Difference between revisions of "Point group"
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| − | <font color="blue">Groupe ponctuel</font> (''Fr'') | + | <font color="orange">زمرة نقطية</font> (''Ar''). <font color="blue">Groupe ponctuel</font> (''Fr''). <font color="red">Punktgruppe</font> (''Ge''). <font color="black">Gruppo punto</font> (''It''). <font color="purple">点群</font> (''Ja''). <font color="brown">Точечная группа симметрии</font> (''Ru''). <font color="green">Grupo puntual</font> (''Sp''). |
==Definition== | ==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 | + | 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 the symmetry operations of these groups 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== | ==Occurrence== | ||
Crystallographic point groups occur: | 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 ( | + | * 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|site-symmetry groups]] of points in lattices or in crystal structures, and as symmetries of atomic groups and coordination | + | * in [[point space]], as [[site symmetry|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== | ==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 '' | + | 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 operation]]s 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== | ||
| − | Chapter | + | *Chapter 3.2.1 of ''International Tables for Crystallography, Volume A'', 6th edition |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Latest revision as of 15:48, 30 November 2018
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 the symmetry operations of these groups 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 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 operations 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 3.2.1 of International Tables for Crystallography, Volume A, 6th edition