International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 9.8, p. 941

## Section 9.8.4.4.3. Structure factor

T. Janssen,a A. Janner,a A. Looijenga-Vosb and P. M. de Wolffc

aInstitute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands,bRoland Holstlaan 908, NL-2624 JK Delft, The Netherlands, and cMeermanstraat 126, 2614 AM, Delft, The Netherlands

#### 9.8.4.4.3. Structure factor

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The scattering from a set of atoms at positions is described in the kinematic approximation by the structure factor: where is the atomic scattering factor. For an incommensurate crystal phase, this structure factor is equal to the structure factor of the crystal structure embedded in 3 + d dimensions, where H is the projection of on . This structure factor is expressed by a sum of the products of atomic scattering factors and phase factors over all particles in the unit cell of the higher-dimensional lattice. For an incommensurate phase, the number of particles in such a unit cell is infinite: for a given atom in space, the embedded positions form a dense set on lines or hypersurfaces of the higher-dimensional space. Disregarding pathological cases, the sum may be replaced by an integral. Including the possibility of an occupation modulation, the structure factor becomes (up to a normalization factor) where the first sum is over the different species, the second over the positions in the unit cell of the basic structure, the integral over a unit cell of the lattice spanned by in ; is the atomic scattering factor of species A, is the probability of atom j being of species A when the internal position is t.

In particular, for a given atomic species, without occupational modulation and a sinusoidal one-dimensional displacive modulation According to (9.8.4.45), the structure factor is For a diffraction vector H = K + mq, this reduces to For a general one-dimensional modulation with occupation modulation function and displacement function , the structure factor becomes Because of the periodicity of and , one can expand the Fourier series: and consequently the structure factor becomes The diffraction from incommensurate crystal structures has been treated by de Wolff (1974), Yamamoto (1982a,b), Paciorek & Kucharczyk (1985), Petricek, Coppens & Becker (1985), Petříček & Coppens (1988), Perez-Mato et al. (1986, 1987), and Steurer (1987).

### References

Paciorek, W. A. & Kucharczyk, D. (1985). Structure factor calculations in refinement of a modulated crystal structure. Acta Cryst. A41, 462–466.
Perez-Mato, J. M., Madariaga, G. & Tello, M. J. (1986). Diffraction symmetry of incommensurate structures. J. Phys. C, 19, 2613–2622.
Perez-Mato, J. M., Madariaga, G., Zuñiga, F. J. & Garcia Arribas, A. (1987). On the structure and symmetry of incommensurate phases. A practical formulation. Acta Cryst. A43, 216–226.
Petříček, V. & Coppens, P. (1988). Structure analysis of modulated molecular crystals. III. Scattering formalism and symmetry considerations: extension to higher-dimensional space groups. Acta Cryst. A44, 235–239.
Petricek, V., Coppens, P. & Becker, P. (1985). Structure analysis of displacively modulated molecular crystals. Acta Cryst. A41, 478–483.
Steurer, W. (1987). (3+1)-dimensional Patterson and Fourier methods for the determination of one-dimensionally modulated structures. Acta Cryst. A43, 36–42.
Wolff, P. M. de (1974). The pseudo-symmetry of modulated crystal structures. Acta Cryst. A30, 777–785.
Yamamoto, A. (1982a). A computer program for the refinement of modulated structures. Report NIRIM, Ibaraki, Japan.
Yamamoto, A. (1982b). Structure factor of modulated crystal structures. Acta Cryst. A38, 87–92.