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 [{\bf r}_n] is described in the kinematic approximation by the structure factor: [S_{\bf H}=\textstyle\sum\limits_n\,f_n({\bf H})\exp(2\pi i{\bf H}\cdot{\bf r}_n), \eqno (9.8.4.44)]where [f_n({\bf H})] is the atomic scattering factor. For an incommensurate crystal phase, this structure factor [S_{\bf H}] is equal to the structure factor [S_{H_S}] of the crystal structure embedded in 3 + d dimensions, where H is the projection of [H_s] on [V_E]. This structure factor is expressed by a sum of the products of atomic scattering factors [f_n] and phase factors [\exp(2\pi iH_s\cdot r_{sn})] 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) [\eqalignno{ S_{\bf H}&=\textstyle\sum\limits_A\textstyle\sum\limits_j\textstyle\int\limits_\Omega{\rm d}{\bf t}\ f_A({\bf H})P_{Aj}({\bf t}) \cr &\quad\times\exp\{2\pi i({\bf H,H}_I)\cdot [{\bf r}_j+{\bf u}_j({\bf t}), {\bf t}]\}, & (9.8.4.45)}]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 [{\bf d}_1,\ldots,{\bf d}_d] in [V_I]; [f_A] is the atomic scattering factor of species A, [P_{Aj}({\bf t})] 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 [P_j(t)=1\semi \quad {\bf u}_j(t)={\bf U}_j\sin[2\pi({\bf q}\cdot {\bf r}_j+t+\varphi_j)]. \eqno (9.8.4.46)]According to (9.8.4.45)[link], the structure factor is [\eqalignno{ S_{\bf H} &=\textstyle\sum\limits_j\textstyle\int\limits^1_0{\rm d} t\ f_j({\bf H})\exp(2\pi i{\bf H}\cdot{\bf r}_j)\exp(2\pi imt) \cr &\quad\times\exp[2\pi i{\bf H}\cdot{\bf U}_j\sin2\pi({\bf q}\cdot{\bf r}_j+t+\varphi_j)]. &(9.8.4.47)}]For a diffraction vector H = K + mq, this reduces to [\eqalignno{ S_{\bf H} &=\textstyle\sum\limits_j\,f_j({\bf H})\exp(2\pi i{\bf K\cdot r}_j)J_{-m}(2\pi{\bf H}\cdot{\bf U}_j) \cr &\quad\times\exp(-2\pi im\varphi_j). & (9.8.4.48)}]For a general one-dimensional modulation with occupation modulation function [p_j(t)] and displacement function [{\bf u}_j(t)], the structure factor becomes [\eqalignno{ S_{\bf H}&=\textstyle\sum\limits_j\textstyle\int\limits^1_0{\rm d} t\ f_j({\bf H})p_j({\bf q}\cdot {\bf r}_j+t+\psi_j)\exp[2\pi i({\bf H}\cdot{\bf r}_j+mt)] \cr&\quad\times\exp[2\pi i{\bf H}\cdot {\bf u}_j({\bf q}\cdot{\bf r}_j+t+\varphi_j)]. & (9.8.4.49)}]Because of the periodicity of [p_j(t)] and [{\bf u}_j(t)], one can expand the Fourier series: [\eqalignno{ &p_j({\bf q}\cdot{\bf r}_j+t+\psi_j)\exp[2\pi i{\bf H}\cdot{\bf u}_j({\bf q}\cdot{\bf r}_j+t+\varphi_j)] \cr&\quad=\textstyle\sum\limits_k\,C_{j,k}({\bf H})\exp[2\pi ik({\bf q}\cdot{\bf r}_j+t)], &(9.8.4.50)}]and consequently the structure factor becomes [S_{\bf H}=\textstyle\sum\limits_j\,f_j({\bf H})\exp(2\pi i{\bf K}\cdot{\bf r}_j)C_{j,-m}({\bf H}), \quad\hbox{where }{\bf H}={\bf K}+m{\bf q}. \eqno (9.8.4.51)]The diffraction from incommensurate crystal structures has been treated by de Wolff (1974[link]), Yamamoto (1982a[link],b[link]), Paciorek & Kucharczyk (1985[link]), Petricek, Coppens & Becker (1985[link]), Petříček & Coppens (1988[link]), Perez-Mato et al. (1986[link], 1987[link]), and Steurer (1987[link]).

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.








































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