Equivalent positions and systematic absences in diffraction patterns

1 Equivalent positions and systematic absences in diffraction patterns 

In Sections 5.1 and 5.2 a molecular crystal was defined as an array of molecular objects related by symmetry operations, including pure translation a system endowed by periodic translational symmetry. The asymmetric unit is the symmetry-independent part of the crystal, and it is repeated within the unit cell into a number of equivalent positions, by as many symmetry operations. The unit cell contents are then repeated in space by pure translation. 

All OkO reflections with k odd have zero net intensity. The explanation is easy. The space group of the succinic anhydride crystal is P2i2i2i, and one of the equivalent positions of this space group is x,y,z —x, 1/2 - - y, 1/2 — z. Consider now equation 5.27 the summation has terms from all atoms in the unit cell, including atoms in equivalent positions. Therefore, for OkO reflections and for any atom j  

Whenever = 2n- -l, with n integer, exp[7riA ] = —1 and the structure factor vanishes. Thus, the above list of observed structure factors is indeed a direct experimental proof that in the crystal of succinic anhydride any scatterer atx,y,z has an equivalent scatterer at -x, 1/2 -b y, 1/2 - z. The same applies to (hOO) and (00/) reflections because of the other two equivalent positions of the space group. Internal symmetry is revealed by destructive interference of scattered waves from symmetry-related objects. In fact, the analysis of systematic absences is the method normally used for determining the space group from diffraction patterns. 

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