Space-groups symmetries identity period

Ultimately we want to know how a crystal diffracts X rays and produces the diffraction pattern that it does, and conversely, how the diffraction pattern can be used to reconstruct the crystal. It will be found useful in this regard to consider the crystal as the combination, or product of two distinct components, or functions. The first of these is the contents of a unit cell, characterized mathematically by the coordinates of the atoms in an asymmetric unit along with their space group symmetry equivalent positions. The second is a point lattice that describes the periodic distribution of the unit cell contents, and is characterized by a, b, and c. A crystal may then be concisely defined as the first component, or function, repeated in identically the same way at every nonzero point of the second. This physical process of repetitive superposition is termed a convolution. It can be formulated mathematically as the product of the two components, or functions as... 

Simple translation is the most obvious symmetry element of the space groups. It brings the pattern into congruence with itself over and over again. The shortest displacement through which this translation brings the pattern into coincidence with itself is the elementary translation or elementary period. Sometimes it is also called the identity period. The presence of translation is seen well in the pattern in Figure 8-1. The symmetry analysis of the whole pattern was called by Budden the analytical approach. The reverse procedure is the... 

An important application of one-dimensional space groups is for polymeric molecules in chemistry. Figure 8-13 illustrates the structure and symmetry elements of an extended polyethylene molecule. The translation, or identity period, is shown, which is the distance between two carbon atoms separated by a third one. However, any portion with this length may be selected as the identity period along the polymeric chain. The translational symmetry of polyethylene is characterized by this identity period. 

Because atomic arrangements in crystal structures are periodic from unit cell to unit cell, one part of a molecule may lie in one unit cell, and another part may lie in an adjacent unit cell. If there is an atom at x. because of this translational symmetry there is another at 1 + x, and another ai n + x, where n is any integer. Values of x, y, and that are reported usually correspond to a complete and distinct molecule so that, for convenience, some atomic coordinates may have negative values, and other atomic coordinates may have values greater than 1.0000. The symmetry of the space group and the identical contents of adjacent unit cells can lead to a diagram of the complete crystal structure and an analysis not only of the molecular structure, but also of its surroundings. 

Thus, a space group will have for its identity element the invariant subgroup of translations and the other elements will be the cosets of this subgroup. The identity element will have all the translations corresponding to the periodicity of the lattice and the other elements (cosets) will correspond to the symmetry of the lattice. 

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