International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 4.6, pp. 486532
https://doi.org/10.1107/97809553602060000568 Chapter 4.6. Reciprocalspace images of aperiodic crystals ^{a}Laboratory of Crystallography, Swiss Federal Institute of Technology, CH8092 Zurich, Switzerland This chapter presents an aid to the characterization of aperiodic crystals based on information from diffraction experiments and gives a survey of aperiodic crystals from the viewpoint of the experimentally accessible reciprocal space. Characteristic features of the diffraction patterns of the different types of aperiodic crystals are shown. A standard way of determining the metrics and finding the optimum nD embedding is described. Structurefactor formulae for general and special cases are given. 
The discovery of materials with icosahedral diffraction symmetry (Shechtman et al., 1984) was the main reason for the reassessment of the definition of crystallinity and for the introduction of the concept of aperiodic crystals. The first aperiodic crystal, i.e. a material with Bragg reflections not located only at reciprocallattice nodes, was identified long before (Dehlinger, 1927). In the following decades a wealth of incommensurately modulated phases and composite crystals were discovered. Nevertheless, only a few attempts have been made to develop a crystallography of aperiodic crystals; the most powerful of these was the higherdimensional approach (see de Wolff, 1974, 1977; Janner & Janssen, 1979, 1980a,b; de Wolff et al., 1981). In fact, incommensurate structures can be easily described using the higherdimensional approach and also, fully equivalently, in a dual way: as a threedimensional (3D) combination of one or more periodic basic structures and one or several modulation waves (de Wolff, 1984). However, with the discovery of quasicrystals and their noncrystalline symmetries, the latter approach failed and geometrical crystallography including the higherdimensional approach received new attention. For more recent reviews of the crystallography of all three types of aperiodic crystals see van Smaalen (1995), of incommensurately modulated structures see Cummins (1990), of quasicrystals see Steurer (1990, 1996), of quasicrystals and their crystalline approximants see Goldman & Kelton (1993) and Kelton (1995). Textbooks on quasicrystals have been written by Janot (1994) and Senechal (1995).
According to the traditional crystallographic definition, an ideal crystal corresponds to an infinite 3D periodic arrangement of identical structure motifs. Its symmetry can be described by one of the 230 3D space groups. Mathematically, a periodic structure can be generated by the convolution of a function representing the structure motif with a lattice function. The structure motif can be given, for instance, by the electrondensity distribution of one primitive unit cell of the structure. The lattice function is represented by a set of δ functions at the nodes of a 3D lattice Λ with basis , , and ( is the set of integer numbers). In reciprocal space, this convolution corresponds to the product of the Fourier transform of the lattice function and the Fourier transform of the structure motif . is represented by the reciprocal lattice decorated with δ functions on the reciprocallattice nodes , with the reciprocalbasis vectors , , defined by and . The product is called the weighted reciprocal lattice; the weights are given by the structure factors . Thus, the characteristic feature of an ideal crystal in direct and reciprocal space is the existence of a lattice. In direct space, this lattice is decorated with identical structure motifs preserving translational and point symmetry in the framework of spacegroup symmetry. In reciprocal space, only the point symmetry between structure factors is maintained. The Fourier spectrum (or Fourier image, i.e. the Fourier transform) of the electrondensity distribution of an ideal crystal consists of a countably infinite set of discrete Bragg peaks with a strictly defined minimum distance.
This crystal definition can be generalized to dimensions. A ddimensional (dD) ideal aperiodic crystal can be defined as a dD irrational section of an ndimensional () crystal with nD lattice symmetry. The intersection of the nD hypercrystal with the dD physical space is equivalent to a projection of the weighted nD reciprocal lattice onto the dD physical space. The resulting set (Fourier module) is countably dense. Countably dense means that the dense set of Bragg peaks can be mapped onetoone onto the set of natural numbers. Hence, the Bragg reflections can be indexed with integer indices on an appropriate basis. The Fourier module of the projected reciprocallattice vectors has the structure of a module of rank n. A module is a free Abelian group, its rank n is given by the number of free generators (rationally independent vectors). The dimension of a module is that of the vector space spanned by it. The vectors are the images of the vectors projected onto the physical space . Thus, by definition, the 3D reciprocal space of an ideal aperiodic crystal consists of a countably dense set of Bragg reflections only. Contrary to an ideal crystal, a minimum distance between Bragg reflections does not exist in an aperiodic one. In summary, it may be stressed that the terms aperiodic and periodic refer to properties of crystal structures in dD space. In nD space, as considered here, lattice symmetry is always present and, therefore, the term crystal is used.
