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 ($R_E,~R_I$) of orthogonal 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.

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.