International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 598621
Section 4.6.3. Reciprocalspace images ^{a}Laboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, WolfgangPauliStrasse 10, CH8093 Zurich, Switzerland 
Onedimensionally modulated structures are the simplest representatives of IMSs. The vast majority of the one hundred or so IMSs known so far belong to this class (Cummins, 1990). However, there is also an increasing number of IMSs with 2D or 3D modulation. The dimension d of the modulation is defined by the number of rationally independent modulation wavevectors (satellite vectors) (Fig. 4.6.3.1). The electrondensity function of a dD modulated 3D crystal can be represented by the Fourier series The Fourier coefficients (structure factors) differ from zero only for reciprocalspace vectors with . The d satellite vectors are given by , with a matrix σ. In the case of an IMS, at least one entry to σ has to be irrational. The wavelength of the modulation function is . The set of vectors H forms a Fourier module of rank , which can be decomposed into a rank 3 and a rank d submodule corresponds to a module of rank 3 in a 3D subspace (the physical space), corresponds to a module of rank d in a dD subspace (perpendicular space). The submodule is identical to the 3D reciprocal lattice of the average structure. results from the projection of the perpendicularspace component of the reciprocal lattice upon the physical space. Owing to the coincidence of one subspace with the physical space, the dimension of the embedding space is given as and not as nD. This terminology points out the special role of the physical space.
Hence the reciprocalbasis vectors , can be considered to be physicalspace projections of reciprocalbasis vectors , spanning a reciprocal lattice : The first vector component of refers to the physical space, the second to the perpendicular space spanned by the mutually orthogonal unit vectors . c is an arbitrary constant which can be set to 1 without loss of generality.
A direct lattice Σ with basis , and , can be constructed according to Consequently, the aperiodic structure in physical space is equivalent to a 3D section of the hypercrystal.
The 3D reciprocal space of a IMS consists of two separable contributions, the set of main reflections and the set of satellite reflections (Fig. 4.6.3.1). In most cases, the modulation is only a weak perturbation of the crystal structure. The main reflections are related to the average structure, the satellites to the difference between the average and actual structure. Consequently, the satellite reflections are generally much weaker than the main reflections and can be easily identified. Once the set of main reflections has been separated, a conventional basis , for is chosen.
The only ambiguity is in the assignment of rationally independent satellite vectors . They should be chosen inside the reciprocalspace unit cell (Brillouin zone) of in such a way as to give a minimal number d of additional dimensions. If satellite vectors reach the Brillouinzone boundary, centred Bravais lattices are obtained. The star of satellite vectors has to be invariant under the pointsymmetry group of the diffraction pattern. There should be no contradiction to a reasonable physical modulation model concerning period or propagation direction of the modulation wave. More detailed information on how to find the optimum basis and the correct setting is given by Janssen et al. (2004) and Janner et al. (1983a,b).
The Laue symmetry group of the Fourier module , is isomorphous to or a subgroup of one of the 11 3D crystallographic Laue groups leaving invariant. The action of the pointgroup symmetry operators R on the reciprocal basis , can be written as
The matrices form a finite group of integral matrices which are reducible, since R is already an orthogonal transformation in 3D physical space. Consequently, R can be expressed as pair of orthogonal transformations in 3D physical and dD perpendicular space, respectively. Owing to their mutual orthogonality, no symmetry relationship exists between the set of main reflections and the set of satellite reflections. is the transpose of which acts on vector components in direct space.
For the directspace (superspace) symmetry operator and its matrix representation on Σ, the following decomposition can be performed: is a matrix, is a matrix and is a matrix. The translation operator consists of a 3D vector and a dD vector . According to Janner & Janssen (1979), can be derived from . has integer elements only as it contains components of primitivelattice vectors of , whereas σ in general consists of a rational and an irrational part: . Thus, only the rational part gives rise to nonzero entries in . With the order of the Laue group denoted by N, one obtains , where , implying that and .
Example
In the case of a 3D IMS with 1D modulation the matrix has the components of the wavevector . because for , q can only be transformed into . Corresponding to , one obtains (modulo ). The row matrix is equivalent to the difference vector between and (Janssen et al., 2004).
For a monoclinic modulated structure with point group for (unique axis ) and satellite vector , with an irrational number, one obtains From the relations , it can be shown that the symmetry operations 1 and 2 are associated with the perpendicularspace transformations , and m and with . The matrix is given by for the operation 2, for instance.
The matrix representations of the symmetry operators R in reciprocal superspace decompose according to Phase relationships between modulation functions of symmetryequivalent atoms can give rise to systematic extinctions of different classes of satellite reflections. The extinction rules may include indices of both main and satellite reflections. A full list of systematic absences is given in the table of superspace groups (Janssen et al., 2004). Thus, once point symmetry and systematic absences are found, the superspace group can be obtained from the tables in a way analogous to that used for regular 3D crystals. A different approach for the symmetry description of IMSs from the 3D Fourierspace perspective has been given by Dräger & Mermin (1996).
The structure factor of a periodic structure is defined as the Fourier transform of the density distribution of its unit cell (UC): The same is valid in the case of the (3 + d)D description of IMSs. The parallel and perpendicularspace components are orthogonal to each other and can be separated. The Fourier transform of the parallelspace component of the electrondensity distribution of a single atom gives the usual atomic scattering factors . For the structurefactor calculation, one does not need to use explicitly. The hyperatoms correspond to the convolution of the electrondensity distribution in 3D physical space with the modulation function in dD perpendicular space. Therefore, the Fourier transform of the (3 + d)D hyperatoms is simply the product of the Fourier transform of the physicalspace component with the Fourier transform of the perpendicularspace component, the modulation function.
