Symmetry point groups Systematic absences

The type of arrangement of pattern-units is called the space-lattice . Secondly, the group of atoms forming a pattern-unit—the group of atoms associated with each lattice point—may have certain symmetries, and some of these symmetries cause further systematic absences of certain types of reflections from the diffraction pattern. The complex of symmetry elements displayed by the complete arrangement is known as the space-group. ... 

Seven crystal systems as described in Table 3.2 occur in the 32 point groups that can be assigned to protein crystals. For crystals with symmetry higher than triclinic, particles within the cell are repeated as a consequence of symmetry operations. The number of asymmetric units within the unit cell is related but not necessarily equal to the number of molecules in a unit cell, depending on how the molecules are related by symmetry operations. From the symmetry in the X-ray diffraction pattern and the systematic absence of specific reflections in the pattern, it is possible to deduce the space group to which the crystal belongs. 

As mentioned earlier, the unit-cell space group can be determined from systematic absences in the the diffraction pattern. With the space group in hand, the crystallographer can determine the space group of the reciprocal lattice, and thus know which orientations of the crystal will give identical data. All reciprocal lattices possess a symmetry element called a center cf symmetry or point of inversion at the origin. That is, the intensity of each reflection hkl is identical to the intensity of reflection -h k -1. To see why, recall from our discussion of lattice indices (Section II.B) that the the index of the (230) planes can also be expressed as (-2 -3 0). In fact, the 230 and the —2 -3 0 reflections come from opposite sides of the same set of planes, and the reflection intensities are identical. (The equivalence of Ihkl and l h k l is called Friedel s law,but there are exceptions. See Chapter 6, Section IV.) This means that half of the reflections in the reciprocal lattice are redundant, and data collection that covers 180° about any reciprocal-lattice axis will capture all unique reflections. 

The diffraction pattern of a crystal has its own syimnetry (known as Lane syrmnetry), related to the symmetry of the stmcture, thns in the absence of systematic errors (particnlarly absorption), reflections with different, bnt related, indices should have equal intensities. According to Friedel s law, the diffraction pattern of any crystal has a center of inversion, whether the crystal itself is centrosymmetric or not, that is, reflections with indices hkl and hkl ( Friedel equivalents ) are equal. Therefore Lane symmetry is equal to the point-group symmetry of a crystal plus the inversion center (if it is not already present). There are 11 Lane symmetry classes. For example, if a crystal is monoclinic (fi 90°), then I hkl) = I hkl) = I hkl) = I hkl) I hkl). For an orthorhombic crystal, reflections hkl, hkl, hkl, hid and their Friedel equivalents are equal. If by chance a monochnic crystal has f 90°, it can be mistaken for an orthorhombic, but Lane symmetry will show the error. 

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