International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 9.8, pp. 941943
Section 9.8.5. Generalizations 
The basic structure of a modulated crystal does not always have spacegroup symmetry. Consider, for example, composite crystals (also called intergrowth crystals). Disregarding modulations, one can describe these crystals as composed of a finite number of subsystems, each with its own spacegroup symmetry. The lattices of these subsystems can be mutually incommensurate. In that case, the overall symmetry is not a space group, the composite crystal is incommensurate and so also is its basic structure. The superspace approach can also be applied to such crystals. Let the subsystems be labelled by an index ν. For the subsystem ν, we denote the lattice by with basis vectors (i = 1, 2, 3), its reciprocal lattice by with basis vectors (i = 1, 2, 3), and the space group by . The atomic positions of the basic structure are given by where is a lattice vector belonging to . In the special case that the subsystems are mutually commensurate, there are three basis vectors a*, b*, c* such that all vectors are integral linear combinations of them. In general, however, more than three basis vectors are needed, but never more than three times the number of subsystems. Suppose that the vectors form a basis set such that every can be expressed as an integral linear combination of them: with and . Then the vectors of the diffraction pattern of the unmodulated system are again of the form (9.8.4.5) and generate a vector module of dimension three and rank , which can be considered as projection of a dimensional lattice .
We now assume that one can choose for i = 1, 2, 3 and we denote by . This corresponds to assuming the existence of a subset of Bragg reflections at the positions of a threedimensional reciprocal lattice *. Then there is a standard basis for the lattice , which is the reciprocal of , given by In order to find the dimensional periodic structure for which this composite crystal is the threedimensional intersection, one associates with a translation t in the internal space threedimensional independent shifts, one for each subsystem. These shifts of the subsystems replace the phase shifts adopted for the modulated structures: is now the space of the variable relative positions of the subsystems. Again, a translation in the superspace can give rise to a nonEuclidean transformation in the threedimensional space of the crystal, because of the variation in the relative positions among subsystems. Each subsystem, however, is rigidly translated. For the basis vectors , the shift of the subsystem ν is defined in terms of projection operators : Then an arbitrary translation in displaces the subsystem ν over a vector . A translation belonging to the dimensional lattice induces for the subsystem ν in ordinary space a relative translation over vector . The statement is that this translation is a vector of the lattice and leaves therefore the subsystem ν invariant. So the lattice translations belonging to form a group of symmetry operations for the composite crystal as a whole.
The proof is as follows. If k belongs to , the vector belongs to . In particular, for , one has, because of (9.8.5.2) and (9.8.5.4), and
Note that one has , for any t from as is a linear operator. Because of the linearity, this holds for every k from as well. Since belongs to and to , one has for their inner product: which implies that is an element of .
In conclusion, one may state that the composite structure is the intersection with the ordinary space (t = 0) of a pattern having atomic position vectors given by Such a pattern is invariant under the dimensional lattice . Again, orthogonal transformations R of O(3) leaving the vector module invariant can be extended to orthogonal transformation of allowing a Euclidean structure to be given to the superspace. One can then consider the superspacegroup symmetry of the basic structure defined by atomic positions as in (9.8.5.6). A superspacegroup element as in (9.8.4.28) induces (in threedimensional space) for the subsystem ν the transformation changing the position into an equivalent one of the composite structure, not necessarily, however, within the same subsystem ν.
Finally, the composite structure can also be modulated. For the case of a onedimensional modulation of each subsystem ν, the positions are Possibly the modulation vectors can also be expressed as integral linear combinations of the . Then, the dimension of is again . In general, however, one has to consider additional vectors, in order to ensure the validity of (9.8.4.5), now for n = 3 + d. We can then write The peaks of the diffraction pattern are at positions defined by a vector module M*, which can be considered as the projection of a (3 + d)dimensional lattice *, the reciprocal of which leaves invariant the pattern of the modulated atomic positions in the superspace given by with for , where is the internal part of the (3 + d)dimensional vector that projects on . Their symmetry is a (3 + d)dimensional superspace group . The transformation induced in the modulated composite crystal by an element under of is now readily written down. For the case and = ({Rv}, {Δ}) , the position is transformed into and the modulation into
This shows how the superspacegroup approach can be applied to a composite (modulated) structure. Note that composite systems do not necessarily have an invariant set of (main) reflections at lattice positions.