For a general displacive modulation one obtains for the ith coordinate of the kth atom in 3D physical space where are the basicstructure coordinates and are the modulation functions with unit periods in their arguments (Fig. 4.6.3.2). The arguments are , where are the coordinates of the kth atom referred to the origin of its unit cell and are the phases of the modulation functions. The modulation functions themselves can be expressed in terms of a Fourier series as where are the orders of harmonics for the jth modulation wave of the ith component of the kth atom and their amplitudes are and .

The relationships between the coordinates and the modulation function in a special section of the space. 
Analogous expressions can be derived for a density modulation, i.e., the modulation of the occupation probability : and for the modulation of the tensor of thermal parameters : The resulting structurefactor formula is for summing over the set (R, t) of superspace symmetry operations and the set of N′ atoms in the asymmetric unit of the unit cell (Yamamoto, 1982). Different approaches without numerical integration based on analytical expressions including Bessel functions have also been developed. For more information see Paciorek & Chapuis (1994), Petricek, Maly & Cisarova (1991), and references therein.
For illustration, some fundamental IMSs will be discussed briefly (see Korekawa, 1967; Böhm, 1977).
Harmonic density modulation. A harmonic density modulation can result on average from an ordered distribution of vacancies on atomic positions. For an IMS with N atoms per unit cell one obtains for a harmonic modulation of the occupancy factor the structurefactor formula for the mth order satellite Thus, a linear correspondence exists between the structurefactor magnitudes of the satellite reflections and the amplitude of the density modulation. Furthermore, only firstorder satellites exist, since the modulation wave consists only of one term. An important criterion for the existence of a density modulation is that a pair of satellites around the origin of the reciprocal lattice exists (Fig. 4.6.3.3).
Symmetric rectangular density modulation. The boxfunctionlike modulated occupancy factor can be expanded into a Fourier series, and the resulting structure factor of the mth order satellite is According to this formula, only oddorder satellites occur in the diffraction pattern. Their structurefactor magnitudes decrease linearly with the order (Fig. 4.6.3.3b).
Harmonic displacive modulation. The displacement of the atomic coordinates is given by the function and the structure factor by The structurefactor magnitudes of the mthorder satellite reflections are a function of the mthorder Bessel functions. The arguments of the Bessel functions are proportional to the scalar products of the amplitude and the diffraction vector. Consequently, the intensity of the satellites will vary characteristically as a function of the length of the diffraction vector. Each main reflection is accompanied by an infinite number of satellite reflections (Figs. 4.6.3.3c and 4.6.3.4).
Composite structures consist of N mutually incommensurate substructures with N basic sublattices , with . The reciprocal sublattices , with , have either only the origin of the reciprocal lattice or one or two reciprocallattice directions in common. Thus, one needs reciprocalbasis vectors for integer indexing of diffraction patterns that show Bragg reflections at positions given by the Fourier module . The CSs discovered to date have at least one lattice direction in common and consist of a maximum number of substructures. They can be divided in three main classes: channel structures, columnar packings and layer packings (see van Smaalen, 1992, 1995).
In the following, the approach of Janner & Janssen (1980b) and van Smaalen (1992, 1995, and references therein) for the description of CSs is used. The set of diffraction vectors of a CS, i.e. its Fourier module , can be split into the contributions of the ν subsystems by employing matrices with integer coefficients , . In the general case, each subsystem will be modulated with the periods of the others due to their mutual interactions. Thus, in general, CSs consist of several intergrown incommensurately modulated substructures. The satellite vectors , referred to the νth subsystem can be obtained from by applying the integer matrices : , . The matrices consisting of the components of the satellite vectors with regard to the reciprocal sublattices can be calculated by , where the subscript 3 refers to the submatrix of physical space and the subscript d to the matrix of the internal space. The juxtaposition of the matrix and the matrix defines the nonsingular matrix , This matrix allows the reinterpretation of the Fourier module as the Fourier module of a ddimensionally modulated subsystem ν. It also describes the coordinate transformation between the superspace basis Σ and .
The superspace description is obtained analogously to that for IMSs (see Section 4.6.3.1) by considering the 3D Fourier module of rank as the projection of a reciprocal lattice upon the physical space. Thus, one obtains for the definition of the direct and reciprocal lattices (Janner & Janssen, 1980b)
The indexing of diffraction patterns of composite structures can be performed in the following way:
The vectors forming a basis for the 3D Fourier module can be chosen such that , and are linearly independent. Then the remaining d vectors can be described as a linear combination of the first three, defining the matrix σ: . This is formally equivalent to the reciprocal basis obtained for an IMS (see Section 4.6.3.1) and one can proceed in an analogous way to that for IMSs.
The symmetry of CSs can be described with basically the same formalism as used for IMSs. This is a consequence of the formally equivalent applicability of the higherdimensional approach, in particular of the superspacegroup theory developed for IMSs [see Janner & Janssen (1980a,b); van Smaalen (1991, 1992); Yamamoto (1992a)].
The structure factor of a composite structure consists of the weighted contributions of the subsystem structure factors :with coefficients similar to those for IMSs.
The weights are the Jacobians of the transformations from to t, and are the reflection indices with respect to the subsystem Fourier modules (van Smaalen, 1995, and references therein). The relative values of , where are related to the volume ratios of the contributing subsystems. The subsystem structure factors correspond to those for IMSs (see Section 4.6.3.1). Besides this formula, based on the publications of Yamamoto (1982) and van Smaalen (1995), different structurefactor equations have been discussed (Kato, 1990; Petricek, Maly, Coppens et al., 1991).
Structures with quasiperiodic order in one dimension and lattice symmetry in the other two dimensions are the simplest representatives of quasicrystals. A few phases of this structure type have been identified experimentally (see Steurer, 1990). Since the Fibonacci chain represents the most important model of a 1D quasiperiodic structure, it will be used in this section to represent the quasiperiodic direction of 3D structures with 1D quasiperiodic order. As discussed in Section 4.6.2.4, 1D quasiperiodic structures are on the borderline between quasiperiodic and incommensurately modulated structures. They can be described using either of the two approaches. In the following, the quasiperiodic description will be preferred to take account of the scaling symmetry.
