International
Tables for
Crystallography
Volume G
Definition and exchange of crystallographic data
Edited by S. R. Hall and B. McMahon

International Tables for Crystallography (2006). Vol. G, ch. 3.4, pp. 138-139

Section 3.4.4.3. Description of the structure

G. Madariagaa*

aDepartamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain
Correspondence e-mail: gotzon.madariaga@ehu.es

3.4.4.3. Description of the structure

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A modulated structure is described by a reference periodic structure and the atomic modulation functions. Such functions are periodic and are normally expanded as Fourier series. The modulated parameters may apply to the atom positions (displacive modulation), the site occupancies (occupational modulation) and/or the temperature factors. In composite structures, each substructure is referred to the crystallographic basis defined by the W matrices [see equation (3.4.4.2[link])]. The simplest case corresponds to a one-dimensional displacive modulated structure. In this case, the atomic modulation functions are given by[u^{\mu}_{\alpha}=\textstyle\sum\limits_{n=1}^{\infty} U^{\mu}_{n \alpha} \cos ( 2 \pi n {{\bf q} \cdot {\bf r}} + \varphi ^{\mu}_{n \alpha} ),\eqno(3.4.4.3)]where [( U^{\mu}_{n \alpha},\varphi ^{\mu}_{n \alpha})] is the complex amplitude of each Fourier term; [\mu] labels the atoms; [\alpha=x, y, z]; r is the average atom position; and [n{\bf q}] represents the successive harmonics of the modulation. At present, displacive modulations along axes other than a, b and c can be calculated with the restriction stated in Section 3.4.2[link].

Atomic displacements can also be expressed as rigid rotations and translations. The incommensurate phase of K2SeO4 is one-dimensional displacive and sinusoidal (at least over a wide range of temperature), i.e. n = 1 in equation (3.4.4.3[link]). The tetrahedral SeO4 groups behave as rigid bodies. Symmetry considerations restrict the possible translations to occur along c where the only allowed rigid rotations are around the a and b axes. The incommensurate structure is then expressed as shown in Example 3.4.4.4[link].

Example 3.4.4.4. Atomic displacements as translations and rigid rotations.

[Scheme scheme42]

Alternatively, equation (3.4.4.3[link]) can be expressed as a Fourier series with real amplitudes. This form is also covered by the msCIF dictionary. Note that there is a global phase, which is irrelevant in the incommensurate case but fixes the space group if the modulated structure is commensurate. Global phases are also defined in the dictionary.








































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