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. 600602
Section 4.6.3.1.3. Structure factor ^{a}Laboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, WolfgangPauliStrasse 10, CH8093 Zurich, Switzerland 
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
Böhm, H. (1977). Eine erweiterte Theorie der Satellitenreflexe und die Bestimmung der modulierten Struktur des Natriumnitrits. Habilitation thesis, University of Munster.Korekawa, M. (1967). Theorie der Satellitenreflexe. Habilitation thesis, University of Munich.
Paciorek, W. A. & Chapuis, G. (1994). Generalized Bessel functions in incommensurate structure analysis. Acta Cryst. A50, 194–203.
Petricek, V., Maly, K. & Cisarova, I. (1991). The computing system `JANA'. In Methods of Structural Analysis of Modulated Structures and Quasicrystals, edited by J. M. PérezMato, F. J. Zuniga & G. Madriaga, pp. 262–267. Singapore: World Scientific.
Yamamoto, A. (1982). Structure factor of modulated crystal structures. Acta Cryst. A38, 87–92.