International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 598602
Section 4.6.3.1. Incommensurately modulated structures (IMSs) ^{a}Laboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, WolfgangPauliStrasse 10, CH8093 Zurich, Switzerland 
Onedimensionally modulated structures are the simplest representatives of IMSs. The vast majority of the one hundred or so IMSs known so far belong to this class (Cummins, 1990). However, there is also an increasing number of IMSs with 2D or 3D modulation. The dimension d of the modulation is defined by the number of rationally independent modulation wavevectors (satellite vectors) (Fig. 4.6.3.1). The electrondensity function of a dD modulated 3D crystal can be represented by the Fourier series The Fourier coefficients (structure factors) differ from zero only for reciprocalspace vectors with . The d satellite vectors are given by , with a matrix σ. In the case of an IMS, at least one entry to σ has to be irrational. The wavelength of the modulation function is . The set of vectors H forms a Fourier module of rank , which can be decomposed into a rank 3 and a rank d submodule corresponds to a module of rank 3 in a 3D subspace (the physical space), corresponds to a module of rank d in a dD subspace (perpendicular space). The submodule is identical to the 3D reciprocal lattice of the average structure. results from the projection of the perpendicularspace component of the reciprocal lattice upon the physical space. Owing to the coincidence of one subspace with the physical space, the dimension of the embedding space is given as and not as nD. This terminology points out the special role of the physical space.
Hence the reciprocalbasis vectors , can be considered to be physicalspace projections of reciprocalbasis vectors , spanning a reciprocal lattice : The first vector component of refers to the physical space, the second to the perpendicular space spanned by the mutually orthogonal unit vectors . c is an arbitrary constant which can be set to 1 without loss of generality.
A direct lattice Σ with basis , and , can be constructed according to Consequently, the aperiodic structure in physical space is equivalent to a 3D section of the hypercrystal.
The 3D reciprocal space of a IMS consists of two separable contributions, the set of main reflections and the set of satellite reflections (Fig. 4.6.3.1). In most cases, the modulation is only a weak perturbation of the crystal structure. The main reflections are related to the average structure, the satellites to the difference between the average and actual structure. Consequently, the satellite reflections are generally much weaker than the main reflections and can be easily identified. Once the set of main reflections has been separated, a conventional basis , for is chosen.
The only ambiguity is in the assignment of rationally independent satellite vectors . They should be chosen inside the reciprocalspace unit cell (Brillouin zone) of in such a way as to give a minimal number d of additional dimensions. If satellite vectors reach the Brillouinzone boundary, centred Bravais lattices are obtained. The star of satellite vectors has to be invariant under the pointsymmetry group of the diffraction pattern. There should be no contradiction to a reasonable physical modulation model concerning period or propagation direction of the modulation wave. More detailed information on how to find the optimum basis and the correct setting is given by Janssen et al. (2004) and Janner et al. (1983a,b).
The Laue symmetry group of the Fourier module , is isomorphous to or a subgroup of one of the 11 3D crystallographic Laue groups leaving invariant. The action of the pointgroup symmetry operators R on the reciprocal basis , can be written as
The matrices form a finite group of integral matrices which are reducible, since R is already an orthogonal transformation in 3D physical space. Consequently, R can be expressed as pair of orthogonal transformations in 3D physical and dD perpendicular space, respectively. Owing to their mutual orthogonality, no symmetry relationship exists between the set of main reflections and the set of satellite reflections. is the transpose of which acts on vector components in direct space.
For the directspace (superspace) symmetry operator and its matrix representation on Σ, the following decomposition can be performed: is a matrix, is a matrix and is a matrix. The translation operator consists of a 3D vector and a dD vector . According to Janner & Janssen (1979), can be derived from . has integer elements only as it contains components of primitivelattice vectors of , whereas σ in general consists of a rational and an irrational part: . Thus, only the rational part gives rise to nonzero entries in . With the order of the Laue group denoted by N, one obtains , where , implying that and .
Example
In the case of a 3D IMS with 1D modulation the matrix has the components of the wavevector . because for , q can only be transformed into . Corresponding to , one obtains (modulo ). The row matrix is equivalent to the difference vector between and (Janssen et al., 2004).
For a monoclinic modulated structure with point group for (unique axis ) and satellite vector , with an irrational number, one obtains From the relations , it can be shown that the symmetry operations 1 and 2 are associated with the perpendicularspace transformations , and m and with . The matrix is given by for the operation 2, for instance.
