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, p. 137

## Section 3.4.4.1. Description of reciprocal space

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

#### 3.4.4.1. Description of reciprocal space

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Modulated and composite structures need more than three reciprocal vectors in order to index the whole set of reflections with integer numbers. Hence a diffraction vector is written aswhere the notation has been chosen according to the core CIF dictionary. In the case of a modulated structure, , and are the reciprocal vectors of the reference structure (and therefore h, k and l index the main reflections). are the modulation wave vectors. They are three-dimensional vectors with some irrational component (if the modulated structure is incommensurate) in the lattice spanned by , and . d is the dimension of the modulation. In the case of composite structures, the diffraction pattern can be indexed using 3 + d (arbitrarily selected) vectors . , and normally span the reciprocal lattice of the main reflections of one of the substructures (notice that this is only one particular, but highly intuitive, choice). The remaining d vectors with are the wave vectors of the modulation [ in equation (3.4.4.1)].

In a composite structure, the (3 + d)-dimensional reciprocal basis of the subsystem is determined by a (3 + d) × (3 + d) matrix [see van Smaalen (1995) and references therein]:where the subscripts i = 1, 2 and 3 label the reciprocal vectors , and , and label the wave vectors of the modulation expressed as linear combinations of , and .

The simplest case corresponds to a one-dimensional (d = 1) modulated structure. Consider for example the incommensurate phase of K2SeO4. The wave vector of the modulation can be chosen to be . Relevant information about the diffraction pattern of this compound is expressed using both the core CIF and msCIF dictionaries as shown in Example 3.4.4.1.

#### Example 3.4.4.1. msCIF description of the diffraction pattern of a one-dimensional modulated structure.

A more complicated example is the composite structure (LaS)1.14NbS2. The two mutually incommensurate subsystems (along the a axis) are (van Smaalen, 1991) NbS2 () and LaS (). The reciprocal basis can be chosen to be , , and . For this particular choice, the two W matrices [see equation (3.4.4.2)] are

This information is transcribed to CIF format as shown in Example 3.4.4.2. (Note that the default values for the wave vector components and the elements of W are 0.)

### References

Smaalen, S. van (1991). Superspace-group approach to the modulated structure of the inorganic misfit layer compound (LaS)1.14NbS2. J. Phys. Condens. Matter, 3, 1247–1263.
Smaalen, S. van (1995). Incommensurate crystal structures. Crystallogr. Rev. 4, 79–202.