Besides the aperiodic crystals mentioned above, other classes of aperiodic structures with strictly defined construction rules exist (see Axel & Gratias, 1995). Contrary to the kind of aperiodic crystals dealt with in this chapter, the Fourier spectra of aperiodic structures considered in the latter reference are continuous and contain only in a few cases additional sharp Bragg reflections (δ peaks).
Experimentally, the borderline between aperiodic crystals and their periodic approximations (crystalline approximants ) is not sharply defined. Finite crystal size, static and dynamic disorder, chemical impurities and defects broaden Bragg peaks and cause diffuse diffraction phenomena. Furthermore, the resolution function of the diffraction equipment is limited.
However, the concept of describing an aperiodic structure as a dD physicalspace section of an nD crystal (see Section 4.6.2) is only useful if it significantly simplifies the description of its structural order. Thus, depending on the shape of the atomic surfaces, which gives information on the atomic ordering, incommensurately modulated structures (IMSs, Sections 4.6.2.2 and 4.6.3.1), composite structures (CSs, Sections 4.6.2.3 and 4.6.3.2), or quasiperiodic structures (QSs, Sections 4.6.2.4 and 4.6.3.3) can be obtained from irrational cuts. The atomic surfaces are continuous dimensional objects for IMSs and CSs, and discrete dimensional objects for QSs. A class of aperiodic crystals with discrete fractal atomic surfaces also exists (Section 4.6.2.5). In this case the Hausdorff dimension (Hausdorff, 1919) of the atomic surface is not an integer number and smaller than . The most outstanding characteristic feature of a fractal is its scale invariance: the object appears similar to itself `from near as from far, that is, whatever the scale' (Gouyet, 1996).
To overcome the problems connected with experimental resolution, the translational symmetry of periodic crystals is used as a hard constraint in the course of the determination of their structures. Hence, spacegroup symmetry is taken for granted and only the local atomic configuration in a unit cell (actually, asymmetric unit) remains to be determined. In reciprocal space, this assumption corresponds to a condensation of Bragg reflections with finite full width at half maximum (FWHM) to δ peaks accurately located at the reciprocallattice nodes. Diffuse diffraction phenomena are mostly neglected. This extrapolation to the existence of an ideal crystal is generally out of the question even if samples of very poor quality (high mosaicity, microdomain structure, defects, …) are investigated.
The same practice is convenient for the determination of real aperiodic structures once the type of idealized aperiodic ordering is `known'. Again, the global ordering principle is taken as a hard constraint. For instance, the question of whether a structure is commensurately or incommensurately modulated can only be answered within a given experimental resolution. Experimentally, the ratio of the wavelength of a modulation to the period of the underlying lattice can always be determined as a rational number only. Saying that a structure is incommensurately modulated, with the above ratio being an irrational number, simply means that the experimental results can be better understood, modelled and interpreted assuming an incommensurate modulation. For example, an incommensurate chargedensity wave can be moved through an ideal crystal without changing the energy of the crystal. This is not so for a commensurate modulation. In some cases, the modulation period changes with temperature in discrete steps (`devil's staircase'), generating a series of commensurate superstructures (`lockin structures'); in other cases, a continuous variation can be observed within the experimental resolution. The latter case will be described best by an incommensurately modulated structure.
However, if only the local structure of an aperiodic crystal is of interest, a structure analysis does not take much more experimental effort than for a regular crystal. In contrast, for the analysis of the global structure, i.e. the characterization of the type of its `aperiodicity', diffraction experiments with the highest possible resolution are essential. Some problems connected with the structure analysis of aperiodic crystals are dealt with in Section 4.6.4.
