics, but that (presumably) is not why chemists study them. Space groups are important to a chemist because they are essential to solving and interpreting crystal structures. Therefore, we conclude this chapter with three topics that relate space group theory directly to the use of X-ray crystallography to obtain chemically useful information. 

Systematic absences—these are essential to solving crystal structures and hence defining molecular and solid state structures. 

Distinguishing Space Groups by Systematic Absences. From the symmetry and metric properties of an X-ray diffraction pattern we can determine which of the 6 crystal systems and, further, which of the 11 Laue symmetries we are dealing with. Since we need to know the specific space group in order to solve and refine a crystal structure, we would still be in a highly unsatisfactory situation were it not for the fact that the X-ray data can tell us still more. 

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. 

In summary, the introduction of the centering points will cause the 2,1 reflection to disappear, but will not cause the 1,1 reflection to disappear. The reasoning used can readily be extended. All sets of planes that pass through the center of the cell (which requires them to have h + k even, that is, h + k - 2n) continue to produce reflections when centering is introduced. All 

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