# Point group

(Difference between revisions)
 Revision as of 09:26, 12 October 2017 (view source)m (lang)← Older edit Revision as of 09:30, 12 October 2017 (view source)m (→Controversy on the nomenclature: typo)Newer edit → Line 10: Line 10: ==Controversy on the nomenclature== ==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 ''localization 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 [[group isomorphism|isomorphic]] to a crystallographic point group. The vector part being in general non-zero, some authors reject the term ''point group'' for the [[site symmetry|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. + 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 ''location 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 [[group isomorphism|isomorphic]] to a crystallographic point group. The vector part being in general non-zero, some authors reject the term ''point group'' for the [[site symmetry|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== ==See also==

## Revision as of 09:30, 12 October 2017

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

## 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 $\bar 1$, $\bar 2$ (= m), $\bar 3$, $\bar 4$, $\bar 6$ 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 polyhedra ('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 location 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 reject 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.