The electrondensitydistribution function of a 1D quasiperiodically ordered 3D crystal can be represented by a Fourier series: The Fourier coefficients (structure factors) differ from zero only for reciprocalspace vectors with , or with integer indexing with . The set of all vectors H forms a Fourier module of rank 4 which can be decomposed into two rank 2 submodules . corresponds to a module of rank 2 in a 1D subspace, corresponds to a module of rank 2 in a 2D subspace. Consequently, the first submodule can be considered as a projection from a 2D reciprocal lattice, , while the second submodule is of the form of a reciprocal lattice, .
Hence, the reciprocalbasis vectors , , can be considered to be projections of reciprocalbasis vectors , , spanning a 4D reciprocal lattice, onto the physical space , with A direct lattice Σ with basis and , can be constructed according to (compare Fig. 4.6.2.8) , with Consequently, the structure in physical space is equivalent to a 3D section of the 4D hypercrystal.
The reciprocal space of the Fibonacci chain is densely filled with Bragg reflections (Figs. 4.6.2.9 and 4.6.3.5). According to the nD embedding method, the shorter the parallelspace distance between two Bragg reflections, the larger the corresponding perpendicularspace distance becomes. Since the structure factor decreases rapidly as a function of (Fig. 4.6.3.6), `neighbouring' reflections of strong Bragg peaks are extremely weak and, consequently, the reciprocal space appears to be filled with discrete Bragg peaks even for lowresolution experiments.
This property allows an unambiguous identification of a correct set of reciprocalbasis vectors. However, infinitely many sets allowing a correct indexing of the diffraction pattern with integer indices exist. Nevertheless, an optimum basis (low indices are assigned to strong reflections) can be derived: the intensity distribution, not the metrics, characterizes the best choice of indexing. Once the minimum distance S in the structure is identified from chemical considerations, the reciprocal basis should be chosen as described in Section 4.6.2.4. It has to be kept in mind, however, that the identification of the metrics is not sufficient to distinguish in the 1D aperiodic case between an incommensurately modulated structure, a quasiperiodic structure or special kinds of structures with fractally shaped atomic surfaces.
A correct set of reciprocalbasis vectors can be identified in the following way:
The possible Laue symmetry group of the Fourier module is any one of the direct product . corresponds to one of the ten crystallographic 2D point groups, in the general case of a quasiperiodic stacking of periodic layers. Consequently, the nine Laue groups , , mmm, 4/m, 4/mmm, , , and are possible. These are all 3D crystallographic Laue groups except for the two cubic ones.
The (unweighted) Fourier module shows only 2D lattice symmetry. In the third dimension, the submodule remains invariant under the scaling symmetry operation with . The scaling symmetry operators form an infinite group of reciprocalbasis transformations in superspace, and act on the reciprocal basis in superspace.
The structure factor of a periodic structure is defined as the Fourier transform of the density distribution of its unit cell (UC): The same is valid in the case of the nD description of a quasiperiodic structure. The parallel and perpendicularspace components are orthogonal to each other and can be separated. In the case of the 1D Fibonacci sequence, the Fourier transform of the parallelspace component of the electrondensity distribution of a single atom gives the usual atomic scattering factor . Parallel to , adopts values only within the interval and one obtains The factor results from the normalization of the structure factors to . With and the integrand can be rewritten as yielding Using gives Thus, the structure factor has the form of the function with x a perpendicular reciprocalspace coordinate. The upper and lower limiting curves of this function are given by the hyperbolae (Fig. 4.6.3.6). The continuous shape of as a function of allows the estimation of an overall temperature factor and atomic scattering factor for reflectiondata normalization (compare Figs. 4.6.3.6 and 4.6.3.7).
In the case of a 3D crystal structure which is quasiperiodic in one direction, the structure factor can be written in the form The sum runs over all n averaged hyperatoms in the 4D unit cell of the structure. The geometric form factor corresponds to the Fourier transform of the kth atomic surface, normalized to , the area of the 2D unit cell projected upon , and , the area of the kth atomic surface.
The atomic temperature factor can also have perpendicularspace components. Assuming only harmonic (static or dynamic) displacements in parallel and perpendicular space one obtains, in analogy to the usual expression (Willis & Pryor, 1975), with The elements of the type represent the average values of the atomic displacements along the ith axis times the displacement along the jth axis on the V basis.
In the following, only the properties of the quasiperiodic component of the 3D structure, namely the Fourier module , are discussed. The intensities of the Fibonacci chain decorated with point atoms are only a function of the perpendicularspace component of the diffraction vector. and are illustrated in Figs. 4.6.3.5 and 4.6.3.6 as a function of and of . The distribution of as a function of their frequencies clearly resembles a centric distribution, as can be expected from the centrosymmetric 2D subunit cell. The shape of the distribution function depends on the radius of the limiting sphere in reciprocal space. The number of weak reflections increases with the square of , that of strong reflections only linearly (strong reflections always have small components).
The weighted reciprocal space of the Fibonacci sequence contains an infinite number of Bragg reflections within a limited region of the physical space. Contrary to the diffraction pattern of a periodic structure consisting of point atoms on the lattice nodes, the Bragg reflections show intensities depending on the perpendicularspace components of their diffraction vectors.
The reciprocal space of a sequence generated from hyperatoms with fractally shaped atomic surfaces (squared Fibonacci sequence) is very similar to that of the Fibonacci sequence (Figs. 4.6.3.8 and 4.6.3.9). However, there are significantly more weak reflections in the diffraction pattern of the `fractal' sequence, caused by the geometric form factor.