To determine the longrange order – whether a real `quasicrystal' is perfectly quasiperiodic, on average quasiperiodic, a crystalline approximant or a nanodomain structure – requires information from experiments that are sensitive to changes of the global structure. Hence, one needs diffraction experiments that allow the accurate determination of the spatial intensity distribution. Consequently, the limiting factors for such experiments are the maximum spatial and intensity resolution of the diffraction and detection equipment, as well as the size and quality of the sample. Nevertheless, the resolution available on stateoftheart standard synchrotronbeamline equipment is sufficient to test whether the ordering of atoms in an aperiodic crystal reaches the same degree of perfection as found in highquality silicon. Of course, the higher the sample quality the more necessary it is to account for dynamical diffraction effects such as reflection broadening and displacement. Otherwise, a misinterpretation may bias the global structure modelling.
The following sections present an aid to the characterization of aperiodic crystals based on information from diffraction experiments and give a survey of aperiodic crystals from the viewpoint of the experimentally accessible reciprocal space. Characteristic features of the diffraction patterns of the different types of aperiodic crystals are shown. A standard way of determining the metrics and finding the optimum nD embedding is described. Structurefactor formulae for general and special cases are given.
An incommensurate modulation of a latticeperiodic structure destroys its translational symmetry in direct and reciprocal space. In the early seventies, a method was suggested by de Wolff (1974) for restoring the lost lattice symmetry by considering the diffraction pattern of an incommensurately modulated structure (IMS) as a projection of an nD reciprocal lattice upon the physical space. n, the dimension of the superspace, is always larger than or equal to d, the dimension of the physical space. This leads to a simple method for the description and characterization of IMSs as well as a variety of new possibilities in their structure analysis. The nD embedding method is well established today and can be applied to all aperiodic crystals with reciprocalspace structure equivalent to a module with finite rank n (Janssen, 1988). The dimension of the embedding space is determined by the rank of the module, i.e. by the number of reciprocalbasis vectors necessary to allow for indexing all Bragg reflections with integer numbers. The point symmetry of the 3D reciprocal space (Fourier spectrum) constrains the point symmetry of the nD reciprocal lattice and restricts the number of possible nD symmetry groups.
In the following sections, the nD descriptions of the four main classes of aperiodic crystals are demonstrated on simple 1D examples of incommensurately modulated phases, composite crystals, quasicrystals and structures with fractally shaped atomic surfaces. The main emphasis is placed on quasicrystals that show scaling symmetry, a new and unusual property in structural crystallography. A detailed discussion of the different types of 3D aperiodic crystals follows in Section 4.6.3.
A periodic deviation of atomic parameters from a reference structure (basic structure, BS) is called a modulated structure (MS). In the case of mutual incommensurability of the basic structure and the modulation period, the structure is called incommensurately modulated. Otherwise, it is called commensurately modulated. The modulated atomic parameters may be one or several of the following:
An incommensurately modulated structure can be described in a dual way by its basic structure and a modulation function . This allows the structurefactor formula to be calculated and a full symmetry characterization employing representation theory to be performed (de Wolff, 1984). A more general method is the nD description: it relates the dD aperiodic incommensurately modulated structure to a periodic structure in nD space. This simplifies the symmetry analysis and structurefactor calculation, and allows more powerful structuredetermination techniques.
The nD embedding method is demonstrated in the following 1D example of a displacively modulated structure. A basic structure , with period a and , is modulated by a function , with the satellite vector , period , and α a rational or irrational number yielding a commensurately or incommensurately modulated structure (Fig. 4.6.2.1).
If the 1D IMS and its 1D modulation function are properly combined in a 2D parameter space , a 2D latticeperiodic structure results (Fig. 4.6.2.2). The actual atoms are generated by the intersection of the 1D physical (external, parallel) space with the continuous hyperatoms. The hyperatoms have the shape of the modulation function along the perpendicular (internal, complementary) space . They result from a convolution of the physicalspace atoms with their modulation functions.
