Superspace group - Revision history

BrianMcMahon: Added German and Spanish translations (U. Mueller)

2017-11-20T09:09:03Z

Added German and Spanish translations (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-17T11:51:49Z

Style edits to align with printed edition

MassimoNespolo: +cat, lang.

2014-12-04T05:51:07Z

+cat, lang.

BrianMcMahon at 12:43, 19 May 2014

2014-05-19T12:43:36Z

BrianMcMahon at 11:28, 8 February 2012

2012-02-08T11:28:11Z

TedJanssen at 09:57, 19 May 2009

2009-05-19T09:57:18Z

TedJanssen at 06:27, 19 May 2009

2009-05-19T06:27:24Z

TedJanssen at 15:53, 18 May 2009

2009-05-18T15:53:43Z

TedJanssen at 15:32, 18 May 2009

2009-05-18T15:32:33Z

New page

[[Superspace group]]

Groupe de superespace (Fr.)

'''Definition'''

A ''superspace group'' is an ''n''-dimensional space group such that its point group
leaves an ''m''-dimensional subspace (''m''=1,2,3) invariant. An aperiodic crystal
structure in ''m''-dimensional physical space may be obtained as the intersection
of the ''m''-dimensional subspace with a lattice periodic structure in the ''n''-dimensional space. Its symmetry group is a superspace group.

'''History'''

Superspace groups were introduced by P.M. de Wolff to describe the incommensurate
modulated structure of γ-Na<sub>2</sub>CO<sub>3</sub>. Later the theory was generalized,
first to modulated structures with more modulation wave vectors, later for
incommensurate composite structures and quasicrystals. The general theory applies
to quasiperiodic crystal structures.

'''Comment'''

Superspace groups in ''n'' dimensions are ''n''-dimensional space groups, but
not all space groups are superspace groups, because not all of them have point groups
leaving a physical space invariant. On the other hand, the equivalence relations
are different. Two ''n''-dimensional space groups may be equivalent as space groups
(they belong to the same space group type), but non-equivalent as superspace groups
when the transformation from one point group to the other does not leave the
physical space invariant. So, the four-dimensional hypercubic group is not a
superspace group, because there is no invariant subspace for its point group. On the
other hand, the groups P2(1) and Pm(-1} are equivalent as four-dimensional
space groups (both are P211), but non-equivalent as superspace groups.

← Older revision		Revision as of 09:09, 20 November 2017
Line 1:		Line 1:
−	<~~Font~~ color="blue">Groupe de superespace </font>(''Fr''). <~~Font~~ color="black">Gruppo di superspazio </font>(''It''). <~~Font~~ color="purple">超空間群 </font>(''Ja'').	+	<font color="blue">Groupe de superespace</font> (''Fr''). <font color="red">Superraumgruppe</font> (''Ge''). <font color="black">Gruppo di superspazio</font> (''It''). <font color="purple">超空間群</font> (''Ja''). <font color="green">Grupo superespacial</font> (''Sp'').
		+

	== Definition ==		== Definition ==

@@ Line 1: / Line 1: @@
-<Font color="blue">Groupe de superespace </font>(Fr.); <Font color="black">Gruppo di superspazio </font>(It.);　<Font color="purple">超空間群 </font>(Ja.)
+<Font color="blue">Groupe de superespace </font>(''Fr''). <Font color="black">Gruppo di superspazio </font>(''It''). <Font color="purple">超空間群 </font>(''Ja'').
 == Definition ==
@@ Line 10: / Line 10: @@
 == History ==
-Superspace groups were introduced by P.M. de Wolff to describe the incommensurate
+Superspace groups were introduced by P. M. de Wolff to describe the incommensurate
 modulated structure of &gamma;-Na<sub>2</sub>CO<sub>3</sub>. Later the theory was generalized,
 first to modulated structures with more modulation wave vectors, later for
@@ Line 26: / Line 26: @@
 physical space invariant. So, the four-dimensional hypercubic group is not a
 superspace group, because there is no invariant subspace for its point group. On the
-other hand, the groups P2(1) and Pm(-1) are equivalent as four-dimensional
+other hand, the groups ''P''2(1) and ''Pm''(-1) are equivalent as four-dimensional
-space groups (both are P211), but non-equivalent as superspace groups.
+space groups (both are ''P''211), but non-equivalent as superspace groups.
 [[Category: Fundamental crystallography]]

@@ Line 1: / Line 1: @@
-<Font color="blue">Groupe de superespace </font>(Fr.)
+<Font color="blue">Groupe de superespace </font>(Fr.); <Font color="black">Gruppo di superspazio </font>(It.);　<Font color="purple">超空間群 </font>(Ja.)
 == Definition ==
@@ Line 28: / Line 28: @@
 other hand, the groups P2(1) and Pm(-1) are equivalent as four-dimensional
 space groups (both are P211), but non-equivalent as superspace groups.

@@ Line 1: / Line 1: @@
 <Font color="blue">Groupe de superespace </font>(Fr.)
- '''Definition'''
+== Definition ==
 An (''m+d'')-dimensional  ''superspace group'' is an ''n''-dimensional space group (with ''n=m+d'') such that its point group
@@ Line 11: / Line 8: @@
 of the ''m''-dimensional subspace with a lattice periodic structure in the ''n''-dimensional space.  Its symmetry group is a superspace group.
- '''History'''
+== History ==
 Superspace groups were introduced by P.M. de Wolff to describe the incommensurate
@@ Line 19: / Line 16: @@
 to quasiperiodic crystal structures.
- '''Comment'''
+== Comment ==
 Superspace groups in ''n'' dimensions are ''n''-dimensional space groups, but

@@ Line 7: / Line 7: @@
 An (''m+d'')-dimensional  ''superspace group'' is an ''n''-dimensional space group (with ''n=m+d'') such that its point group
-leaves an ''m''-dimensional subspace (''m''=1,2,3) invariant. An aperiodic crystal
+leaves an ''m''-dimensional (real) subspace (''m''=1,2,3) invariant. An aperiodic crystal
 structure in ''m''-dimensional physical space may be obtained as the intersection
 of the ''m''-dimensional subspace with a lattice periodic structure in the ''n''-dimensional space.  Its symmetry group is a superspace group.