Superspace point group

Groupe ponctuel de superespace (Fr.); Gruppo puntuale di superspazio (It.); 超空間の点群 (Ja.)

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 ($R_E,~R_I$) of orthogobal transformations. Both $R_E$ and $R_I$ may themselves be R-reducible in turn. They form the m-dimensional point group $K_E$, and the d-dimensional point group $K_I$, respectively.

Comments

On a lattice basis the point group elements are represented by integral matrices $\Gamma (R)$. The action of the point group on the reciprocal lattice is given by the integral matrix $\Gamma^*(R)$, which is the inverse transpose of $\Gamma (R)$.

The diffraction spots of an aperiodic crystal belong to a vector module $M^*$ that is the projection of the n-dimensional reciprocal lattice $\Sigma^*$ on the physical space. The projections of the basis vectors $a_{si}^*$ of $\Sigma^*$ are the basis vectors $a_{si}^*$ of the vector module $M^*$. Therefore, the action of the n-dimensional point group of the superspace group on the basis of $M^*$ is

$R_E a_i^* ~=~ \sum_{j=1}^n \Gamma^*(R)_{ij} a_j^* ,~~(i=1,\dots,n).$

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 $\Gamma_E^*(K)$ and $\Gamma_I^*(K)$ are integral representations of K, as are their conjugates $\Gamma_E(K)$ and $\Gamma_I(K)$.

Points in direct space, with lattice coordinates $x_1,\dots,x_n$ transform according to

In direct space the internal space $V_I$ is left invariant, and this subspace contains a d-dimensional lattice, that is left invariant.