4.6.3.3.1.5. Relationships between structure factors at symmetryrelated points of the Fourier image
The two possible pointsymmetry groups in the 1D quasiperiodic case, and , relate the structure factors to A 3D structure with 1D quasiperiodicity results from the stacking of atomic layers with distances following a quasiperiodic sequence. The point groups describing the symmetry of such structures result from the direct product corresponds to one of the ten crystallographic 2D point groups, can be or . Consequently, 18 3D point groups are possible.
Since 1D quasiperiodic sequences can be described generically as incommensurately modulated structures, their possible point and space groups are equivalent to a subset of the superspace groups for IMSs with satellite vectors of the type , i.e. , for the quasiperiodic direction [001] (Janssen et al., 2004).
From the scaling properties of the Fibonacci sequence, some relationships between structure factors can be derived. Scaling the physicalspace structure by a factor , , corresponds to a scaling of the perpendicular space by the inverse factor . For the scaling of the corresponding reciprocal subspaces, the inverse factors compared to the direct spaces have to be applied.
The set of vectors r, defining the vertices of a Fibonacci sequence , multiplied by a factor τ coincides with a subset of the vectors defining the vertices of the original sequence (Fig. 4.6.3.10). The residual vertices correspond to a particular decoration of the scaled sequence, i.e. the sequence . The Fourier transform of the sequence then can be written as the sum of the Fourier transforms of the sequences and ; In terms of structure factors, this can be reformulated as
Hence, phases of structure factors that are related by scaling symmetry can be determined from each other.
Further scaling relationships in reciprocal space exist: scaling a diffraction vectorwith the matrixincreases the magnitudes of structure factors assigned to this particular diffraction vector H,
This is due to the shrinking of the perpendicularspace component of the diffraction vector by powers of while expanding the parallelspace component by according to the eigenvalues τ and of S acting in the two eigenspaces and :Thus, for scaling n times we obtainwithyielding eventually The scaling of the diffraction vectors H by corresponds to a hyperbolic rotation (Janner, 1992) with angle , where (Fig. 4.6.3.11):
A structure quasiperiodic in two dimensions, periodic in the third dimension and with decagonal diffraction symmetry is called a decagonal phase. Its holohedral Laue symmetry group is . All reciprocalspace vectors can be represented on a basis (V basis) and (Fig. 4.6.3.12) as . The vector components refer to a Cartesian coordinate system in physical (parallel) space. Thus, from the number of independent reciprocalbasis vectors necessary to index the Bragg reflections with integer numbers, the dimension of the embedding space has to be at least five. This can also be shown in a different way (Hermann, 1949).

Reciprocal basis of the decagonal phase in the 5D description projected upon (above left) and (above right). Below, a perspective physicalspace view is shown. 
The set of all vectors H remains invariant under the action of the symmetry operators of the point group . The symmetryadapted matrix representations for the pointgroup generators, the tenfold rotation , the reflection plane (normal of the reflection plane along the vectors – with modulo 5) and the inversion operation may be written in the form
By blockdiagonalization, these reducible symmetry matrices can be decomposed into nonequivalent irreducible representations. These can be assigned to the two orthogonal subspaces forming the 5D embedding space , the 3D parallel (physical) subspace and the perpendicular 2D subspace . Thus, using , we obtainwhere The column vectors of the matrix W give the parallel (above the partition line) and perpendicularspace components (below the partition line) of a reciprocal basis in V space. Thus, W can be rewritten using the physicalspace reciprocal basis defined above as yielding the reciprocal basis , in the 5D embedding space (D space): The symmetry matrices can each be decomposed into a matrix and a matrix. The first one, , acts on the parallelspace component, the second one, , on the perpendicularspace component. In the case of , the coupling factor between a rotation in parallel and perpendicular space is 3. Thus, a rotation in physical space is related to a rotation in perpendicular space (Fig. 4.6.3.12).
With the condition , a basis in direct 5D space is obtained: The metric tensors G, are of the type with for the reciprocal space and for the direct space. Thus, for the lattice parameters in reciprocal space we obtain , ; ; , ; , , and for those in direct space , ; ; , ; . The volume of the 5D unit cell can be calculated from the metric tensor G:
Since decagonal phases are only quasiperiodic in two dimensions, it is sufficient to demonstrate their characteristics on a 2D example, the canonical Penrose tiling (Penrose, 1974). It can be constructed from two unit tiles: a skinny (acute angle ) and a fat (acute angle ) rhomb with equal edge lengths and areas , (Fig. 4.6.3.13). The areas and frequencies of these two unit tiles in the Penrose tiling are both in a ratio 1 to τ. By replacing each skinny and fat rhomb according to the inflation rule, a τinflated tiling is obtained. Inflation (deflation) means that the number of tiles is inflated (deflated), their edge lengths are decreased (increased) by a factor τ. By infinite repetition of this inflation operation, an infinite Penrose tiling is generated. Consequently, this substitution operation leaves the tiling invariant.
From Fig. 4.6.3.13 it can be seen that the sets of vertices of the deflated tilings are subsets of the set of vertices of the original tiling. The τdeflated tiling is dual to the original tiling; a further deflation by a factor τ gives the original tiling again. However, the edge lengths of the tiles are increased by a factor , and the tiling is rotated around . Only the fourth deflation of the original tiling yields the original tiling in its original orientation but with all lengths multiplied by a factor .
