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. 487494
Section 4.6.2. The ndimensional description of aperiodic crystals ^{a}Laboratory of Crystallography, Swiss Federal Institute of Technology, CH8092 Zurich, Switzerland 
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).
References
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Janssen, T. (1995). From quasiperiodic to more complex systems. In Beyond quasicrystals, edited by F. Axel & D. Gratias, pp. 75–140. Les Ulis: Les Editions de Physique and Berlin: SpringerVerlag.
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.
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Wolff, P. M. de (1974). The pseudosymmetry of modulated crystal structures. Acta Cryst. A30, 777–785.
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