Tables for
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. 135-136

Section Modulation functions as Fourier series

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: Modulation functions as Fourier series

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Data items in these categories are as follows:

(a) ATOM_SITE_DISPLACE_FOURIER [Scheme scheme17]



(d) ATOM_SITE_OCC_FOURIER [Scheme scheme20]

(e) ATOM_SITE_OCC_FOURIER_PARAM [Scheme scheme21]

(f) ATOM_SITE_ROT_FOURIER [Scheme scheme22]

(g) ATOM_SITE_ROT_FOURIER_PARAM [Scheme scheme23]

(h) ATOM_SITE_U_FOURIER [Scheme scheme24]

(i) ATOM_SITE_U_FOURIER_PARAM [Scheme scheme25]


(k) ATOM_SITES_MODULATION [Scheme scheme27]

(l) ATOM_SITES_ROT_FOURIER [Scheme scheme28]

The bullet ([\bullet]) indicates a category key. The arrow ([\rightarrow]) is a reference to a parent data item.

It is common to represent a modulated structure using a reference periodic structure on which are superimposed atomic modulation functions expanded as Fourier series. (A full discussion of this is given in Section[link].) The msCIF dictionary provides separate categories for listing the modulated parameters that apply to atom positions, site occupancies and thermal parameters. The structuring of the data items within each of these categories follows a similar pattern.

For example, consider the modulation of the atomic displacements. The ATOM_SITE_DISPLACE_FOURIER category allows a listing of the axis along which the displacement occurs (*_axis) and the wave vectors contributing to that displacement component (*_wave_vector_seq_id) for each relevant atom site (labelled by _atom_site_displace_Fourier_atom_site_label). *_wave_vector_seq_id is a pointer to the description of the separate modulation wave vectors and must match one of the identifiers _atom_site_Fourier_wave_vector_seq_id listed separately in the ATOM_SITE_FOURIER_WAVE_VECTOR category. Likewise, the *_atom_site_label data item must match a value of _atom_site_label in the main list of atom positions. This is how the modulation is linked to the atom list. The item _atom_site_displace_Fourier_id is the formal key for the ATOM_SITE_DISPLACE_FOURIER category. It is used to locate the matching Fourier coefficients in the ATOM_SITE_DISPLACE_FOURIER_PARAM category. The coefficients may be reported in a sine–cosine ( _atom_site_displace_Fourier_param_sin, *_cos) or modulus–argument (*_mod, *_phase) representation.

Where a group of atoms is treated as a rigid group, the categories above describe only the translational part of the positional distortion. ATOM_SITE_ROT_FOURIER and ATOM_SITE_ROT_FOURIER_PARAM are used to describe the rotational components.

ATOM_SITE_OCC_FOURIER and ATOM_SITE_U_FOURIER, and their associated *_PARAM categories, are the analogous categories for the modulation of site occupation and thermal parameters.

All the categories above describe the properties of individual atom sites. Larger-scale descriptions of the displacive modulation or of the rotational component of a rigid group are covered by the categories ATOM_SITES_DISPLACE_FOURIER and ATOM_SITES_ROT_FOURIER, each of which at present contains one descriptive data item.

The ATOM_SITES_MODULATION category contains data items describing the initial phases of the modulation waves, which are essential for determining the space group of the commensurate superstructure. More details are given in the dictionary.

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