Superspace point group - Revision history

BrianMcMahon: Tidied translations and added German and Spanish (U. Mueller)

2017-11-20T09:17:19Z

Tidied translations and added German and Spanish (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-17T11:54:26Z

Style edits to align with printed edition

MassimoNespolo: +cat, lang.

2014-12-04T05:54:41Z

+cat, lang.

BrianMcMahon at 11:29, 8 February 2012

2012-02-08T11:29:13Z

TedJanssen at 10:11, 19 May 2009

2009-05-19T10:11:25Z

TedJanssen at 10:10, 19 May 2009

2009-05-19T10:10:48Z

TedJanssen at 07:56, 19 May 2009

2009-05-19T07:56:41Z

TedJanssen at 07:01, 19 May 2009

2009-05-19T07:01:40Z

New page

[[superspace point group]]

Groupe ponctuel de superespace (Fr.)

'''Definition'''

An (''m+d'')-dimensional superspace group is a space group with a point group ''K'' that leaves
an ''m''-dimensional subspace invariant. Therefore, ''K'' is R-reducible and its elements are
pairs (<math>R_E,~R_I</math>) of orthogobal transformations. Both <math>R_E</math> and <math>R_I</math> may themselves be R-reducible in turn. They form the ''m''-dimensional point group <math>K_E</math>, and the ''d''-dimensional point group <math>K_I</math>, respectively.

'''Comments'''

On a lattice basis the point group elements are represented by integral matrices <math>\Gamma (R)</math>.
The action of the point group on the [[reciprocal lattice]] is given by the integral matrix <math>\Gamma^*(R)</math>, which is the inverse transpose of <math>\Gamma (R)</math>.

The diffraction spots of an aperiodic crystal belong to a [[vector module]] <math>M^*</math> that is the
projection of the ''n''-dimensional reciprocal lattice <math>\Sigma^*</math> on the physical space. The projections
of the basis vectors <math>a_{si}^*</math> of <math>\Sigma^*</math> are the basis vectors <math>a_{si}^*</math> of the vector module <math>M^*</math>. Therefore,
the action of the n-dimensional point group of the superspace group on the basis of <math>M^*</math> is

<math>R_E a_i^* ~=~ \sum_{j=1}^n \Gamma^*(R)_{ij} a_j^* ,~~(i=1,\dots,n).</math>

For an [[incommensurate modulated structure]], the submodule of the main reflections is invariant. As a consequence, the elements of the point group in superspace in this case is Z-reducible. There is a basis
such that the point group elements are represented by the integral matrices

<math>\Gamma^* (R) ~=~\left( \begin{array}{ll} \Gamma_E^*(R) & \Gamma_M^*(R) \\ 0 & \Gamma_I^* \end{array} \right).</math>

Both <math>\Gamma_E^*(K)</math> and <math>\Gamma_I^*(K)</math> are integral representations of ''K'',
as are their conjugates <math>\Gamma_E(K)</math> and <math>\Gamma_I(K)</math>.

← Older revision		Revision as of 09:17, 20 November 2017
Line 1:		Line 1:
−	<~~Font~~ color="blue">Groupe ponctuel de superespace</font> (''Fr''). <~~Font~~ color="black">Gruppo puntuale di superspazio</font> (''It''). <~~Font~~ color="purple">超空間の点群</font> (''Ja'').	+	<font color="blue">Groupe ponctuel de superespace</font> (''Fr''). <font color="red">Punktgruppe des Superraums</font> (''Ge''). <font color="black">Gruppo puntuale di superspazio</font> (''It''). <font color="purple">超空間の点群</font> (''Ja''). <font color="green">Grupo puntual del superespacio</font> (''Sp'').
		+