Contrary to the reciprocalspace scaling behaviour of , the set of vertices of the Penrose tiling is not invariant by scaling the length scale simply by a factor τ using the scaling matrix S: The square of S, however, maps all vertices of the Penrose tiling upon other ones: corresponds to a hyperbolic rotation with in superspace (Janner, 1992). However, only operations of the type , , scale the Penrose tiling in a way which is equivalent to the (4nth) substitutional operations discussed above. The rotoscaling operation , also a symmetry operation of the Penrose tiling, leaves a pentagram invariant as demonstrated in Fig. 4.6.3.14 (Janner, 1992). Blockdiagonalization of the scaling matrix S decomposes it into two nonequivalent irreducible representations which give the scaling properties in the two orthogonal subspaces of the 4D embedding space, , the 2D parallel (physical) subspace and the perpendicular 2D subspace . Thus, using , we obtain where
The 2D Penrose tiling can also be embedded canonically in the 5D space. Canonically means that the 5D lattice is hypercubic and that the projection of one unit cell upon the 3D perpendicular space , giving a rhombicosahedron, defines the atomic surface. However, the parallelspace image , , with , of the 5D basis is not linearly independent. Consequently, the atomic surface consists of only a subset of the points contained in the rhombicosahedron: five equidistant pentagons (one with diameter zero) resulting as sections of the rhombicosahedron with five equidistant parallel planes (Fig. 4.6.3.15). The linear dependence of the 5D basis allows the embedding in the 4D space. The resulting hyperrhombohedral hyperlattice is spanned by the basis , , discussed above. The atomic surfaces occupy the positions , , on the body diagonal of the 4D unit cell. Neighbouring pentagons are in an anti position to each other (Fig. 4.6.3.16). Thus the 4D unit cell is decorated centrosymmetrically. The edge length of a Penrose rhomb is related to the length of physicalspace basis vectors by , with the smallest distance . The point density (number of vertices per unit area) of a Penrose tiling with Penrose rhombs of edge length can be calculated from the ratio of the relative number of unit tiles in the tiling to their area: This is equivalent to the calculation from the 4D description, where and are the area of the atomic surface and the volume of the 4D unit cell, respectively. The pentagon radii are and for the atomic surfaces in with and . A detailed discussion of the properties of Penrose tiling is given in the papers of Penrose (1974, 1979), Jaric (1986) and Pavlovitch & Kleman (1987).
The indexing of the submodule of the diffraction pattern of a decagonal phase is not unique. Since corresponds to a module of rank 4 with decagonal point symmetry, it is invariant under scaling by : . Nevertheless, an optimum basis (low indices are assigned to strong reflections) can be derived: not the metrics, as for regular periodic crystals, but the intensity distribution characterizes the best choice of indexing.
A correct set of reciprocalbasis vectors can be identified experimentally in the following way:
The diffraction symmetry of decagonal phases can be described by the Laue groups or . The set of all vectors H forms a Fourier module of rank 5 in physical space which can be decomposed into two submodules . corresponds to a module of rank 4 in a 2D subspace, corresponds to a module of rank 1 in a 1D subspace. Consequently, the first submodule can be considered as a projection from a 4D reciprocal lattice, , while the second submodule is of the form of a regular 1D reciprocal lattice, . The diffraction pattern of the Penrose tiling decorated with equal point scatterers on its vertices is shown in Fig. 4.6.3.17. All Bragg reflections within are depicted. Without intensitytruncation limit, the diffraction pattern would be densely filled with discrete Bragg reflections. To illustrate their spatial and intensity distribution, an enlarged section of Fig. 4.6.3.17 is shown in Fig. 4.6.3.18. This picture shows all Bragg reflections within . The projected 4D reciprocallattice unit cell is drawn and several reflections are indexed. All reflections are arranged along lines in five symmetryequivalent orientations. The perpendicularspace diffraction patterns (Figs. 4.6.3.19 and 4.6.3.20) show a characteristic starlike distribution of the Bragg reflections. This is a consequence of the pentagonal shape of the atomic surfaces: the Fourier transform of a pentagon has a starlike distribution of strong Fourier coefficients.

Schematic diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs a_{r} = 4.04 Å). All reflections are shown within and . 

The perpendicularspace diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs a_{r} = 4.04 Å). All reflections are shown within and . 
The 5D decagonal space groups that may be of relevance for the description of decagonal phases are listed in Table 4.6.3.1. These space groups are a subset of all 5D decagonal space groups fulfilling the condition that the 5D point groups they are associated with are isomorphous to the 3D point groups describing the diffraction symmetry. Their structures are comparable to 3D hexagonal groups. Hence, only primitive lattices exist. The orientation of the symmetry elements in the 5D space is defined by the isomorphism of the 3D and 5D point groups. However, the action of the tenfold rotation is different in the subspaces and : a rotation of in is correlated with a rotation of in . The reflection and inversion operations are equivalent in both subspaces.

The structure factor for the decagonal phase corresponds to the Fourier transform of the 5D unit cell, with 5D diffraction vectors , N hyperatoms, parallelspace atomic scattering factor , temperature factor and perpendicularspace geometric form factor . is equivalent to the conventional Debye–Waller factor and describes random fluctuations along the perpendicularspace coordinate. These fluctuations cause characteristic jumps of vertices in physical space (phason flips). Even random phason flips map the vertices onto positions which can still be described by physicalspace vectors of the type . Consequently, the set of all possible vectors forms a module. The shape of the atomic surfaces corresponds to a selection rule for the positions actually occupied. The geometric form factor is equivalent to the Fourier transform of the atomic surface, i.e. the 2D perpendicularspace component of the 5D hyperatoms.
For example, the canonical Penrose tiling corresponds to the Fourier transform of pentagonal atomic surfaces: where is the area of the 5D unit cell projected upon and is the area of the kth atomic surface. The area can be calculated using the formula Evaluating the integral by decomposing the pentagons into triangles, one obtains with running over the five triangles, where the radii of the pentagons are , , and the vectors
As shown by Ishihara & Yamamoto (1988), the Penrose tiling can be considered to be a superstructure of a pentagonal tiling with only one type of pentagonal atomic surface in the nD unit cell. Thus, for the Penrose tiling, three special reflection classes can be distinguished: for and the class of strong main reflections is obtained, and for the classes of weaker first and secondorder satellite reflections are obtained (see Fig. 4.6.3.18).