As said earlier, it sometimes makes sense also to use the description as developed for incommensurate crystal phases for a (commensurate) superstructure. In fact, very often the modulation wavevector also shows, in addition to continuously varying (incommensurate) values, several rational values at various phase transitions of a given crystal or in different compounds of a given structural family. In these cases, there is threedimensional spacegroup symmetry. Generally, the space groups of the various phases are different. The description as used for incommensurate phases then gives the possibility of a more unified characterization for the symmetry of related modulated crystal phases. In fact, the theory of higherdimensional space groups for modulated structures is largely independent of the assumption of irrationality. Only some of the statements need to be adapted. The most important change is that there is no longer a onetoone correspondence between the points of the reciprocal lattice * and its projection on V defining the positions of the Bragg peaks. Furthermore, the projection of the lattice on the space forms a discrete set. The latter means the following. For an incommensurate modulation, the incommensurate structure, which is the intersection of a periodic structure with the hyperplane , is also the intersection of the same periodic structure with a hyperplane = constant , where this constant is of the form Because for an incommensurate structure these vectors form a dense set in , the phase of the modulation function with respect to the basic structure is not determined. For a commensurate modulation, however, the points (9.8.5.12) form a discrete set, even belong to a lattice, and the phase (or the phases) of the modulation are determined within vectors of this lattice. Notice that the grid of this lattice becomes finer as the denominators in the rational components become larger.
When is a (3 + d)dimensional superspace group, its elements, in general, do not leave the ordinary space V invariant. The subgroup of all elements that do leave V invariant, when restricted to V, is a group of distancepreserving transformations in three dimensions and thus a subgroup of E(3), the threedimensional Euclidean group. In general, this subgroup is not a threedimensional space group. It is so when the modulation wavevectors all have rational components only, i.e. when σ is a matrix with rational entries. Because the phase of the modulation function is now determined (within a given rational number smaller than 1), the space group depends in general on this phase.
As an example, consider a onedimensional modulation of a basic structure with orthorhombic space group Pcmn. Suppose that the modulation wavevector is γc*. Then the mirror perpendicular to the c axis is combined with = = −1. Suppose, furthermore, that the glide reflection perpendicular to the a axis and the b mirror are both combined with a phase shift . In terms of the coordinates x, y, z with respect to the a, b and c axes, and internal coordinate t, the generators of the (3 + 1)dimensional superspace group Pcmn(00γ)ss0 act as Note that these positions are referred to a split basis (i.e. of basis vectors lying either in V or in ) and not to a basis of the lattice . When the superstructure is the intersection of a periodic structure with the plane at , its threedimensional space group follows from equation (9.8.5.13) by the requirement . When γ has the rational value r/s with r and s relatively prime, the conditions for each of the generators to give an element of the threedimensional space group are, respectively: for m, n, r, s integers and t real. These conditions are never satisfied simultaneously. It depends on the parity of both r and s which element occurs, and for the elements with = −1 it also depends on the value of the `phase' t, or more precisely on the product τ = 4st. The translation group is determined by the first condition as in (9.8.5.14a). Its generators are where the last vector is the external part of the lattice vector s(c, −r/s) + r(0, 1). The other spacegroup elements can be derived in the same way. The possible space groups are:

In general, the threedimensional space groups compatible with a given (3 + d)dimensional superspace group can be determined using analogous equations.
As one can see from the table above, the orthorhombic (3 + d)dimensional superspace group leads in several cases to monoclinic threedimensional space groups. The lattice of main reflections, however, still has orthorhombic pointgroup symmetry. Description in the conventional way by means of threedimensional groups then neglects some of the structural features present. Even if the orthorhombic symmetry is slightly broken, the orthorhombic basic structure is a better characterization than a monoclinic one. Note that in that case the superspacegroup symmetry is also only an approximation.
When the denominators of the wavevector components become small, additional symmetry operations become possible. Because the onetoone correspondence between * and M* is no longer present, there may occur symmetry elements with trivial action in V but with nontrivial transformation in . For d = 1, these possibilities have been enumerated. The corresponding Bravais classes are given in Table 9.8.3.2(b).