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. 907913
Section 9.8.1. Introduction 
Lattice periodicity is a fundamental concept in crystallography. This property is widely considered as essential for the characterization of the concept of a crystal. In recent decades, however, more and more longrangeordered condensedmatter phases have been observed in nature that do not have lattice periodicity, but nevertheless only differ from normal crystal phases in very subtle and not easy to observe structural and physical properties.
It is convenient to extend the concept of a crystal in such a way that it includes these ordered phases as well. If properly applied, crystallographic symmetry concepts are then still valid and of relevance for structural (and physical) investigation. Instead of limiting the validity of crystallographic notions, these developments show how rich the crystallographic ordering can be.
Incommensurate crystals are characterized by Bragg reflection peaks in the diffraction pattern that are well separated but do not belong to a lattice and cannot be indexed by three integer indices. This means that the description in terms of threedimensional point and space groups, as given in IT A (2005), breaks down. It is possible, however, to generalize the crystallographic concepts on which the usual description is based. The ensuing symmetry groups and their equivalence classes are different from those of threedimensional crystallography. The main properties of these new groups and the specific problems posed by their application to incommensurate crystal structures form the subject of the following sections.
The tables presented here, and their explanation, have been restricted to cover mainly the simplest case: modulated crystal structures (in particular, those with a single modulation wave). Nevertheless, most of the basic concepts also apply for the other classes of incommensurate crystal structures.
A modulation is here considered to be a periodic deformation of a `basic structure' having spacegroup symmetry. If the periodicity of the modulation does not belong to the periodicities of the basic structure, the modulated crystal structure is called incommensurate.
That description is based on the observation in the diffraction pattern of modulated structures of main reflections, situated on a reciprocal lattice, and additional reflections, generally of weaker intensity, called satellites.
As far as we know, the first indications that complex crystal structures could exist not having normal lattice periodicity came, at the end of the 19th century, from studies of the morphology of the mineral calaverite. It was found that the faces of this compound do not obey the law of rational indices (Smith, 1903; Goldschmidt, Palache & Peacock, 1931). Donnay (1935) showed that faces that could not be indexed correspond to additional reflections in the Xray diffraction pattern. From the experience with manufacturing of optical gratings, it was already known that periodic perturbation gives rise to additional diffraction spots. Dehlinger (1927) used this theory of `Gittergeister' (lattice ghosts) to explain line broadening in Debye–Scherrer diagrams of metals and alloys. Preston (1938) called the `Gittergeister' he found in diffraction from an aluminium–copper alloy `satellites'. Periodic displacement (modulation) waves were considered by Daniel & Lipson (1943) as the origin of satellites in a copper–iron–nickel alloy. Complex magnetic ordering giving rise to satellites was found in magnetic crystals (Herpin, Meriel & Villain, 1960; Koehler, Cable, Wollan & Wilkinson, 1962). In the early 1970's, the importance of the irrationality was realized and the term `incommensurate phase' was introduced for modulated crystal phases with wavevectors that have irrational components. Such phases were found in insulators as well (Tanisaki, 1961, 1963; Brouns, Visser & de Wolff, 1964). The remarkable morphological properties of calaverite could also be related to the existence of an incommensurate modulation (Dam, Janner & Donnay, 1985). Structures that are well ordered, show sharp incommensurate spots in the diffraction pattern, but cannot be described as a modulation of a lattice periodic system, were also discovered, indicating that incommensurability is not restricted to modulated crystals. Examples are the composite (or intergrowth) crystals (Jellinek, 1972) and, more recently, quasicrystals (Shechtman, Blech, Gratias & Cahn, 1984). The latter not only lack threedimensional lattice periodicity, but also show noncrystallographic symmetry elements, such as five, eight or twelvefold axes. For a review, see Janssen & Janner (1987).
We denote a basis for the lattice of main reflections by a*, b*, c*. The satellites are separated from the main reflections by vectors that are integral linear combinations of some basic modulation vectors denoted by q_{j} . Accordingly, the positions of the Bragg reflections are given by If the modulation is incommensurate, the set of all positions (9.8.1.1) does not form a lattice. Mathematically, it has the structure of a Zmodule, and its elements are threedimensional vectors (of the reciprocal space). Accordingly, we call it a (reciprocal) vector module.