The matrix representations of the symmetry operators R in reciprocal superspace decompose according to Phase relationships between modulation functions of symmetryequivalent atoms can give rise to systematic extinctions of different classes of satellite reflections. The extinction rules may include indices of both main and satellite reflections. A full list of systematic absences is given in the table of superspace groups (Janssen et al., 2004). Thus, once point symmetry and systematic absences are found, the superspace group can be obtained from the tables in a way analogous to that used for regular 3D crystals. A different approach for the symmetry description of IMSs from the 3D Fourierspace perspective has been given by Dräger & Mermin (1996).
The structure factor of a periodic structure is defined as the Fourier transform of the density distribution of its unit cell (UC): The same is valid in the case of the (3 + d)D description of IMSs. The parallel and perpendicularspace components are orthogonal to each other and can be separated. The Fourier transform of the parallelspace component of the electrondensity distribution of a single atom gives the usual atomic scattering factors . For the structurefactor calculation, one does not need to use explicitly. The hyperatoms correspond to the convolution of the electrondensity distribution in 3D physical space with the modulation function in dD perpendicular space. Therefore, the Fourier transform of the (3 + d)D hyperatoms is simply the product of the Fourier transform of the physicalspace component with the Fourier transform of the perpendicularspace component, the modulation function.
For a general displacive modulation one obtains for the ith coordinate of the kth atom in 3D physical space where are the basicstructure coordinates and are the modulation functions with unit periods in their arguments (Fig. 4.6.3.2). The arguments are , where are the coordinates of the kth atom referred to the origin of its unit cell and are the phases of the modulation functions. The modulation functions themselves can be expressed in terms of a Fourier series as where are the orders of harmonics for the jth modulation wave of the ith component of the kth atom and their amplitudes are and .

The relationships between the coordinates and the modulation function in a special section of the space. 
Analogous expressions can be derived for a density modulation, i.e., the modulation of the occupation probability : and for the modulation of the tensor of thermal parameters : The resulting structurefactor formula is for summing over the set (R, t) of superspace symmetry operations and the set of N′ atoms in the asymmetric unit of the unit cell (Yamamoto, 1982). Different approaches without numerical integration based on analytical expressions including Bessel functions have also been developed. For more information see Paciorek & Chapuis (1994), Petricek, Maly & Cisarova (1991), and references therein.
For illustration, some fundamental IMSs will be discussed briefly (see Korekawa, 1967; Böhm, 1977).
Harmonic density modulation. A harmonic density modulation can result on average from an ordered distribution of vacancies on atomic positions. For an IMS with N atoms per unit cell one obtains for a harmonic modulation of the occupancy factor the structurefactor formula for the mth order satellite Thus, a linear correspondence exists between the structurefactor magnitudes of the satellite reflections and the amplitude of the density modulation. Furthermore, only firstorder satellites exist, since the modulation wave consists only of one term. An important criterion for the existence of a density modulation is that a pair of satellites around the origin of the reciprocal lattice exists (Fig. 4.6.3.3).
Symmetric rectangular density modulation. The boxfunctionlike modulated occupancy factor can be expanded into a Fourier series, and the resulting structure factor of the mth order satellite is According to this formula, only oddorder satellites occur in the diffraction pattern. Their structurefactor magnitudes decrease linearly with the order (Fig. 4.6.3.3b).
Harmonic displacive modulation. The displacement of the atomic coordinates is given by the function and the structure factor by The structurefactor magnitudes of the mthorder satellite reflections are a function of the mthorder Bessel functions. The arguments of the Bessel functions are proportional to the scalar products of the amplitude and the diffraction vector. Consequently, the intensity of the satellites will vary characteristically as a function of the length of the diffraction vector. Each main reflection is accompanied by an infinite number of satellite reflections (Figs. 4.6.3.3c and 4.6.3.4).
References
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Dräger, J. & Mermin, N. D. (1996). Superspace groups without the embedding: the link between superspace and Fourierspace crystallography. Phys. Rev. Lett. 76, 1489–1492.
Janner, A. & Janssen, T. (1979). Superspace groups. Physica A, 99, 47–76.
Janner, A., Janssen, T. & de Wolff, P. M. (1983a). Bravais classes for incommensurate crystal phases. Acta Cryst. A39, 658–666.
Janner, A., Janssen, T. & de Wolff, P. M. (1983b). Determination of the Bravais class for a number of incommensurate crystals. Acta Cryst. A39, 671–678.
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