Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 600-602   | 1 | 2 |

Section Structure factor

W. Steurera* and T. Haibacha

aLaboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich, Switzerland
Correspondence e-mail: Structure factor

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The structure factor of a periodic structure is defined as the Fourier transform of the density distribution [\rho ({\bf r})] of its unit cell (UC): [{F} ({\bf H}) = {\textstyle\int\limits_{\rm UC}} \rho ({\bf r}) \exp (2 \pi i {\bf H} \cdot {\bf r})\ \hbox{d}{\bf r}.]The same is valid in the case of the (3 + d)D description of IMSs. The parallel- and perpendicular-space components are orthogonal to each other and can be separated. The Fourier transform of the parallel-space component of the electron-density distribution of a single atom gives the usual atomic scattering factors [f_{k} ({\bf H}^{\parallel})]. For the structure-factor calculation, one does not need to use [\rho ({\bf r})] explicitly. The hyperatoms correspond to the convolution of the electron-density 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 [f_{k} ({\bf H}^{\parallel})] of the physical-space component with the Fourier transform of the perpendicular-space component, the modulation function.

For a general displacive modulation one obtains for the ith coordinate [x_{ik}] of the kth atom in 3D physical space [x_{ik} = \bar{x}_{ik} + u_{ik} (\bar{x}_{4}, \ldots, \bar{x}_{3 + d}),\ i = 1, \ldots, 3,]where [\bar{x}_{ik}] are the basic-structure coordinates and [u_{ik} (\bar{x}_{4}, \ldots, \bar{x}_{3 + d})] are the modulation functions with unit periods in their arguments (Fig.[link]. The arguments are [\bar{x}_{3 + j} = \alpha_{ij} \bar{x}_{ik}^{0} + t_{j},\ j = 1, \ldots, d], where [\bar{x}_{ik}^{0}] are the coordinates of the kth atom referred to the origin of its unit cell and [t_{j}] are the phases of the modulation functions. The modulation functions [u_{ik} (\bar{x}_{4}, \ldots, \bar{x}_{3 + d})] themselves can be expressed in terms of a Fourier series as [\eqalign{&u_{ik} \left(\bar{x}_{4}, \ldots, \bar{x}_{3 + d}\right)\cr &\quad = {\textstyle\sum\limits_{n_{1} = 1}^{\infty}} \ldots {\textstyle\sum\limits_{n_{d} = 1}^{\infty}} \left\{^{u} C_{ik}^{\,n_{1} \ldots n_{d}}\cos \left[2 \pi \left(n_{1} \bar{x}_{4} + \ldots + n_{d} \bar{x}_{3 + d}\right)\right]\right.\cr &\quad\quad \left. +\, ^{u}S_{ik}^{\,n_{1} \ldots n_{d}}\sin \left[2 \pi \left(n_{1} \bar{x}_{4} + \ldots + n_{d} \bar{x}_{3 + d}\right)\right]\right\},}]where [n_{j}] are the orders of harmonics for the jth modulation wave of the ith component of the kth atom and their amplitudes are [^{u}C_{ik}^{\,n_{1} \ldots n_{d}}] and [^{u}S_{ik}^{\,n_{1} \ldots n_{d}}].


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The relationships between the coordinates [x_{1k}, x_{4k}, \bar{x}_{1}, \bar{x}_{4}] and the modulation function [u_{1k}] in a special section of the [(3 + d)\hbox{D}] space.

Analogous expressions can be derived for a density modulation, i.e., the modulation of the occupation probability [p_{k} \left(\bar{x}_{4}, \ldots, \bar{x}_{3 + d}\right)]: [\eqalign{&p_{k} \left(\bar{x}_{4}, \ldots, \bar{x}_{3 + d}\right)\cr &\quad= {\textstyle\sum\limits_{n_{1} = 1}^{\infty}} \ldots {\textstyle\sum\limits_{n_{d} = 1}^{\infty}} \left\{^{p} C_{k}^{\,n_{1} \ldots n_{d}} \cos \left[2 \pi \left(n_{1} \bar{x}_{4} + \ldots + n_{d} \bar{x}_{3 + d}\right)\right] \right.\cr &\quad \quad\left. + \,\,^{p}S_{k}^{\,n_{1} \ldots n_{d}} \sin \left[2 \pi \left(n_{1} \bar{x}_{4} + \ldots + n_{d} \bar{x}_{3 + d}\right)\right]\right\},}]and for the modulation of the tensor of thermal parameters [B_{ijk} \left(\bar{x}_{4}, \ldots, \bar{x}_{3 + d}\right)]: [\eqalign{&B_{ijk} \left(\bar{x}_{4}, \ldots, \bar{x}_{3 + d}\right)\cr &= {\textstyle\sum\limits_{n_{1} = 1}^{\infty}} \ldots {\textstyle\sum\limits_{n_{d} = 1}^{\infty}} \left\{^{B} C_{ijk}^{\,n_{1} \ldots n_{d}} \cos \left[2 \pi \left(n_{1} \bar{x}_{4} + \ldots + n_{d} \bar{x}_{3 + d}\right)\right]\right.\cr &\quad \left. + \,\,^{B}S_{ijk}^{\,n_{1} \ldots n_{d}} \sin \left[2 \pi \left(n_{1} \bar{x}_{4} + \ldots + n_{d} \bar{x}_{3 + d}\right)\right]\right\}.}]The resulting structure-factor formula is [\eqalign{F({\bf H}) &= \textstyle\sum\limits_{k = 1}^{N'} \textstyle\sum\limits_{(R, \, t)} \textstyle\int\limits_{0}^{1} {\rm d} \bar{x}_{4, \, k} \ldots \textstyle\int\limits_{0}^{1} {\rm d} \bar{x}_{3 + d, \, k}f_{k}({\bf H}^{\parallel})p_{k}\cr &\quad \times \exp \left(- \textstyle\sum\limits_{i, \, j = 1}^{3 + d} h_{i} \left[RB_{ijk} R^{T}\right]h_{j} + 2 \pi i \textstyle\sum\limits_{j = 1}^{3 + d} h_{j} Rx_{jk} + h_{j}t_{j}\right)}]for summing over the set (R, t) of superspace symmetry operations and the set of N′ atoms in the asymmetric unit of the [(3 + d)\hbox{D}] unit cell (Yamamoto, 1982[link]). Different approaches without numerical integration based on analytical expressions including Bessel functions have also been developed. For more information see Paciorek & Chapuis (1994)[link], Petricek, Maly & Cisarova (1991)[link], and references therein.