The modulation that gives rise to the satellites can be:
Mixed forms also occur. Other modulation phenomena found are, for example, outofphase structures and translation interface modulated structures. The satellite reflections of modulated structures for the various types of modulation were systematically studied by Korekawa (1967).
For an incommensurate structure, at least one component of a basic modulation wavevector q_{j} with respect to the lattice of main reflections is irrational. As argued later, even in the case of a commensurate modulation, i.e. when all components of these modulation wavevectors are rational, it is sometimes convenient to adopt the description of equation (9.8.1.1) and to apply the formalism developed for the incommensurate crystal case.
The modulated crystal case presented above is only the simplest one giving rise to incommensurate crystal structures. Another class is represented by the socalled composite or intergrowth crystal structures. Here the basic structure consists of two or more subsystems each with its own spacegroup symmetry (neglecting mutual interaction) such that, in general, the corresponding lattices are incommensurate. This means that these lattices are not sublattices of a common one. Actually, a mutual interaction is in general present giving rise to periodic deviations, i.e. to modulations. These crystals are different from the previous ones, in the sense that they show more than one set of main reflections. In the case of an intergrowth crystal, the Bragg reflections can be labelled by a finite set of integral indices as well and the Fourier wavevectors form a vector module as defined above. The difference from the previous case is that the basic structure itself is incommensurate, even if one disregards modulation.
A third class of incommensurate crystals found in nature is represented by the quasicrystals (examples of which are the icosahedral phases). In that case also, the basic structure is incommensurate, the difference from the previous case being that the splitting in subsystems of main reflections and/or in main reflections and satellites is not a natural one, even in the lowest approximation.
It is justified to assume that crystal phases allow for an even wider range of ordered structures than those mentioned above. Nevertheless, all the cases studied so far allow a common approach based on higherdimensional crystallography, i.e. on crystallographic properties one obtains after embedding the crystal structure in a higherdimensional Euclidean space.
To present that approach in its full generality would require a too abstract language. On the other hand, restriction to the simplest case of a modulated crystal would give rise to misleading views. It what follows, the basic ideas are formulated as simply as possible, but still within a general point of view. Their application and the illustrative examples are mainly restricted to the modulated crystal case. The tables presented here, even if applicable to more general cases, cover essentially the onedimensional modulated case [d = 1 in equation (9.8.1.1)].
Incommensurate modulated crystals are systems that do not obey the classical requirements for crystals. Nevertheless, their longrange order is as perfect as that of ordinary crystals. In the diffraction pattern they also show sharp, well separated spots, and in the morphology flat faces. Dendritic crystallization with typical pointgroup symmetry is observed in both commensurate and incommensurate materials. Therefore, we shall consider both as crystalline phases and generalize for that reason the concept of a crystal. The positions of the Bragg diffraction peaks given in (9.8.1.1) are a special case. In general, they are elements of a vector module M* and can be written as This leads to the following definition of crystal.
An ideal crystal is considered to be a matter distribution having Fourier wavevectors expressible as an integral linear combination of a finite number (say n) of them and such that its diffraction pattern is characterized by a discrete set of resolved Bragg peaks, which can be indexed accordingly by a set of n integers .
Implicit in this definition is the possibility of neglecting diffraction intensities below a given threshold, allowing one to identify and to label individual Bragg peaks even when n is larger than the dimension m of the crystal, which is usually three. Actually, for incommensurate crystals, the unresolved Bragg peaks of arbitrarily small intensities form a dense set, because they may come arbitrarily close to each other.
Here some typical examples are indicated. In the normal crystal case, n = m = 3 and , , are conventionally denoted by a*, b*, c* and the indices by h, k, l.
In the case of a onedimensionally modulated crystal, which can be described as a periodic plane wave deformation of a normal crystal (defining the basic structure), one has n = 4. Conventionally, , , are chosen to be a*, b*, and c* generating the positions of the main reflections, whereas = q is the wavevector of the modulation. The corresponding indices are usually denoted by h, k, l and m. Also, crystals having two and threedimensional modulation are known: in those cases, n = 5 and n = 6, respectively.