A basis (D basis) of the 2D hyperlattice is given by where a is the translation period of the BS and c is an arbitrary constant. The components of the basis vectors are given on a 2D orthogonal coordinate system (V basis). The components of the basis vector are simply the parallelspace period a of the BS and α times the perpendicularspace component of the basis vector . The vector is always parallel to the perpendicular space and its length is one period of the modulation function in arbitrary units (this is expressed by the arbitrary factor ). An atom at position r of the BS is displaced by an amount given by the modulation function , with . Hence, the perpendicularspace variable t has to adopt the value for the physicalspace variable r. This can be achieved by assigning the slope α to the basis vector . The choice of the parameter c has no influence on the actual MS, i.e. the way in which the 2D structure is cut by the parallel space (Fig. 4.6.2.2 c).
The basis of the lattice , reciprocal to Σ, can be obtained from the condition : with . The metric tensors for the reciprocal and direct 2D lattices for are The choice of an arbitrary number for c has no influence on the metrics of the physicalspace components of the IMS in direct or reciprocal space.
The Fourier transform of the hypercrystal depicted in Fig. 4.6.2.2 gives the weighted reciprocal lattice shown in Fig. 4.6.2.3. The 1D diffraction pattern in physical space is obtained by a projection of the weighted 2D reciprocal lattice along as the Fourier transform of a section in direct space corresponds to a projection in reciprocal space and vice versa: Reciprocallattice points lying in physical space are referred to as main reflections, all others as satellite reflections. All Bragg reflections can be indexed with integer numbers in the 2D description . In the physicalspace description, the diffraction vector can be written as , with for the satellite vector and the order of the satellite reflection. For a detailed discussion of the embedding and symmetry description of IMSs see, for example, Janssen et al. (2004).
A commensurately modulated structure with and , , and with , can be generated by shearing the 2D lattice Σ with a shear matrix : The subscript D (V) following the shear matrix indicates that it is acting on the D (V) basis. The shear matrix does not change the distances between the atoms in the basic structure. In reciprocal space, using the inverted and transposed shear matrix, one obtains
In the simplest case, a composite structure (CS) consists of two intergrown periodic structures with mutually incommensurate lattices. Owing to mutual interactions, each subsystem may be modulated with the period of the other. Consequently, CSs can be considered as coherent intergrowths of two or more incommensurately modulated substructures. The substructures have at least the origin of their reciprocal lattices in common. However, in all known cases, at least one common reciprocallattice plane exists. This means that at least one particular projection of the composite structure exhibits full lattice periodicity.
The unmodulated (basic) 1D subsystems of a 1D incommensurate intergrowth structure can be related to each other in a 2D parameter space (Fig. 4.6.2.4). The actual atoms result from the intersection of the physical space with the hypercrystal. The hyperatoms correspond to a convolution of the real atoms with infinite lines parallel to the basis vectors and of the 2D hyperlattice .
An appropriate basis is given by where and are the lattice parameters of the two substructures and c is an arbitrary constant. Taking into account the interactions between the subsystems, each one becomes modulated with the period of the other. Consequently, in the 2D description, the shape of the hyperatoms is determined by their modulation functions (Fig. 4.6.2.5).
A basis of the reciprocal lattice can be obtained from the condition : The metric tensors for the reciprocal and the direct 2D lattices for are The Fourier transforms of the hypercrystals depicted in Figs. 4.6.2.4 and 4.6.2.5 correspond to the weighted reciprocal lattices illustrated in Figs. 4.6.2.6 and 4.6.2.7. The 1D diffraction patterns in physical space are obtained by a projection of the weighted 2D reciprocal lattices upon . All Bragg reflections can be indexed with integer numbers in both the 2D description and in the 1D physicalspace description with two parallel basis vectors .
The reciprocallattice points and , , , on the main axes and are the main reflections of the two substructures. All other reflections are referred to as satellite reflections. Their intensities differ from zero only in the case of modulated substructures. Each reflection of one subsystem coincides with exactly one reflection of the other subsystem.
