InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 344-348
## Section 2.5.3.4.1. General remarks M. Tanaka
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Incommensurately modulated crystals do not have three-dimensional lattice periodicity. The crystals, however, recover lattice periodicity in a space higher than three dimensions. de Wolff (1974, 1977) showed that one-dimensional displacive and substitutionally modulated crystals can be described as a three-dimensional section of a (3 + 1)-dimensional periodic crystal. Janner & Janssen (1980*a*,*b*) developed a more general approach for describing a modulated crystal with *n* modulations as (3 + *n*)-dimensional periodic crystals (*n* = 1, 2, …). Yamamoto (1982) derived a general structure-factor formula for *n*-dimensionally modulated crystals (*n* = 1, 2, …), which holds for both displacive and substitutionally modulated crystals. Tables of the (3 + 1)-dimensional space groups for one-dimensional incommensurately modulated crystals were given by de Wolff *et al.* (1981), where the wavevector of the modulation was assumed to lie in the *c* direction. Later, some corrections to the tables were made by Yamamoto *et al.* (1985). The analysis of incommensurately modulated crystals using (3 + 1)-dimensional space groups has become familiar in the field of X-ray structure analysis.

Fung *et al.* (1980) applied the CBED method to the study of incommensurately modulated transition-metal dichalcogenides. Steeds *et al*. (1985) applied the LACBED method (Tanaka *et al.*, 1980) to the study of incommensurately modulated crystals of NiGe_{1−x}P_{x}. Tanaka *et al.* (1988, pp. 74–81) examined the symmetries of the incommensurate and fundamental reflections appearing in the CBED patterns obtained from the incommensurately modulated crystals of Sr_{2}Nb_{2}O_{7} and Mo_{8}O_{23}. Terauchi & Tanaka (1993) clarified theoretically the interrelation between the symmetries of CBED patterns and the (3 + 1)-dimensional point-group symbols for incommensurately modulated crystals and verified experimentally the theoretical results for Sr_{2}Nb_{2}O_{7} and Mo_{8}O_{23}. Terauchi *et al.* (1994) investigated dynamical extinction for the (3 + 1)-dimensional space groups. They clarified that approximate dynamical extinction lines appear in CBED discs of the reflections caused by incommensurate modulations when the amplitudes of the incommensurate modulation waves are small. They tabulated the dynamical extinction lines appearing in the CBED discs for all the (3 + 1)-dimensional space groups of the incommensurately modulated crystals. The tables were stored in the British Library Document Supply Centre as Supplementary Publication No. SUP 71810 (65 pp.). They showed an example of the dynamical extinction lines obtained from Sr_{2}Nb_{2}O_{7}. The point- and space-group determinations of the (3 + 1)-dimensional crystals are described compactly in the book by Tanaka *et al.* (1994, pp. 156–205).

Fig. 2.5.3.17 illustrates (3 + 1)-dimensional descriptions of a crystal structure without modulation (*a*), a one-dimensional displacive modulated structure (*b*) and a one-dimensional substitutionally modulated structure (*c*). The arrows labelled **a**_{1}–**a**_{3} (**a**, **b** and **c**) and **a**_{4} indicate the (3 + 1)-dimensional crystal axes. The horizontal line labelled **R**_{3} represents the three-dimensional space (external space). In the (3 + 1)-dimensional description, an atom is not located at a point as in the three-dimensional space, but extends as a string along the fourth direction **a**_{4} perpendicular to the three-dimensional space **R**_{3}. The shaded parallelogram is a unit cell in the (3 + 1)-dimensional space. The unit cell contains two atom strings in this case. In the case of no modulations, the atoms are shown as straight strings, as shown in Fig. 2.5.3.17(*a*). For a displacive modulation, atoms are expressed by wavy strings periodic along the fourth direction **a**_{4} as shown in Fig. 2.5.3.17(*b*). The width of the atom strings indicates the spread of the atoms in **R**_{3}. The atom positions of the modulated structure in **R**_{3} are given as a three-dimensional (**R**_{3}) section of the atom strings in the (3 + 1)-dimensional space. A substitutional modulation, which is described by a modulation of the atom form factor, is expressed by atom strings with a density modulation along the direction **a**_{4} as shown in Fig. 2.5.3.17(*c*).

The diffraction vector **G** is written aswhere a set of *h*_{1}*h*_{2}*h*_{3}*h*_{4} is a (3 + 1)-dimensional reflection index, and **a***, **b*** and **c*** are the reciprocal-lattice vectors of the real-lattice vectors **a**, **b** and **c** of the average structure. The modulation vector **k** is written aswhere one coefficient *k*_{i} (*i* = 1–3) is an irrational number and the others are rational. Fig. 2.5.3.18(*a*) shows a diffraction pattern of a crystal with an incommensurate modulation wavevector *k*_{1}**a*** (*k*_{2} and *k*_{3} = 0). Large and small black spots show the fundamental reflections and incommensurate reflections, respectively, only the first-order incommensurate reflections being shown. It should be noted that the diffraction pattern of a modulated crystal is obtained by a projection of the Fourier transform of the (3 + 1)-dimensional periodic structure. Fig. 2.5.3.18(*b*) is assumed to be the Fourier transform of Fig. 2.5.3.17(*b*). Incommensurate reflections are obtained by a projection of the reciprocal-lattice points onto .

The displacive modulation is expressed by the atom displacement with *x*_{4}. The structure factor for the (3 + 1)-dimensional crystal with a displacive modulation is given by de Wolff (1974, 1977) as follows:where and (*i* = 1–3) are, respectively, the atom form factor and the *i*th component of the position of the *μ*th atom in the unit cell of the average structure. The symbol is the *i*th component of the displacement of the *μ*th atom. Since the atom in the (3 + 1)-dimensional space is continuous along **a**_{4} and discrete along **R**_{3}, the structure factor is expressed by summation in **R**_{3} and integration along **a**_{4} as seen in equation (2.5.3.6). The integration implies that the sum for the atoms with displacements is taken over the infinite number of unit cells of the average structure. That is, equation (2.5.3.6) is the structure factor for a unit cell with the lattice parameter of an infinite length in **R**_{3} along the direction of the modulation wavevector **k**.

CBED patterns are obtained from a finite area of a specimen crystal. For the symmetry analysis of CBED patterns obtained from modulated structures, the effect of the finite size was considered by Terauchi & Tanaka (1993). The integration over a unit-cell length along **a**_{4} in equation (2.5.3.6) is rewritten in the following way with the summation over a finite number of three-dimensional sections of the atom strings:where , and , being the number of unit cells of the average structure included in a specimen volume from which CBED patterns are taken.

The substitutional modulation arises from a periodic variation of the site-occupation probability of the atoms. This modulation is expressed by a modulation of the atom form factor with *x*_{4}. The structure factor for a finite-size crystal is written aswhere .

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