In the case of a composite crystal, one can identify two or more subsystems, each with its own spacegroup symmetry. As an example, consider the case where two subsystems share a* and b* of their reciprocallattice basis, whereas they differ in periodicity along the c axis and have, respectively, and as third basis vector. Then again, n = 4 and can be chosen as , , , and . The indices can be denoted by h, k, l_{1}, l_{2}. In general, the subsystems interact, giving rise to modulations and possibly (but not necessarily) to a larger value of n. In addition to the main reflections from the undistorted subsystems, satellite reflections then occur.
In the case of a quasicrystal, the Bragg reflections require more than three indices, but they do not arise from structurally different subsystems having lattice periodicity. So, for the icosahedral phase of AlMn alloy, n = 6, and one can take for , , a cubic basis a*, b*, and c* and for , , then τa*, τb*, and τc*, respectively, with τ the golden number .
The Laue point group of a crystal is the point symmetry group of its diffraction pattern. This subgroup of the orthogonal group O(3) is of finite order. The finite order of the group follows from the discreteness of the (resolved) Bragg peak positions, implying a finite number only of peaks of the same intensity lying at a given distance from the origin.
Under a symmetry rotation R, the indices of each reflection are transformed into those of the reflection at the rotated position. Therefore, a symmetry rotation is represented by an nonsingular matrix Γ(R) with integral entries. Accordingly, a Laue point group admits an ndimensional faithful integral representation . Because any finite group of matrices is equivalent with a group of orthogonal matrices of the same dimension, it follows that:

Such a procedure is called the superspace embedding of the crystal.
Note that this procedure only involves a reinterpretation of the structural data (expressed in terms of Fourier coefficients): the structural information in the ndimensional space is exactly the same as that in the threedimensional description.
The symmetry group of a crystal is then defined as the Euclidean symmetry group of the crystal structure embedded in the superspace.
Accordingly, the symmetry of a crystal whose diffraction pattern is labelled by n integral indices is an ndimensional space group.
The equivalence relation of these symmetry groups follows from the requirement of invariance of the equivalence class with respect to the various possible choices of bases and embeddings. Because of that, the equivalence relation is not simply the one valid for ndimensional crystallography.
In the case of modulated crystals, such an equivalence relation has been worked out explicitly, giving rise to the concept of the (3 + d)dimensional superspace group.
The concepts of point group, lattice holohedry, Bravais classes, systems of nonprimitive translations, and so on, then follow from the general properties of ndimensional space groups together with the (appropriate) equivalence relations.
A glossary of symbols is given in Appendix 9.8.1 and a list of definitions in Appendix 9.8.2.
To introduce what follows, the simple case of a displacively modulated crystal structure is considered. The pointatom approximation is adopted and the modulation is supposed to be a sinusoidal plane wave.
This means that the structure can be described in terms of atomic positions of a basic structure with threedimensional spacegroup symmetry, periodically displaced according to the modulation wave. Writing for the position of the jth particle in the unit cell of the basic structure given by the lattice vector n: the position of the same particle in the modulated structure is given by where q is the wavevector of the modulation and is the polarization vector for the jth particle's modulation. (This is not the most general sinusoidal modulation, because different components may have different phases .) In general, the symmetry of the modulated structure is different from that of the basic structure and the only translations that leave the modulated structure invariant are those lattice translations m of the basic structure satisfying the condition = integer . If the components α, β and γ of the wavevector q with respect to the basis a*, b*, c* are all rational numbers, there is a full lattice of such translations, the structure then is a superstructure of the original one and has spacegroup symmetry. If at least one of the α, β, and γ is irrational, the structure does not have threedimensional lattice translation symmetry. Nevertheless, the crystal structure is by no means disordered: it is fully determined by the basic structure and the modulation wave(s).
The crystalline order is reflected in the structure factor, which is given by the expression where is the atomic scattering factor (which still, in general, depends on H). Using the Jacobi–Anger relation, one can rewrite (9.8.1.5) as where is the mthorder Bessel function. The summation over n results in a sum of δ functions on the positions of the reciprocal lattice: Consequently, the structure factor vanishes unless there are integers h, k, l, and m such that If q is incommensurate, i.e. if there is no integer N such that Nq belongs to the reciprocal lattice spanned by a*, b*, c*, one needs more than three integers (in the present case four) for indexing H. This is characteristic for incommensurate crystal phases.