The Fibonacci sequence, the best investigated example of a 1D quasiperiodic structure, can be obtained from the substitution rule σ: , , replacing the letter S by L and the letter L by the word LS (see e.g. Luck et al., 1993). Applying the substitution matrix associated with σ, this rule can be written in the form S gives the sum of letters, , and not their order. Consequently, the same substitution matrix can also be applied, for instance, to the substitution : , . The repeated action of S on the alphabet yields the words and as illustrated in Table 4.6.2.1. The frequencies of letters contained in the words and can be calculated by applying the nth power of the transposed substitution matrix on the unit vector. From it follows that In the case of the Fibonacci sequence, gives the total number of letters S and L, and the number of letters L.

An infinite Fibonacci sequence, i.e. a word with , remains invariant under inflation (deflation). Inflation (deflation) means that the number of letters L, S increases (decreases) under the action of the (inverted) substitution matrix S. Inflation and deflation represent selfsimilarity (scaling) symmetry operations on the infinite Fibonacci sequence. A more detailed discussion of the scaling properties of the Fibonacci chain in direct and reciprocal space will be given later.
The Fibonacci numbers form a series with {the golden mean }. The ratio of the frequencies of L and S in the Fibonacci sequence converges to τ if the sequence goes to infinity. The continued fraction expansion of the golden mean τ, contains only the number 1. This means that τ is the `most irrational' number, i.e. the irrational number with the worst truncated continued fraction approximation to it. This might be one of the reasons for the stability of quasiperiodic systems, where τ plays a role. The strong irrationality may impede the lockin into commensurate systems (rational approximants).
By associating intervals (e.g. atomic distances) with length ratio τ to 1 to the letters L and S, a quasiperiodic structure (Fibonacci chain) can be obtained. The invariance of the ratio of lengths is responsible for the invariance of the Fibonacci chain under scaling by a factor . Owing to a minimum atomic distance S in real crystal structures, the full set of inverse symmetry operators does not exist. Consequently, the set of scaling operators forms only a semigroup, i.e. an associative groupoid. Groupoids are the most general algebraic sets satisfying only one of the group axioms: the associative law. The scaling properties of the Fibonacci sequence can be derived from the eigenvalues of the scaling matrix S. For this purpose the equation with eigenvalue λ and unit matrix I has to be solved. The evaluation of the determinant yields the characteristic polynomial yielding in turn the eigenvalues , and the eigenvectors , . Rewriting the eigenvalue equation gives for the first (i.e. the largest) eigenvalue Identifying the eigenvector with shows that an infinite Fibonacci sequence remains invariant under scaling by a factor τ. This scaling operation maps each new lattice vector upon a vector r of the original lattice: Considering periodic lattices, these eigenvalues are integer numbers. For quasiperiodic `lattices' (quasilattices) they always correspond to algebraic numbers (Pisot numbers). A Pisot number is the solution of a polynomial equation with integer coefficients. It is larger than one, whereas the modulus of its conjugate is smaller than unity: and (Luck et al., 1993). The total lengths and of the words can be determined from the equations and with the eigenvalue . The left Perron–Frobenius eigenvector of S, i.e. the left eigenvector associated with , determines the ratio S:L to 1:. The right Perron–Frobenius eigenvector of S associated with gives the relative frequencies, 1 and τ, for the letters S and L (for a definition of the Perron–Frobenius theorem see Luck et al., 1993, and references therein).
The general case of an alphabet with k letters (intervals) , of which at least two are on incommensurate length scales and which transform with the substitution matrix S, can be treated analogously. S is a matrix with nonnegative integer coefficients. Its eigenvalues are solutions of a polynomial equation of rank k with integer coefficients (algebraic or Pisot numbers). The dimension n of the embedding space is generically equal to the number of letters (intervals) k involved by the substitution rule. From substitution rules, infinitely many different 1D quasiperiodic sequences can be generated. However, their atomic surfaces in the nD description are generically of fractal shape (see Section 4.6.2.5).