This section deals with the reciprocalspace characteristics of the 2D quasiperiodic component of the 3D structure, namely the Fourier module . The radial structurefactor distributions of the Penrose tiling decorated with point scatterers are plotted in Figs. 4.6.3.21 and 4.6.3.22 as a function of parallel and perpendicular space. The distribution of as a function of their frequencies clearly resembles a centric distribution, as can be expected from the centrosymmetric 4D subunit cell. The shape of the distribution function depends on the radius of the limiting sphere in reciprocal space. The number of weak reflections increases to the power of four, that of strong reflections only quadratically (strong reflections always have small components). The radial distribution of the structurefactor amplitudes as a function of perpendicular space clearly shows three branches, corresponding to the reflection classes with , and (Fig. 4.6.3.23).
The weighted reciprocal space of the Penrose tiling contains an infinite number of Bragg reflections within a limited region of the physical space. Contrary to the diffraction pattern of a periodic structure consisting of point atoms on the lattice nodes, the Bragg reflections show intensities depending on the perpendicularspace components of their diffraction vectors (Figs. 4.6.3.19, 4.6.3.20 and 4.6.3.22).
4.6.3.3.2.5. Relationships between structure factors at symmetryrelated points of the Fourier image
Scaling the Penrose tiling by a factor by employing the matrix scales at the same time its reciprocal space by a factor : Since this operation increases the lengths of the diffraction vectors by the factor τ in parallel space and decreases them by the factor in perpendicular space, the following distribution of structurefactor magnitudes (for point atoms at rest) is obtained: The scaling operations , , the rotoscaling operations (Fig. 4.6.3.14) and the tenfold rotation , where connect all structure factors with diffraction vectors pointing to the nodes of an infinite series of pentagrams. The structure factors with positive signs are predominantly on the vertices of the pentagram while the ones with negative signs are arranged on circles around the vertices (Figs. 4.6.3.24 to 4.6.3.27).
A structure that is quasiperiodic in three dimensions and exhibits icosahedral diffraction symmetry is called an icosahedral phase. Its holohedral Laue symmetry group is . All reciprocalspace vectors can be represented on a basis , , where , and , the angle between two neighbouring fivefold axes (Fig. 4.6.3.28). This can be rewritten as where are Cartesian basis vectors. Thus, from the number of independent reciprocalbasis vectors needed to index the Bragg reflections with integer numbers, the dimension of the embedding space has to be six. The vector components refer to a Cartesian coordinate system (V basis) in the physical (parallel) space.

Perspective (a) parallel and (b) perpendicularspace views of the reciprocal basis of the 3D Penrose tiling. The six rationally independent vectors point to the edges of an icosahedron. 
The set of all diffraction vectors remains invariant under the action of the symmetry operators of the icosahedral point group . The symmetryadapted matrix representations for the pointgroup generators, one fivefold rotation α, a threefold rotation β and the inversion operation γ, can be written in the form
Blockdiagonalization of these reducible symmetry matrices decomposes them into nonequivalent irreducible representations. These can be assigned to the two orthogonal subspaces forming the 6D embedding space , the 3D parallel (physical) subspace and the perpendicular 3D subspace . Thus, using , we obtainwhere , , , . The column vectors of the matrix W give the parallel (above the partition line) and perpendicularspace components (below the partition line) of a reciprocal basis in V. Thus, W can be rewritten using the physicalspace reciprocal basis defined above and an arbitrary constant c, yielding the reciprocal basis , in the 6D embedding space (D space) The symmetry matrices can each be decomposed into two matrices. The first one, , acts on the parallelspace component, the second one, , on the perpendicularspace component. In the case of , the coupling factor between a rotation in parallel and perpendicular space is 2. Thus a rotation in physical space is related to a rotation in perpendicular space (Figs. 4.6.3.28 and 4.6.3.29).

Schematic representation of a rotation in 6D space. The point P is rotated to P′. The component rotations in parallel and perpendicular space are illustrated. 
With the condition , the basis in direct 6D space is obtained: The metric tensors G, are of the type with for the reciprocal space and , for the direct space. For we obtain hypercubic direct and reciprocal 6D lattices.
The lattice parameters in reciprocal and direct space are and with , respectively. The volume of the 6D unit cell can be calculated from the metric tensor G. For it is simply
The best known example of a 3D quasiperiodic structure is the canonical 3D Penrose tiling (see Janssen, 1986). It can be constructed from two unit tiles: a prolate and an oblate rhombohedron with equal edge lengths (Fig. 4.6.3.30). Each face of the rhombohedra is a rhomb with acute angles . Their volumes are , and their frequencies ::1. The resulting point density (number of vertices per unit volume) is . Ten prolate and ten oblate rhombohedra can be packed to form a rhombic triacontahedron. The icosahedral symmetry of this zonohedron is broken by the many possible decompositions into the rhombohedra. Removing one zone of the triacontahedron gives a rhombicosahedron consisting of five prolate and five oblate rhombohedra. Again, the singular fivefold axis of the rhombicosahedron is broken by the decomposition into rhombohedra. Removing one zone again gives a rhombic dodecahedron consisting of two prolate and two oblate rhombohedra. Removing the last remaining zone leads finally to a single prolate rhombohedron. Using these zonohedra as elementary clusters, a matching rule can be derived for the 3D construction of the 3D Penrose tiling (Levine & Steinhardt, 1986; Socolar & Steinhardt, 1986).

The two unit tiles of the 3D Penrose tiling: a prolate and an oblate rhombohedron with equal edge lengths . 
