Difference between revisions of "Space group"
From Online Dictionary of Crystallography
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The symmetry group of a three-dimensional [[crystal pattern]] is called its '''space group'''. In ''E''<sup>2</sup>, the symmetry group of a two-dimensional crystal pattern is called its '''plane group'''. In ''E''<sup>1</sup>, the symmetry group of a one-dimensional crystal pattern is called its '''line group'''. | The symmetry group of a three-dimensional [[crystal pattern]] is called its '''space group'''. In ''E''<sup>2</sup>, the symmetry group of a two-dimensional crystal pattern is called its '''plane group'''. In ''E''<sup>1</sup>, the symmetry group of a one-dimensional crystal pattern is called its '''line group'''. | ||
| − | To each crystal pattern belongs an infinite set of translations '''T''', which are symmetry operations of that pattern. The set of all '''T''' forms a group known as the '''translation subgroup''' T of the space group G of the crystal pattern. T is an [[Abelian group]] and a [[normal subgroup]] of the space group. | + | To each crystal pattern belongs an infinite set of translations '''T''', which are symmetry operations of that pattern. The set of all '''T''' forms a group known as the '''translation subgroup''' T of the space group G of the crystal pattern. T is an [[Abelian group]] and a [[normal subgroup]] of the space group. The [[factor group]] G/T of a space group G and its translation subgroup is isomorphic to the [[point group]] P of G. |
==See also== | ==See also== | ||
Revision as of 12:22, 14 April 2017
Groupe d'espace (Fr); Raumgruppe (Ge); Gruppo spaziale (It); 空間群 (Ja).
The symmetry group of a three-dimensional crystal pattern is called its space group. In E2, the symmetry group of a two-dimensional crystal pattern is called its plane group. In E1, the symmetry group of a one-dimensional crystal pattern is called its line group.
To each crystal pattern belongs an infinite set of translations T, which are symmetry operations of that pattern. The set of all T forms a group known as the translation subgroup T of the space group G of the crystal pattern. T is an Abelian group and a normal subgroup of the space group. The factor group G/T of a space group G and its translation subgroup is isomorphic to the point group P of G.
See also
- Fixed-point-free space groups
- Symmorphic space groups
- Section 1.3 of International Tables for Crystallography, Volume A, 6th edition