The quasiperiodic 1D density distribution of the Fibonacci chain can be represented by the Fourier series with (the set of real numbers). The Fourier coefficients form a Fourier module equivalent to a module of rank 2. Thus a periodic function in 2D space can be defined by where and are, by construction, direct and reciprocal lattice vectors (Figs. 4.6.2.8 and 4.6.2.9): The 1D Fibonacci chain results from the cut of the parallel (physical) space with the 2D lattice Σ decorated with line elements for the atomic surfaces (acceptance domains). In this description, the atomic surfaces correspond simply to the projection of one 2D unit cell upon the perpendicularspace coordinate. This satisfies the condition that each unit cell contributes exactly to one point of the Fibonacci chain (primitive unit cell). The physical space is related to the eigenspace of the substitution matrix S associated with its eigenvalue . The perpendicular space corresponds to the eigenspace of the substitution matrix S associated with its eigenvalue . Thus, the physical space scales to powers of τ and the perpendicular space to powers of .
By blockdiagonalization, the reducible substitution (scaling) matrix S can be decomposed into two nonequivalent irreducible representations. These can be assigned to the two 1D orthogonal subspaces and forming the 2D embedding space . Thus, using , where one obtains the scaling operations and in parallel and in perpendicular space as indicated by the partition lines.
The metric tensors for the reciprocal and the direct 2D square lattices read The short distance S of the Fibonacci sequence is related to by with the projectors and onto and . The point density of the Fibonacci chain, i.e. the number of vertices per unit length, can be calculated using the formula where and are the areas of the atomic surface and of the 2D unit cell, respectively.
For an infinite Fibonacci sequence generated from the intervals S and L an average distance d can be calculated: Therefrom, the point density can also be calculated:
An approximant structure of the Fibonacci sequence with a unit cell containing m intervals L and n intervals S can be generated by shearing the 2D lattice Σ by the shear matrix , where : This shear matrix does not change the magnitudes of the intervals L and S. In reciprocal space the inverted and transposed shear matrix is applied on the reciprocal basis, where :
The point of the nth interval L or S of an infinite Fibonacci sequence is given by where t is the phase of the modulation function (Janssen, 1986). Thus, the Fibonacci sequence can also be dealt with as an incommensurately modulated structure. This is a consequence of the fact that for 1D structures only the crystallographic point symmetries 1 and allow the existence of a periodic average structure.
The embedding of the Fibonacci chain as an incommensurately modulated structure can be performed as follows:
One possible way of indexing based on the same as defined above is illustrated in Fig. 4.6.2.10. The scattering vector is given by , where , or, in the 2D representation, , where and , with the direct basis The modulation function is sawtoothlike (Fig. 4.6.2.11).
A 1D structure with a fractal atomic surface (Hausdorff dimension 0.9157…) can be derived from the Fibonacci sequence by squaring its substitution matrix S: corresponding to the substitution rule , as well as two other nonequivalent ones (see Janssen, 1995). The eigenvalues are obtained by calculating The evaluation of the determinant gives the characteristic polynomial with the solutions , with and , and the same eigenvectors as for the Fibonacci sequence. Rewriting the eigenvalue equation gives Identifying the eigenvector with shows that the infinite 1D sequence multiplied by powers of its eigenvalue (scaling operation) remains invariant (each new lattice point coincides with one of the original lattice): The fractal sequence can be described on the same reciprocal and direct bases as the Fibonacci sequence. The only difference in the 2D directspace description is the fractal character of the perpendicularspace component of the hyperatoms (Fig. 4.6.2.12) (see Zobetz, 1993).
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 wave vectors (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 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. 4.6.3.3 b)
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. 4.6.3.3 c 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 vector with the matrix increases 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 obtain with yielding 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 obtain where 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 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 obtain where . 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).
Theoretically, aperiodic crystals show an infinite number of reflections within a given diffraction angle, contrary to periodic crystals. The number of reflections to be included in a structure analysis of a periodic crystal may be very high (one million for virus crystals, for instance) but there is no ambiguity in the selection of reflections to be collected: all Bragg reflections within a limiting sphere in reciprocal space, usually given by , are used. All reflections, observed and unobserved, are included to fit a reliable structure model.