	== Definition ==		== Definition ==

@@ Line 1: / Line 1: @@
-<Font color="blue">Groupe ponctuel de superespace</font> (''Fr''.); <Font color="black">Gruppo puntuale di superspazio</font> (''It''.); <Font color="purple">超空間の点群</font> (''Ja''.)
+<Font color="blue">Groupe ponctuel de superespace</font> (''Fr''). <Font color="black">Gruppo puntuale di superspazio</font> (''It''). <Font color="purple">超空間の点群</font> (''Ja'').
 == Definition ==
@@ Line 5: / Line 5: @@
 An (''m+d'')-dimensional superspace group is a space group with a point group ''K'' that leaves
 an ''m''-dimensional (real) subspace invariant. Therefore, ''K'' is R-reducible and its elements are
-pairs (<math>R_E,~R_I</math>) of orthogobal transformations. Both <math>R_E</math> and <math>R_I</math> may themselves be R-reducible in turn. They form the ''m''-dimensional point group <math>K_E</math>, and the ''d''-dimensional point group <math>K_I</math>, respectively.
+pairs (<math>R_E,~R_I</math>) of orthogonal transformations. Both <math>R_E</math> and <math>R_I</math> may themselves be R-reducible in turn. They form the ''m''-dimensional point group <math>K_E</math>, and the ''d''-dimensional point group <math>K_I</math>, respectively.
 == Comments ==
@@ Line 15: / Line 15: @@
 projection of the ''n''-dimensional reciprocal lattice <math>\Sigma^*</math> on the physical space. The projections
 of the basis vectors <math>a_{si}^*</math> of <math>\Sigma^*</math> are the basis vectors <math>a_{si}^*</math> of the vector module <math>M^*</math>. Therefore,
-the action of the n-dimensional point group of the superspace group on the basis of <math>M^*</math> is
+the action of the ''n''-dimensional point group of the superspace group on the basis of <math>M^*</math> is
 <math>R_E a_i^* ~=~ \sum_{j=1}^n \Gamma^*(R)_{ij} a_j^* ,~~(i=1,\dots,n).</math>

@@ Line 1: / Line 1: @@
-<Font color="blue">Groupe ponctuel de superespace</font> (Fr.)
+<Font color="blue">Groupe ponctuel de superespace</font> (''Fr''.); <Font color="black">Gruppo puntuale di superspazio</font> (''It''.); <Font color="purple">超空間の点群</font> (''Ja''.)
 == Definition ==
@@ Line 32: / Line 32: @@
 In direct space the internal space  <math>V_I</math> is left invariant, and this subspace contains a       ''d''-dimensional lattice, that is left invariant.

← Older revision		Revision as of 10:10, 19 May 2009
Line 29:		Line 29:
	Both <math>\Gamma_E^(K)</math> and <math>\Gamma_I^(K)</math> are integral representations of ''K'',		Both <math>\Gamma_E^(K)</math> and <math>\Gamma_I^(K)</math> are integral representations of ''K'',
	as are their conjugates <math>\Gamma_E(K)</math> and <math>\Gamma_I(K)</math>.		as are their conjugates <math>\Gamma_E(K)</math> and <math>\Gamma_I(K)</math>.
		+
		+	Points in direct space, with lattice coordinates <math>x_1,\dots,x_n</math> transform according to
		+
		+	[[Image:EmbIncDir.gif]]
		+
		+	In direct space the internal space <math>V_I</math> is left invariant, and this subspace contains a ''d''-dimensional lattice, that is left invariant.

@@ Line 1: / Line 1: @@
 [[superspace point group]]
-Groupe ponctuel de superespace (Fr.)
+<Font color="blue">Groupe ponctuel de superespace</font> (Fr.)
   '''Definition'''
@@ Line 21: / Line 21: @@
 <math>R_E a_i^* ~=~ \sum_{j=1}^n \Gamma^*(R)_{ij} a_j^* ,~~(i=1,\dots,n).</math>
 For an [[incommensurate modulated structure]], the submodule of the main reflections is invariant. As a consequence, the elements of the point group in superspace in this case is Z-reducible. There is a basis
 such that the point group elements are represented by the integral matrices
-<math>\Gamma^* (R) ~=~\left( \begin{array}{ll} \Gamma_E^*(R) & \Gamma_M^*(R) \\ 0 & \Gamma_I^* \end{array} \right).</math>
+[[Image:GammaDecomp.gif]]
 Both <math>\Gamma_E^*(K)</math> and <math>\Gamma_I^*(K)</math> are integral representations of ''K'',
 as are their conjugates <math>\Gamma_E(K)</math> and <math>\Gamma_I(K)</math>.