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:

**Identity**. The identity*I*is a member of any crystallographic site-symmetry group because it leaves any point fixed. It is the only operation whose order is 1, its*Hermann-Mauguin symbol*(*HM symbol*) is also 1. [The symbols have been introduced by CARL HERMANN and CHARLES MAUGUIN around 1930. There are symbols for point-group and for space-group*operations*, as well as for site-symmetry, space, and point*groups*. In IT A the symbols form the standard nomenclature].**Inversion**. If the inversion is a member of the site-symmetry group, then the point is the center of inversion. The order of the inversion is 2, its symbol is .**Rotations**. The point is placed on the rotation axis. Due to the periodicity of the crystals, the rotation angles of crystallographic rotations are restricted to multiples of and ,*i.e.*to , , , , , , and . All these angles are of the form , where =2, 3, 4, or 6, and is an integer which is relative prime to . (This restriction does not hold for the symmetry of molecules which may display,*e.g.*, a rotation angle of and its multiples.) Moreover, the angles between different axes of crystallographic rotations are limited to a small number of values only.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 .**Rotoinversions**. The point is placed in the inversion point on the rotoinversion axis. The restrictions on the angles of the rotational parts are the same as for rotations. If , the rotoinversion is called an*-fold*rotoinversion. The symbol for such a rotoinversion is . In crystals can occur:( 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.

**Reflections**. The point is situated on the mirror plane. There are only a few possible angles between the normals of different mirror planes belonging to the reflections of a site-symmetry group: , , , and . The symbol of a reflection is (mirror, miroir) instead of . The order of a reflection is 2, because its 2-fold application yields the identity operation.

**Copyright © 2002 International Union of
Crystallography**