In the diffraction pattern of such a modulated phase, one distinguishes between main reflections (for which m = 0) and satellites (for which . The intensities of the satellites fall off rapidly for large m so that the observed diffraction spots remain separated, although vectors of the form (9.8.1.7) may come arbitrarily close to each other.
In the commensurate case also, there are main reflections and satellites, but, since here there is an integer N such that Nq belongs to the reciprocal lattice, one may restrict the values of m in (9.8.1.7) to the range from 0 to N − 1.
There is more than one way for expressing the longrange order present in an incommensurate crystal in terms of symmetry. One natural way is to adopt the point of view that the measuring process limits the precision in the determination of a modulation wavevector. Accordingly, one can try an approximation of the modulation wavevector q by a commensurate one: an irrational number can be approximated arbitrarily well by a rational one.
There are two main disadvantages in this approach. Firstly, a good approximation implies, in general, a large unit cell for the corresponding superstructure, which involves a large number of parameters. Secondly, the space group one finds may depend essentially on the rational approximation adopted. Consider, for example, an orthorhombic basic structure with space group and modulation with wavevector , polarization along the b direction and positions of the particles given by where for convenience the polarization has been taken as independent of j. Under the glide reflection (integer m) present in the basic structure, the transformed positions are Consider the rational approximation of γ given by P/Q, with P and Q relatively prime integers. Then such a glide transformation as given above is (for certain values of m) a symmetry of the modulated structure only if P is even and Q odd, because only in that case does the equation have a solution for integers l and m. Analogously, the mirror only occurs as a glide plane if P/Q is odd/even, whereas the screw axis along the z axis requires the case odd/odd. Hence, if, for example, one approximates the same irrational number successively by 5/12, 7/17 or 8/19, one finds different symmetry groups. Furthermore, in all the cases, the point group is monoclinic and the information contained in the orthorhombic pointgroup symmetry of the main reflections is lost.
These difficulties are avoided if one embeds the modulated crystal according to the basic ideas expressed in the previous section. Conceptually, it corresponds (de Wolff, 1974, 1977; Janner & Janssen, 1977) to enlarging the class of symmetry transformations admitted, from Euclidean in three dimensions to Euclidean in n dimensions (here n = 4). A fourdimensional Euclidean operation transforms the threedimensional structure but does not leave distances invariant in general. To illustrate this phenomenon, consider the effect of a lattice translation of the basic structure on the modulation: it gives rise to a shift of the phase. Combining such a translation with an appropriate (compensating) shift of the phase, one obtains a transformation that leaves the basic structure and the modulation invariant. Such a transformation corresponds precisely to a lattice translation in four dimensions for the embedded crystal structure. In the real space, however, a shift of the phase is not an admitted Euclidean transformation as it does not leave invariant the distance between the particles, so that the combination of lattice translation and (compensating) phase shift also is not admitted as such, even if for an incommensurate modulation it simply corresponds to a relabelling of the atoms leaving the global structure invariant.
From what has been said, it should be apparent that the fourdimensional embedding of the modulated structure according to the superspace approach can also be obtained directly by considering as additional coordinate the phase t of the modulation. The positions in three dimensions are then the intersection of the lines of the atomic positions (parametrized by t) with the hyperplane t = 0. For the modulated structure of (9.8.1.4), these lines are given by and for a more general, not necessarily sinusoidal, modulation by where is a periodic vector function: . A general point in fourdimensional space is r_{s} = (r, t) with r = (x, y, z).
