View source for Superspace point group

<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 ==

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 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 ==

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

[[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>.

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.

[[Category: Fundamental crystallography]]