Structure factor

Janssen, T.; Janner, A.; Looijenga-Vos, A.; Wolff, P. M. de

doi:10.1107/97809553602060000624

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

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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-Vos^b and P. M. de Wolff^c ^‡

^a Institute 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), 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), Yamamoto (1982a,b), Paciorek & Kucharczyk (1985), Petricek, Coppens & Becker (1985), Petříček & Coppens (1988), Perez-Mato et al. (1986, 1987), and Steurer (1987).

References

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