The notation adopted here for the (3 + 1)dimensional embedding of the modulated structure is a shorthand notation intended to stress the double role of the variable t as a parameter in the threedimensional description of the structure and as a coordinate in the (3 + 1)dimensional embedding. In the latter case, it implicitly assumes the choice of a fourth basis vector d perpendicular to the physical space of the crystal, so that the following fourdimensional vector notations are considered to be equivalent:
The pattern of lines (9.8.1.10a, b) has lattice periodicity. Indeed, it is invariant under the shift and, for every lattice translation n from the basic structure, there is a compensating phase shift: the pattern is left invariant under the combination of the translation n and the phase shift . Therefore, (9.8.1.10a, b) is invariant under translations from a fourdimensional lattice with basis In accordance with the equivalent descriptions given for , equivalent descriptions for are = (0, 1) = (0, d) with dual basis vector = (q, 1) = (q, d*), where d* is reciprocal to d in the fourth direction. With respect to the axes (9.8.1.11), a general point in four dimensions can be written as with Some, or sometimes all, transformations of the space group of the basic structure give rise to a symmetry transformation for the modulated structure when combined with an appropriate phase shift, possibly together with an inversion of the phase. If is an element of the space group of the basic structure, Δ is a phase shift and ɛ = ±1, then the point (r, t) is transformed to (Rr + v, ɛt + Δ). The pattern (9.8.1.10b) is left invariant by ({Rv}, {ɛΔ}) if The position is the transform of in the basic structure and therefore also belongs to the basic structure.
One can conclude that the pattern has as symmetry transformations all elements ({Rv}, {ɛΔ}) satisfying (9.8.1.13). These form a space group in four dimensions.
The reciprocal to the basis (9.8.1.11) is A general reciprocallattice vector is now The projection of this reciprocallattice vector is of the form (9.8.1.7), as it should be. Moreover, the projection of the fourdimensional Fourier transform on the hyperplane t = 0 is exactly the Fourier transform of the structure in this hyperplane. If, as a consequence of the fourdimensional spacegroup symmetry, the fourdimensional diffraction pattern shows systematic extinctions, the same extinctions are present in the diffraction pattern of the threedimensional structure.
There are also other ways to describe the symmetry relations of incommensurate modulated structures. One is based on representation theory. This method has in particular been used when the modulation occurs as a consequence of a soft mode. Then the irreducible representation of the space group to which the soft mode belongs gives information on the modulation function as well. Heine & McConnell have treated the symmetry with a method related to the Landau theory of phase transitions (Heine & McConnell, 1981; McConnell & Heine, 1984). PerezMato et al. have given a formulation in terms of threedimensional structures parametrized by what are here the additional coordinates (PerezMato, Madariaga & Tello, 1984, 1986; PerezMato, Madariaga, Zuñiga & Garcia Arribas, 1987). Other treatments of the problem can be found in Koptsik (1978), Lifshitz (1996), and de Wolff (1984).
For a modulated crystal structure with a onedimensional modulation, the positions of the diffraction spots are given by vectors This set of vectors is a vector module M*. The vectors form a basis of the reciprocal lattice Λ* of the basic structure and q is the modulation wavevector. The choice of the basis of Λ* has the usual freedom, the wavevector q is only determined up to a sign and up to a reciprocallattice vector of the basic structure.
A vector module M* has pointgroup symmetry K, which is the subgroup of all elements R of O(3) leaving it invariant.
In the case of an incommensurate onedimensional modulation, M* is generated by the lattice Λ* of main reflections and the modulation wavevector q. It then follows that K is characterized by the following properties:

An element R of K then transforms the basic vectors , , , q into ones of the form (9.8.1.15). If one denotes, as in (9.8.1.2), q by , this implies with Γ*(R) a 4 × 4 matrix with integral entries. In the case of an incommensurate modulated crystal structure, only two vectors with the same length as q are q and −q. As Λ* is left invariant, it follows that for a onedimensionally modulated structure Γ*(R) has the form This matrix represents the orthogonal transformation R when referred to the basis vectors (i = 1, 2, 3, 4) of the vector module M*. As in the case of lattices, two vector modules of modulated crystals are equivalent if they have bases (i.e. a basis for the reciprocal lattice Λ* of the basic structure together with a modulation wavevector q) such that the set of matrices Γ*(K) representing their symmetry is the same for both vector modules. Equivalent vector modules form a Bravais class.
