# Superspace group

### From Online Dictionary of Crystallography

##### Revision as of 15:32, 18 May 2009 by TedJanssen (talk | contribs)

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_{2}CO_{3}. 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.