Symmetry elements in crystals

Microscopic Symmetry Elements in Crystals Stacking sequence of Billiard balls... 

Microscopic Symmetry Elements in Crystals 39 Table 5.1. Eleven different types of screw axes of symmetries... 

The axes, planes and centre of symmetry are known as the elements of symmetry. All these elements intersect at one point, the centre of gravity of the molecule, which does not change during these operations. Hence the designation point symmetry, in contrast to translational symmetry observed in crystals. Let us take the simple molecule, say, water to understand some of these terminologies (Figure 2.12). 

The simplest symmetry operations and elements needed to describe unitcell symmetry are translation, rotation (element rotation axis), and reflection (element mirror plane). Combinations of these elements produce more complex symmetry elements, including centers of symmetry, screw axes, and glide planes (discussed later). Because proteins are inherently asymmetric, mirror planes and more complex elements involving them are not found in unit cells of proteins. All symmetry elements in protein crystals are translations, rotations, and screw axes, which are rotations and translations combined. 

A single crystal, considered as a finite object, may possess a certain combination of point symmetry elements in different directions, and the symmetry operations derived from them constitute a group in the mathematical sense. The self-consistent set of symmetry elements possessed by a crystal is known as a crystal class (or crystallographic point group). Hessel showed in 1830 that there are thirty-two self-consistent combinations of symmetry elements n and n (n = 1,2,3,4, and 6), namely the thirty-two crystal classes, applicable to the description of the external forms of crystalline compounds. This important... 

The seven crystal systems in three-dimensional space are listed with their defining symmetry elements in Table 7.4. 

Note that only those symmetry elements which intersect the asjnnmetric part of the unit cell are independent, exactly in the same way as only those atoms that are found in the asymmetric part of the unit cell are independent (see Figure 1.6). Once the locations of independent atoms and symmetry elements in the unit cell are known, the whole crystal can be easily... 

Table 1.20. Selected symmetry elements in trigonal and hexagonal crystal systems, their orientation and corresponding symmetry operations in the algebraic form as augmented matrices (see Figure 1.51). ...

Similar kinds of relationships would arise among equivalent reflections for threefold or sixfold axes in the crystal as well. This tells us that symmetry elements in real space, the crystal, may be identified by searching the appropriate zones, or planes of reciprocal space for symmetrical patterns of diffraction intensities. If we see fourfold or threefold or sixfold distributions of reflections in the diffraction patterns, then they must imply corresponding symmetry relationships in the crystal. 

As described previously, the position of the symmetry element in the overall symbol has a structural significance that varies from one crystal system to another as set out in Table 5.4. 

Two different values of REC for particular rings of benzene and phenanthrene result from different measurements of their molecular geometry. The different REC values for symmetrically equivalent rings in other benzenoid hydrocarbons result from the method of X-ray structure determination. If the symmetrical molecule does not lie at the symmetry element in the crystal lattice, each of the rings is measured independently and its geometry (and in turn its REC value) is biased by the another error of measurement. The differences in REC values for symmetrically equivalent... 

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