Symmetry operations, crystallography

Only 230 different arrangements of these symmetry operations are possible in three-dimensional space and the properties of the corresponding 230 space groups are listed in International Tables for Crystallography, Vol. A (1996) supplemented by the information given in Appendix 2. Every crystalline solid must necessarily satisfy the constraints imposed by one of these space groups even if it is only PI, the space group that has no internal symmetry. 

Symmetry Elements and Symmetry Operations 46 Point Croups and Molecular Symmetry 53 Irreducible Representations and Character Tables 59 Uses of Point Croup Symmetry 63 Crystallography 74... 

In crystallography, we are concerned with translational symmetry as well as point group symmetry, and this means that we must add two additional translational symmetry operators (Figure 8.14) ... 

In crystallography one is accustomed to the idea that a structure either has a particular symmetry element or it does not. The membership is thus either 1 or zero. In morphological analysis the symmetry has a value of zero through 1 depending upon how closely the profile approaches the symmetry being considered. The definitions of the symmetry operations are shown below ... 

Before going on to discuss the molecular symmetry groups in more detail we note one feature that they all possess. A symmetry operation which rotates or reflects a molecule into itself must leave the centre of mass (centre of gravity) of the molecule unmoved if the molecule has a plane or axis of symmetry, the centre of mass must lie on this plane or axis. It follows that all the axes and planes of symmetry of a molecule must intersect in at least one common point and that at least one point remains fixed under all the symmetry operations of the molecule. For this reason, the symmetry group of molecule is generally referred to as its point group and we shall use this name, which is taken over from crystallography, from now on. 

One such approach, the symmorphy group approach [43,108], is based on the extension of the family of point symmetry operators to a much richer family of operations which preserve the general morphology of objects. (Note that the term "symmorphy" is used in a different sense in the crystallography literature, with reference to the symmorphic space groups of crystallography, also called semi-direct... 

This description formalizes symmetry operations by using the coordinates of the resulting points and, therefore, it is broadly used to represent both symmetry operations and equivalent positions in the International Tables for Crystallography (see Table 1.18). The symbolic description of symmetry operations, however, is not formal enough to enable easy manipulations involving crystallographic symmetry operations. 

Therefore, symmetrical transformations in the crystal are formalized as algebraic (matrix-vector) operations - an extremely important feature used in all crystallographic calculations in computer software. The partial list of symmetry elements along with the corresponding augmented matrices that are used to represent symmetry operations included in each symmetry element is provided in Table 1.19 and Table 1.20. For a complete list, consult the Intemational Tables for Crystallography, vol. A. 

Alternatively, an approach closer to traditional crystallography can be considered in some cases. The Fourier component k of the magnetic moment of atom j, which transforms to the atom js when the symmetry operator of Gk is applied (xy, = = S x i + L), is transformed as ... 

The equivalence of the important rotoreflection axes with rotoinversion axes or other symmetry operators is given in Table 4.1. In crystallography the rotoinversion operation is always preferred,... 

It is of interest to note that one may change the translation lattice of Fig. 5.3 by replacing the translation lattice vector c with the molecular helix lattice, keeping the translation symmetries a and This would lead to a match of the molecular helix symmetry with the crystal symmetry and even for irrational helices, a crystal stracture symmetry would be recognized. In fact, a whole set of new lattices can be generated replacing all three translation symmetry operations by helix symmetry operations [5]. Since a 1 1/1 hehx has a translational symmetry, this new space lattice description with helices would contain the traditional crystallography as a special case. 

Knowledge of basic crystallography starts from the conception of symmetry, symmetry planes and symmetry operations necessary to identify the parameters, like lattice, lattice planes, crystal lattices (i.e. Bravis lattice) describing different crystal symmetries. Miller indices h, k, 1) to identify crystal planes etc. are explained here. 

Thus the combination of 3-D translational and point symmetry operations leads to an infinite number of sets of symmetry operations. Mathematically, each of these sets forms a group, and they are called space groups. It can be shown that all possible periodic crystals can be described by only 230 space groups. These 230 space groups are described in tables, for example the International Tables for Crystallography [3.8]. 

Note also that, although they are often named as crystallographic classes the eombinations from Table 2.6 correspond in fact to group notion, since classes are only sub-units of the mathematical groups. Therefore, the name of symmetry classes doesn t refer to crystallography in strict mathematical meaning of the classes, but to the elassify-ing of the symmetries eonsidering the combinations of the symmetry operations. 

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