Superspace point group
From Online Dictionary of Crystallography
Groupe ponctuel de superespace (Fr). Punktgruppe des Superraums (Ge). Gruppo puntuale di superspazio (It). 超空間の点群 (Ja). Grupo puntual del superespacio (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 () of orthogonal transformations. Both RE and RI may themselves be R-reducible in turn. They form the m-dimensional point group KE, and the d-dimensional point group KI, respectively.
Comments
On a lattice basis the point group elements are represented by integral matrices Γ(R). The action of the point group on the reciprocal lattice is given by the integral matrix Γ * (R), which is the inverse transpose of Γ(R).
The diffraction spots of an aperiodic crystal belong to a vector module M * that is the
projection of the n-dimensional reciprocal lattice Σ * on the physical space. The projections
of the basis vectors of Σ * are the basis vectors
of the vector module M * . Therefore,
the action of the n-dimensional point group of the superspace group on the basis of M * is
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
Both and
are integral representations of K,
as are their conjugates ΓE(K) and ΓI(K).
Points in direct space, with lattice coordinates transform according to
In direct space the internal space VI is left invariant, and this subspace contains a d-dimensional lattice, that is left invariant.