The 3D Penrose tiling can be embedded in the 6D space as shown above. The 6D hypercubic lattice is decorated on the lattice nodes with 3D triacontahedra obtained from the projection of a 6D unit cell upon the perpendicular space (Fig. 4.6.3.31). Thus the edge length of the rhombs covering the triacontahedron is equivalent to the length of the perpendicularspace component of the vectors spanning the 6D hypercubic lattice .

Atomic surface of the 3D Penrose tiling in the 6D hypercubic description. The projection of the 6D hypercubic unit cell upon gives a rhombic triacontahedron. 
There are several indexing schemes in use. The generic one uses a set of six rationally independent reciprocalbasis vectors pointing to the corners of an icosahedron, , , , , with , the angle between two neighbouring fivefold axes (setting 1) (Fig. 4.6.3.28). In this case, the physicalspace basis corresponds to a simple projection of the 6D reciprocal basis . Sometimes, the same set of six reciprocalbasis vectors is referred to a differently oriented Cartesian reference system (C basis, with basis vectors along the twofold axes) (Bancel et al., 1985). The reciprocal basis is
An alternate way of indexing is based on a 3D set of cubic reciprocalbasis vectors (setting 2) (Fig. 4.6.3.32): The Cartesian C basis is related to the V basis by a rotation around , yielding , followed by a rotation around : Thus, indexing the diffraction pattern of an icosahedral phase with integer indices, one obtains for setting 1 . These indices transform into the second setting to with the fractional cubic indices , , . The transformation matrix is
The diffraction symmetry of icosahedral phases can be described by the Laue group . The set of all vectors H forms a Fourier module of rank 6 in physical space. Consequently, it can be considered as a projection from a 6D reciprocal lattice, . The parallel and perpendicular reciprocalspace sections of the 3D Penrose tiling decorated with equal point scatterers on its vertices are shown in Figs. 4.6.3.33 and 4.6.3.34. The diffraction pattern in perpendicular space is the Fourier transform of the triacontahedron. All Bragg reflections within are depicted. Without intensitytruncation limit, the diffraction pattern would be densely filled with discrete Bragg reflections.
The 6D icosahedral space groups that are relevant to the description of icosahedral phases (six symmorphous and five nonsymmorphous groups) are listed in Table 4.6.3.2. These space groups are a subset of all 6D icosahedral space groups fulfilling the condition that the 6D point groups they are associated with are isomorphous to the 3D point groups and 235 describing the diffraction symmetry. From 826 6D (analogues to) Bravais groups (Levitov & Rhyner, 1988), only three fulfil the condition that the projection of the 6D hypercubic lattice upon the 3D physical space is compatible with the icosahedral point groups : the primitive hypercubic Bravais lattice P, the bodycentred Bravais lattice I with translation 1/2(111111), and the facecentred Bravais lattice F with translations further cyclic permutations. Hence, the I lattice is twofold primitive (i.e. it contains two vertices per unit cell) and the F lattice is 16fold primitive. The orientation of the symmetry elements in the 6D space is defined by the isomorphism of the 3D and 6D point groups. The action of the fivefold rotation, however, is different in the subspaces and : a rotation of in is correlated with a rotation of in . The reflection and inversion operations are equivalent in both subspaces.

The structure factor of the icosahedral phase corresponds to the Fourier transform of the 6D unit cell, with 6D diffraction vectors , parallelspace atomic scattering factor , temperature factor and perpendicularspace geometric form factor . is equivalent to the conventional Debye–Waller factor and describes random fluctuations in perpendicular space. These fluctuations cause characteristic jumps of vertices (phason flips) in the physical space. Even random phason flips map the vertices onto positions that can still be described by physicalspace vectors of the type . Consequently, the set of all possible vectors forms a module. The shape of the atomic surfaces corresponds to a selection rule for the positions actually occupied. The geometric form factor is equivalent to the Fourier transform of the atomic surface, i.e. the 3D perpendicularspace component of the 6D hyperatoms.
For the example of the canonical 3D Penrose tiling, corresponds to the Fourier transform of a triacontahedron: where is the volume of the 6D unit cell projected upon and is the volume of the triacontahedron. and are equal in the present case and amount to the volumes of ten prolate and ten oblate rhombohedra: . Evaluating the integral by decomposing the triacontahedron into trigonal pyramids, each one directed from the centre of the triacontahedron to three of its corners given by the vectors , one obtains with running over all sitesymmetry operations R of the icosahedral group, , , , , and the volume of the parallelepiped defined by the vectors (Yamamoto, 1992b).
The radial structurefactor distributions of the 3D Penrose tiling decorated with point scatterers are plotted in Fig. 4.6.3.35 as a function of parallel and perpendicular space. The distribution of as a function of their frequencies clearly resembles a centric distribution, as can be expected from the centrosymmetric unit cell. The shape of the distribution function depends on the radius of the limiting sphere in reciprocal space. The number of weak reflections increases as the power 6, that of strong reflections only as the power 3 (strong reflections always have small components).
The weighted reciprocal space of the 3D Penrose tiling contains an infinite number of Bragg reflections within a limited region of the physical space. Contrary to the diffraction pattern of a periodic structure consisting of point atoms on the lattice nodes, the Bragg reflections show intensities depending on the perpendicularspace components of their diffraction vectors.
4.6.3.3.3.5. Relationships between structure factors at symmetryrelated points of the Fourier image
The weighted 3D reciprocal space exhibits the icosahedral point symmetry . It is invariant under the action of the scaling matrix : The scaling transformation leaves a primitive 6D reciprocal lattice invariant as can easily be seen from its application on the indices: The matrix leaves invariant, for any with all even or all odd, corresponding to a 6D facecentred hypercubic lattice. In a second case the sum is even, corresponding to a 6D bodycentred hypercubic lattice. Blockdiagonalization of the matrix S decomposes it into two irreducible representations. With we obtain the scaling properties in the two 3D subspaces: scaling by a factor τ in parallel space corresponds to a scaling by a factor in perpendicular space. For the intensities of the scaled reflections analogous relationships are valid, as discussed for decagonal phases (Figs. 4.6.3.36 and 4.6.3.37, Section 4.6.3.3.2.5).