Again, as in the case of threedimensional lattices, it is sometimes convenient to consider a vector module that includes as subset the one spanned by all diffraction spots as in (9.8.1.15). Within such a larger vector module, the actual diffraction peaks then obey centring conditions. For a vector module associated with a modulated structure, centring may involve main reflections (the basic structure then has a centred lattice), or satellites, or both. For example, if in a structure with primitive orthorhombic basic structure the modulation wavevector is given by , one may describe the diffraction spots by means of the nonprimitive lattice basis , , and by the modulation wavevector .
Crystallographic point groups are denoted generally by the same letter K.
The matrices Γ*(R) form a faithful integral representation of the threedimensional point group K. It is also possible to consider them as fourdimensional orthogonal transformations leaving a lattice with basis vectors (9.8.1.14) invariant. Indeed, one can consider the vectors (9.8.1.15) as projections of fourdimensional lattice vectors , which can be written as where [cf. (9.8.1.14)] m has now been replaced by and
As will be explained in Section 9.8.4, these vectors span the fourdimensional reciprocal lattice for a periodic structure having as threedimensional intersection (say defined by the hyperplane t = 0) the modulated crystal structure (a specific example has been given in Subsection 9.8.1.3). In direct space, the point group in four dimensions with elements of O(4) then acts on the corresponding dual basis vectors (9.8.1.11) of the fourdimensional direct lattice as where Γ(R) is the transpose of the matrix appearing in (9.8.1.17) and therefore for incommensurate onedimensionally modulated structures it has the form
Let us summarize the results obtained in the previous paragraph. The matrices Γ(R) form a faithful integral representation of the threedimensional point group K with a fourdimensional carrier space . It is a reducible representation having as invariant subspaces the physical threedimensional space, denoted by V (or sometimes also by ), and the additional onedimensional space, denoted by . In V, the fourdimensional pointgroup transformation acts as R (sometimes also denoted by ), in it acts as one of the two onedimensional pointgroup transformations: the identity or the inversion. Therefore, the space can be made Euclidean with Γ(R) defining a fourdimensional pointgroup transformation , which is an element of a crystallographic subgroup of O(4). The fourdimensional pointgroup transformations are of the form (R, ɛ), with ɛ = ±1 and they act on the fourdimensional lattice basis as where ɛ stands for ɛ(R) as in (9.8.1.20b). So the pointgroup symmetry operations are crystallographic and given by pairs of a threedimensional crystallographic pointgroup transformation and a onedimensional ɛ = ±1, respectively. The case ɛ = −1 corresponds to an inversion of the phase of the modulation function.
As in the threedimensional case, one can define equivalence classes among those fourdimensional point groups.
Two point groups and belong to the same geometric crystal class if their threedimensional (external) parts (forming the point group and , respectively) are in the same threedimensional crystal class [i.e. are conjugated subgroups of O(3)] and their onedimensional internal parts (forming the point groups and , respectively) are equal. The latter condition implies that corresponding pointgroup elements have the same value of ɛ.
Such a geometric crystal class can then be denoted by the symbol of the threedimensional crystal class together with the values of ɛ that correspond to the generators.
Also, the notion of arithmetic equivalence can be generalized to these fourdimensional 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 reciprocalvector 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 threedimensional group [i.e. by ] and by the components of the modulation wavevector with respect to the corresponding reciprocallattice 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 fourdimensional crystallography, as the relation between the higherdimensional space and the threedimensional physical space V plays a fundamental role.
The embedded structures in four dimensions have lattice periodicity. So the symmetry groups are fourdimensional space groups, called superspace groups. The new name has been introduced because of the privileged role played by the threedimensional subspace V. A superspacegroup element consists of a pointgroup transformation (R, ɛ) and a translation (v, Δ). The action of such an element on the fourdimensional space is then given by It is important to realize that a superspacegroup symmetry of an embedded crystal induces threedimensional 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 threedimensional symmetry relations, but of course not Euclidean ones.
In threedimensional Euclidean space, the types of spacegroup transformation are translations, rotations, rotoinversions, reflections, central inversion, screw rotations, and glide planes. Only the latter two transformations have intrinsic nonprimitive translations. For superspace groups, the types of transformation are determined by the pointgroup 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 pointgroup 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 superspacegroup element can be denoted by a symbol that consists of a symbol for the threedimensional 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.