For illustration, some fundamental IMSs will be discussed briefly (see Korekawa, 1967[link]; Böhm, 1977[link]).

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 [p_{k} = (p_{k}^{0}/2) \left\{1 + \cos \left[2 \pi \left(\bar{x}_{4, \, k} + \varphi_{k}\right)\right]\right\},\quad 0 \leq p_{k}^{0} \leq 1,]the structure-factor formula for the mth order satellite [(0 \leq m \leq 1)][\eqalign{F_{0}({\bf H}) &= (1/2) {\textstyle\sum\limits_{k = 1}^{N}}\, f_{k}({\bf H}^{\parallel})T_{k}({\bf H}^{\parallel}) \exp (2 \pi i{\bf H} \cdot {\bf r}_{k}),\cr F_{m}({\bf H}) &= (1/2) {\textstyle\sum\limits_{k = 1}^{N}}\, f_{k}({\bf H}^{\parallel})T_{k}({\bf H}^{\parallel})(p_{k}^{0}/2)^{|m|} \exp \left[2 \pi i \left({\textstyle\sum\limits_{i = 1}^{3}} h_{i}x_{ik} + m \varphi_{k}\right)\right].}]Thus, a linear correspondence exists between the structure-factor magnitudes of the satellite reflections and the amplitude of the density modulation. Furthermore, only first-order 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.[link].


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Schematic diffraction patterns for 3D IMSs with (a) 1D harmonic and (b) rectangular density modulation. The modulation direction is parallel to [{\bf a}_{2}]. In (a) only first-order satellites exist; in (b), all odd-order satellites can be present. In (c), the diffraction pattern of a harmonic displacive modulation along [{\bf a}_{1}] with amplitudes parallel to [{\bf a}_{2}^{*}] is depicted. Several reflections are indexed. The areas of the circles are proportional to the reflection intensities.

Symmetric rectangular density modulation. The box-function-like modulated occupancy factor can be expanded into a Fourier series, [\displaylines{ p_{k} = p_{k}^{0} (4/\pi) \left\{{\textstyle\sum\limits_{n = 1}^{\infty}} [(-1)^{n + 1}/(2n - 1)] \cos \left[2\pi (2n - 1) (\bar{x}_{4, \, k} + \varphi_{k})\right]\right\},\cr \hfill 0 \leq p_{k}^{0} \leq 1,\cr}]and the resulting structure factor of the mth order satellite is [\eqalign{F_{0} ({\bf H}) &= (1/2) {\textstyle\sum\limits_{k = 1}^{N}}\, f_{k} ({\bf H}^{\parallel}) T_{k} ({\bf H}^{\parallel}) \exp \left(2\pi i {\textstyle\sum\limits_{i = 1}^{3}} h_{i} x_{ik}\right),\cr F_{m} ({\bf H}) &= (1/\pi m) \sin (m\pi / 2) {\textstyle\sum\limits_{k = 1}^{N}}\, f_{k} ({\bf H}^{\parallel}) T_{k} ({\bf H}^{\parallel}) p_{k}^{0} \cr&\quad\times{\rm exp}\left[2\pi i \left({\textstyle\sum\limits_{i = 1}^{3}} h_{i} x_{ik} + m \varphi_{k}\right)\right].}]According to this formula, only odd-order satellites occur in the diffraction pattern. Their structure-factor magnitudes decrease linearly with the order [|m|] (Fig.[link]b).

Harmonic displacive modulation. The displacement of the atomic coordinates is given by the function [x_{ik} = x_{ik}^{0} + A_{ik} \cos \left[2\pi (\bar{x}_{4, \, k} + \varphi_{k})\right],\quad i = 1, \ldots, 3,]and the structure factor by [\eqalign{F_{0} ({\bf H}) &= {\textstyle\sum\limits_{k = 1}^{N}} \,f_{k} ({\bf H}^{\parallel}) T_{k} ({\bf H}^{\parallel}) J_{0} (2\pi {\bf H}^{\parallel} \cdot {\bf A}_{k}) \exp \left(2\pi i {\textstyle\sum\limits_{i = 1}^{3}} h_{i} x_{ik}\right),\cr F_{m} ({\bf H}) &= {\textstyle\sum\limits_{k = 1}^{N}}\, f_{k} ({\bf H}^{\parallel}) T_{k} ({\bf H}^{\parallel}) J_{m} (2\pi {\bf H}^{\parallel} \cdot {\bf A}_{k}) \cr&\quad\times {\rm exp}\left[2\pi i \left({\textstyle\sum\limits_{i = 1}^{3}} h_{i} x_{ik} + m \varphi_{k}\right)\right].}]The structure-factor magnitudes of the mth-order satellite reflections are a function of the mth-order 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.[link]c and[link]).


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The relative structure-factor magnitudes of mth-order satellite reflections for a harmonic displacive modulation are proportional to the values of the mth-order Bessel function [J_{m}(x)].


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. Pérez-Mato, 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.

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