Superspace point group

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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 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 a_{si}^* of Σ * 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

GammaDecomp.gif

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

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

EmbIncDir.gif

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