International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 9.8, p. 941

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 higherdimensional 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 higherdimensional 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 onedimensional 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 onedimensional 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), PerezMato 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.PerezMato, J. M., Madariaga, G. & Tello, M. J. (1986). Diffraction symmetry of incommensurate structures. J. Phys. C, 19, 2613–2622.
PerezMato, 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 higherdimensional 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 onedimensionally modulated structures. Acta Cryst. A43, 36–42.
Wolff, P. M. de (1974). The pseudosymmetry 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.