References
Bancel, P. A., Heiney, P. A., Stephens, P. W., Goldman, A. I. & Horn, P. M. (1985). Structure of rapidly quenched Al–Mn. Phys. Rev. Lett. 54, 2422–2425.Böhm, H. (1977). Eine erweiterte Theorie der Satellitenreflexe und die Bestimmung der modulierten Struktur des Natriumnitrits. Habilitation thesis, University of Munster.
Cummins, H. Z. (1990). Experimental studies of structurally incommensurate crystal phases. Phys. Rep. 185, 211–409.
Dräger, J. & Mermin, N. D. (1996). Superspace groups without the embedding: the link between superspace and Fourierspace crystallography. Phys. Rev. Lett. 76, 1489–1492.
Hermann, C. (1949). Kristallographie in Räumen beliebiger Dimensionszahl. I. Die Symmetrieoperationen. Acta Cryst. 2, 139–145.
Ishihara, K. N. & Yamamoto, A. (1988). Penrose patterns and related structures. I. Superstructure and generalized Penrose patterns. Acta Cryst. A44, 508–516.
Janner, A. (1992). Decagrammal symmetry of decagonal Al_{78}Mn_{22} quasicrystal. Acta Cryst. A48, 884–901.
Janner, A. & Janssen, T. (1979). Superspace groups. Physica A, 99, 47–76.
Janner, A. & Janssen, T. (1980a). Symmetry of incommensurate crystal phases. I. Commensurate basic structures. Acta Cryst. A36, 399–408.
Janner, A. & Janssen, T. (1980b). Symmetry of incommensurate crystal phases. II. Incommensurate basic structures. Acta Cryst. A36, 408–415.
Janner, A., Janssen, T. & de Wolff, P. M. (1983a). Bravais classes for incommensurate crystal phases. Acta Cryst. A39, 658–666.
Janner, A., Janssen, T. & de Wolff, P. M. (1983b). Determination of the Bravais class for a number of incommensurate crystals. Acta Cryst. A39, 671–678.
Janssen, T. (1986). Crystallography of quasicrystals. Acta Cryst. A42, 261–271.
Janssen, T., Janner, A., LooijengaVos, A. & de Wolff, P. M. (2004). Incommensurate and commensurate modulated crystal structures. In International Tables for Crystallography, Vol. C, edited by E. Prince, ch. 9.8. Dordrecht: Kluwer Academic Publishers.
Jaric, M. V. (1986). Diffraction from quasicrystals: geometric structure factor. Phys. Rev. B, 34, 4685–4698.
Kato, K. (1990). Strukturverfeinerung des Kompositkristalls im mehrdimensionalen Raum. Acta Cryst. B46, 39–44.
Korekawa, M. (1967). Theorie der Satellitenreflexe. Habilitation thesis, University of Munich.
Levine, D. & Steinhardt, P. J. (1986). Quasicrystals. I. Definition and structure. Phys. Rev. B, 34, 596–616.
Levitov, L. S. & Rhyner, J. (1988). Crystallography of quasicrystals; application to icosahedral symmetry. J. Phys. France, 49, 1835–1849.
Paciorek, W. A. & Chapuis, G. (1994). Generalized Bessel functions in incommensurate structure analysis. Acta Cryst. A50, 194–203.
Pavlovitch, A. & Kleman, M. (1987). Generalized 2D Penrose tilings: structural properties. J. Phys. A Math. Gen. 20, 687–702.
Penrose, R. (1974). The role of aesthetics in pure and applied mathematical research. Bull. Math. Appl. 10, 266–271.
Penrose, R. (1979). Pentaplexity. A class of nonperiodic tilings of the plane. Math. Intell. 2, 32–37.
Petricek, V., Maly, K. & Cisarova, I. (1991). The computing system `JANA'. In Methods of Structural Analysis of Modulated Structures and Quasicrystals, edited by J. M. PérezMato, F. J. Zuniga & G. Madriaga, pp. 262–267. Singapore: World Scientific.
Petricek, V., Maly, K., Coppens, P., Bu, X., Cisarova, I. & FrostJensen, A. (1991). The description and analysis of composite crystals. Acta Cryst. A47, 210–216.
Rabson, D. A., Mermin, N. D., Rokhsar, D. S. & Wright, D. C. (1991). The space groups of axial crystals and quasicrystals. Rev. Mod. Phys. 63, 699–733.
Smaalen, S. van (1991). Symmetry of composite crystals. Phys. Rev. B, 43, 11330–11341.
Smaalen, S. van (1992). Superspace description of incommensurate intergrowth compounds and the application to inorganic misfit layer compounds. Mater. Sci. Forum, 100 & 101, 173–222.
Smaalen, S. van (1995). Incommensurate crystal structures. Crystallogr. Rev. 4, 79–202.
Socolar, J. E. S. & Steinhardt, P. J. (1986). Quasicrystals. II. Unitcell configurations. Phys. Rev. B, 34, 617–647.
Steurer, W. (1990). The structure of quasicrystals. Z. Kristallogr. 190, 179–234.
Willis, B. T. M. & Pryor, A. W. (1975). Thermal Vibrations in Crystallography. Cambridge University Press.
Yamamoto, A. (1982). Structure factor of modulated crystal structures. Acta Cryst. A38, 87–92.
Yamamoto, A. (1992a). Unified setting and symbols of superspace groups for composite crystals. Acta Cryst. A48, 476–483.
Yamamoto, A. (1992b). Ideal structure of icosahedral Al–Cu–Li quasicrystals. Phys. Rev. B, 45, 5217–5227.