Superspace group

Groupe de superespace (Fr.)

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-Na₂CO₃. 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.