In Section 9.8.4, the theory is extended to structures containing d modulations, with . In this case, each pointgroup transformation in internal space is given by and the associated internal translation by the (ddimensional) vector . Thus, The transformations R and are represented by the matrices and , respectively. In the following discussion, this nomenclature (but with v_{I} rather than v_{I}) is sometimes also applied for the (3 + 1)dimensional case. The usual formulae are obtained by replacing R_{I} by ɛ and v_{I} by Δ.
Fourdimensional space groups were obtained in the (3 + 1)reducible case by Fast & Janssen (1969) and in the general case by Brown, Bülow, Neubüser, Wondratschek & Zassenhaus (1978). The groups were determined on the basis of algorithms developed by Zassenhaus (1948), Janssen, Janner & Ascher (1969a,b), Brown (1969), and Fast & Janssen (1971). In the book by Brown, Bülow, Neubüser, Wondratschek & Zassenhaus, quoted above, a mathematical characterization of the basic crystallographic concepts is given together with corresponding tables for the dimensions one, two, three, and four. One finds there, in particular, a full list of fourdimensional space groups. The list by Fast & Janssen is restricted to space groups with (3 + 1)reducible point groups. The fourdimensional groups in the work of Brown et al. are labelled by numbers. For these same groups, alternative symbols have been developed by Weigel, Phan, Veysseyre and Grebille generalizing the principles of the notation adopted by International Tables for Crystallography, Volume A , for the threedimensional space groups (Weigel, Phan & Veysseyre, 1987; Veysseyre & Weigel, 1989; Grebille, Weigel, Veysseyre & Phan, 1990).
The difference in the listing of fourdimensional crystallographic groups one finds in Brown et al. and in Weigel et al. with respect to that in the present tables is not simply a matter of notation. In the first place, here only those groups appear that can occur as symmetry groups of onedimensional incommensurate modulated phases (there are 371 such space groups). Furthermore, as already mentioned, a finer equivalence relation has been considered that reflects the freedom one has in embedding a threedimensional modulated structure in a fourdimensional Euclidean space. Instead of 371, one then obtains 775 inequivalent groups for which the name superspace group has been introduced. A (3 + 1)dimensional superspace group is thus a fourdimensional space group having some additional properties. In Section 9.8.4, the precise definitions are given.
In the commensurate onedimensionally modulated case, 3833 fourdimensional space groups may occur, out of which 320 already belong to the previous 371. The corresponding additional (3 + 1)dimensional superspace groups are also present in the listing by Fast & Janssen (1969) and have been considered again (and applied to structure determination) by van Smaalen (1987). The Bravais classes for the commensurate (3 + 1)dimensional case are given in Table 9.8.3.2(b).
The relation between modulated crystals and the superspace groups is treated in a textbook by Opechowski (1986). That between the superspacegroup symbols of the present tables and those of Weigel et al. is discussed in Grebille et al. (1990).
Note that no new names have been introduced for the underlying crystallographic concepts like Bravais classes, geometric and arithmetic crystal classes, even if in those cases also the equivalence relation is not simply that of fourdimensional Euclidean crystallography, an explicit distinction always being possible by specifying the dimension as (3 + 1) instead of four.
Another type of modulation, the occupation modulation, can be treated in a way similar to the displacive modulation. As an example consider an alloy where the positions of the basic structure have spacegroup symmetry, but are statistically occupied by either of two types of atoms. Suppose that the position r is occupied by an atom of type A with probability and by one of type B with probability and that p is periodic. The probability of finding an A atom at site is with . In this case, the structure factor becomes where and are the atomic scattering factors. Because of the periodicity, one has Hence, where Δ(H) is the sum of δ functions over the reciprocal lattice of the basic structure: Consequently, the diffraction peaks occur at positions H given by (9.8.1.7). For a simple sinusoidal modulation [m = ±1 in (9.8.1.29)], there are only main reflections and firstorder satellites (m = ±1). One may introduce an additional coordinate t and generalize (9.8.1.27) to which has (3 + 1)dimensional spacegroup symmetry. Generalization to more complex modulation cases is then straightforward.
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