Section 9.8.1.4.3. Four-dimensional crystallography

Let us summarize the results obtained in the previous paragraph. The matrices Γ(R) form a faithful integral representation of the three-dimensional point group K with a four-dimensional carrier space . It is a reducible representation having as invariant subspaces the physical three-dimensional space, denoted by V (or sometimes also by ), and the additional one-dimensional space, denoted by . In V, the four-dimensional point-group transformation acts as R (sometimes also denoted by ), in it acts as one of the two one-dimensional point-group transformations: the identity or the inversion. Therefore, the space can be made Euclidean with Γ(R) defining a four-dimensional point-group transformation , which is an element of a crystallographic subgroup of O(4). The four-dimensional point-group transformations are of the form (R, ɛ), with ɛ = ±1 and they act on the four-dimensional lattice basis as where ɛ stands for ɛ(R) as in (9.8.1.20b). So the point-group symmetry operations are crystallographic and given by pairs of a three-dimensional crystallographic point-group transformation and a one-dimensional ɛ = ±1, respectively. The case ɛ = −1 corresponds to an inversion of the phase of the modulation function.

As in the three-dimensional case, one can define equivalence classes among those four-dimensional point groups.

Two point groups and belong to the same geometric crystal class if their three-dimensional (external) parts (forming the point group and , respectively) are in the same three-dimensional crystal class [i.e. are conjugated subgroups of O(3)] and their one-dimensional internal parts (forming the point groups and , respectively) are equal. The latter condition implies that corresponding point-group elements have the same value of ɛ.

Such a geometric crystal class can then be denoted by the symbol of the three-dimensional crystal class together with the values of ɛ that correspond to the generators.

Also, the notion of arithmetic equivalence can be generalized to these four-dimensional point groups, as they admit the same faithful integral representation Γ(K) given above. This means that two such groups are arithmetically equivalent if there is a basis transformation for the reciprocal-vector module, which transforms main reflections into main reflections and satellites into satellites and which transforms one of the matrix groups into the other. The arithmetic classes are determined by the arithmetic equivalence class of the three-dimensional group [i.e. by ] and by the components of the modulation wavevector with respect to the corresponding reciprocal-lattice basis. This is because the elements ɛ are fixed by the relation Note that these (3 + 1)-dimensional equivalence classes are not simply those one obtains in four-dimensional crystallography, as the relation between the higher-dimensional space and the three-dimensional physical space V plays a fundamental role.

The embedded structures in four dimensions have lattice periodicity. So the symmetry groups are four-dimensional space groups, called superspace groups. The new name has been introduced because of the privileged role played by the three-dimensional subspace V. A superspace-group element consists of a point-group transformation (R, ɛ) and a translation (v, Δ). The action of such an element on the four-dimensional space is then given by It is important to realize that a superspace-group symmetry of an embedded crystal induces three-dimensional transformations leaving the original modulated structure invariant. Corresponding to (9.8.1.23), one obtains the following relations [cf. (9.8.1.13)]: with These are purely three-dimensional symmetry relations, but of course not Euclidean ones.

In three-dimensional Euclidean space, the types of space-group transformation are translations, rotations, rotoinversions, reflections, central inversion, screw rotations, and glide planes. Only the latter two transformations have intrinsic non-primitive translations. For superspace groups, the types of transformation are determined by the point-group transformations. By an appropriate choice of the basis in , each of the latter can be brought into the form By a choice of origin, each translational part can be reduced to its intrinsic part, which in combination with the point-group element (R, ɛ) gives one of the transformations in V indicated above together with the inversion, or the identity, or a shift in . So, for phase inversion (when ɛ = −1), the intrinsic shift in vanishes. When ɛ = +1, the intrinsic shift in is given by τ. It will be shown in Subsection 9.8.3.3 that the value of τ is one of Therefore, a superspace-group element can be denoted by a symbol that consists of a symbol for the three-dimensional part following the conventions given in Volume A of International Tables for Crystallography, a symbol that determines ɛ, and one for the corresponding intrinsic internal translation τ.