Point group

Groupe ponctuel (Fr); Punktgruppe (Ge); Grupo puntual (Sp); Gruppo punto (It); Точечная группа симметрии (Ru); 点群 (Ja).

Definition

A point group is a group of symmetry operations all of which leave at least one point unmoved. A crystallographic point group is a point group that maps a point lattice onto itself: in three dimensions rotations and rotoinversions are restricted to 1, 2, 3, 4, 6 and [math]\bar 1[/math], [math]\bar 2[/math] (= m), [math]\bar 3[/math], [math]\bar 4[/math], [math]\bar 6[/math] respectively.

Occurrence

Crystallographic point groups occur:

• in vector space, as symmetries of the external shapes of crystals (morphological symmetry), as well as symmetry of the physical properties of the crystal ("vector point group");
• in point space, as site-symmetry groups of points in lattices or in crystal structures, and as symmetries of atomic groups and coordination polyedra ("point point group").

Controversy on the nomenclature

The matrix representation of a symmetry operation consists of a linear part, which represents the rotation or rotoinversion component of the operation, and a vector part, which gives the shift to be added once the linear part of the operation has been applied. The vector part is divided into two components: the intrinsic component, which represents the screw and glide component of the operation, and the localisation component, which is non-zero when the symmetry element does not pass through the origin. The set of the linear parts of the matrices representing the symmetry operations of a space group is a representation of the point group of the crystal. On the other hand, the set of matrix-vector pairs representing the symmetry operations of a site symmetry group form a group which is isomorphic to a crystallographic point group. The vector part being in general non-zero, some authors refuse the term point group for the site-symmetry groups. On the other hand, all the symmetry elements of a site symmetry group do leave invariant at least one point, albeit not necessarily the origin, satisfying the above definition of point group.

See also

Chapter 10 in International Tables for Crystallography, Volume A