However, for aperiodic crystals it is not possible to collect the infinite number of dense Bragg reflections within . The number of observable reflections within this limiting sphere depends only on the spatial and intensity resolution.
What happens if not all reflections are included in a structure analysis? How important is the contribution of reflections with large perpendicularspace components of the diffraction vector which are weak but densely distributed? These problems are illustrated using the example of the Fibonacci sequence. An infinite model structure consisting of Al atoms with isotropic thermal parameter B = 1 Å^{2}, and distances S = 2.5 Å and , was used for the calculations (Table 4.6.4.1).

It turns out that 92.6% of the total diffracted intensity of 161 322 reflections is included in the 44 strongest reflections and 99.2% in the strongest 425 reflections. It is remarkable, however, that in all the experimental data for icosahedral and decagonal quasicrystals collected so far, rarely more than 20 to 50 reflections along reciprocallattice lines corresponding to net planes with Fibonaccisequencelike distances could be observed. The consequences for structure determinations with such truncated data sets are primarily a lower resolution in perpendicular space than in physical space. This corresponds to a smearing of the hyperatoms in the perpendicular space. For the derivation of the local structurebuilding elements (clusters) of aperiodic crystals this is only a minor problem: the smeared hyperatoms give rise to split atoms and a biased electrondensity distribution. The information on the global aperiodic structure, however, which is contained in the detailed shape of the atomic surfaces, is severely reduced when using a lowresolution diffraction data set. A combination of highresolution electron microscopy, lattice imaging and diffraction techniques allows a good characterization of the local and global order even in these cases. For a more detailed analysis of these problems see Steurer (1995).
The question whether an aperiodic crystal is really aperiodic or rather a highorder approximant is of different importance depending on the point of view. As far as real finite crystals are considered, definitions of periodic and aperiodic real crystals and of periodic and aperiodic perfect crystals have to be given first. Real crystals, despite periodicity or aperiodicity, are the actual samples under investigation. Partial information about their actual structure can be obtained today by imaging methods (scanning tunnelling microscopy, atomic force microscopy, highresolution transmission electron microscopy, …). Basically, the real crystal structure can be determined using full diffraction information from Bragg and diffuse scattering. In practice, however, only `Bragg reflections' are included in a structure analysis. `Bragg reflections' result from the integration of diffraction intensities from extended volumes around a limited number of Braggreflection positions ( module). This process of intensity condensation at Bragg points corresponds in direct space to an averaging process. The real crystal structure is projected upon one unit cell in direct space defined by the module in reciprocal space. Generally, the identification of appropriate reciprocalspace metrics is not a problem in the case of crystals. It can be problematic, however, in the case of aperiodic crystals, in particular quasicrystals (see Lancon et al., 1994). The metrics, and to some extent the global order in the case of quasicrystals, are fixed by assigning the reciprocal basis. The spatial resolution of a diffraction experiment defines the accuracy of the resulting metrics. The decision whether the rational number obtained for the relative length of a satellite vector indicates a commensurate or an incommensurate modulation can only be made considering temperature and pressuredependent chemical and physical properties of the material. The same is valid for other types of aperiodic crystals.
Highorder approximants of quasicrystals often occur in orientationally twinned form or, on a smaller scale, as oriented nanodomain structures. These structures can be identified by electron microscopy, and, in certain cases, also by highresolution Xray diffractometry (Kalning et al., 1994). If the intensity and spatial resolution is sufficient, characteristic reflection splitting and diffuse diffraction phenomena can be observed. It has been demonstrated that for the determination of the local structure (structurebuilding elements) it does not matter greatly whether one uses a data set for a real quasicrystal or one for a twinned approximant (Estermann et al., 1994). Examples of reciprocalspace images of an approximant, a twinned approximant and the related decagonal phase are shown schematically in Fig. 4.6.4.1 and for real samples in Fig. 4.6.4.2.

Simulated diffraction patterns of (a) the singlecrystal approximant of decagonal Al–Co–Ni, (b) the fivefold twinned approximant, and (c) the decagonal phase itself (Estermann et al., 1994). 
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