# Superspace group

### From Online Dictionary of Crystallography

Groupe de superespace (Fr.); Gruppo di superspazio (It.); 超空間群 (Ja.)

## Definition

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 (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.

## 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.