In theoretical and practical work one frequently needs to know the symmetry around a position in a molecule or in a crystal structure. The symmetry of the surroundings of an atom or of the center of gravity of a (more or less complex) group of atoms (ion, molecule, etc.) is determined, among others, by chemical bonds. The surroundings of such a constituent strongly influence the physical and chemical properties of a substance. A striking example is the pair `graphite and diamond', which both are chemically carbon but display different surroundings of the carbon atoms and thus extremely different chemical and physical properties.
The symmetry of the surroundings of a point , called the site symmetry or point symmetry of , is determined by the symmetry of the whole molecule or crystal and by the locus of in the molecule or crystal. Here, we are interested in crystallographic site symmetries only, i.e. the local symmetries around points in a crystal (better, in a crystal pattern). Strictly, one defines:
Definition (D 3.2.2) The set of all symmetry operations of a crystal pattern is called the space group of the crystal pattern. The set of all elements of , i.e. of the space group, which leave a given point fixed, is called the site symmetry, site-symmetry group, point-symmetry group, or point group of with regard to the space group .
In this manuscript the term site-symmetry group or, for short, site symmetry, is preferred for reasons which will become clear in Section 3.4.
Because of its periodicity each crystal has an infinite number of translations as symmetry operations, i.e. is an infinite set. However, a translation can not be an element of a site-symmetry group because a translation has no fixed point at all. The same holds for screw rotations and glide reflections.
For the description of the crystallographic symmetry operations, it is convenient to have available the notion of the `order of an isometry'.
Definition (D 3.2.2) An isometry W has the (or: is of) order , if holds, where is the identity operation, and is the smallest number, for which this equation is fulfilled.
Remark. The different isometries , = 1, ..., , form a group with elements. See also the definition (D 3.4.2) of the group order.
The following types of isometries may be elements of crystallographic site symmetries:
A rotation with the rotation angle is called an -fold rotation. Its symbol is . The symbols of the crystallographic rotations are = 2, =3, =, =4, =, =, =, (and 1 ) for the identity). The order of the rotation is .
( inversion), , , , , , and . The rotoinversion is identical with a reflection, see next item.
Question. Which isometry is , , , and ? The answer to this question is found at the end of this chapter.
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