Structure factor

Steurer, W.; Haibach, T.

doi:10.1107/97809553602060000568

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 4.6, pp. 506-507 | 1 | 2 |

Section 4.6.3.3.2.3. Structure factor

W. Steurer^a ^* and T. Haibach^a

^aLaboratory of Crystallography, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland
Correspondence e-mail: w.steurer@kristall.erdw.ethz.ch

4.6.3.3.2.3. Structure factor

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The structure factor for the decagonal phase corresponds to the Fourier transform of the 5D unit cell, $[F({\bf H}) = {\textstyle\sum\limits_{k = 1}^{N}} \;f_{k}({\bf H}^{\parallel})T_{k}({\bf H}^{\parallel}, {\bf H}^{\perp})g_{k}({\bf H}^{\perp}) \exp (2\pi i {\bf H}\cdot {\bf r}_{k}),]$ with 5D diffraction vectors $[{\bf H} = {\textstyle\sum_{i = 1}^{5}} h_{i}{\bf d}_{i}^{*}, N]$ hyperatoms, parallel-space atomic scattering factor $[f_{k}({\bf H}^{\parallel})]$ , temperature factor $[T_{k}({\bf H}^{\parallel}, {\bf H}^{\perp})]$ and perpendicular-space geometric form factor $[g_{k}({\bf H}^{\perp})]$ . $[T_{k}({\bf H}^{\parallel}, {\bf 0})]$ is equivalent to the conventional Debye–Waller factor and $[T_{k}({\bf 0},{\bf H}^{\perp})]$ describes random fluctuations along the perpendicular-space coordinate. These fluctuations cause characteristic jumps of vertices in physical space (phason flips). Even random phason flips map the vertices onto positions which can still be described by physical-space vectors of the type $[{\bf r} = {\textstyle\sum_{i = 1}^{5}} n_{i}{\bf a}_{i}]$ . Consequently, the set $[M = \{{\bf r} = {\textstyle\sum_{i = 1}^{5}} n_{i}{\bf a}_{i} |n_{i} \in {\bb Z}\}]$ of all possible vectors forms a $[{\bb Z}]$ module. The shape of the atomic surfaces corresponds to a selection rule for the positions actually occupied. The geometric form factor $[g_{k}({\bf H}^{\perp})]$ is equivalent to the Fourier transform of the atomic surface, i.e. the 2D perpendicular-space component of the 5D hyperatoms.

For example, the canonical Penrose tiling $[g_{k}({\bf H}^{\perp})]$ corresponds to the Fourier transform of pentagonal atomic surfaces: $[g_{k}({\bf H}^{\perp}) = (1/A_{\rm UC}^{\perp}) {\textstyle\int\limits_{A_{k}}} \exp (2\pi i {\bf H}^{\perp} \cdot {\bf r}) \hbox{ d}{\bf r},]$ where $[A_{\rm UC}^{\perp}]$ is the area of the 5D unit cell projected upon $[{\bf V}^{\perp}]$ and $[A_{k}]$ is the area of the kth atomic surface. The area $[A_{\rm UC}^{\perp}]$ can be calculated using the formula $[A_{\rm UC}^{\perp} = (4/25a_{i}^{*2}) [(7 + \tau) \sin (2\pi/5) + (2 + \tau) \sin (4\pi/5)].]$ Evaluating the integral by decomposing the pentagons into triangles, one obtains $[\eqalign{ g_{k}({\bf H}^{\perp}) &={1 \over A_{\rm UC}^{\perp}} \sin \left({2\pi \over 5}\right)\cr &\quad\times\sum\limits_{j = 0}^{4} {A_{j} [\exp (iA_{j + 1}\lambda_{k}) - 1] - A_{j + 1} [\exp (iA_{j}\lambda_{k}) - 1] \over A_{j}A_{j + 1} (A_{j} - A_{j + 1})}}]$ with $[j = 0, \ldots, 4]$ running over the five triangles, where the radii of the pentagons are $[\lambda_{j}]$ , $[A_{j} = 2\pi {\bf H}^{\perp}{\bf e}_{j}]$ , $[{\bf H}^{\perp} = \pi^{\perp}({\bf H}) = {\textstyle\sum\limits_{j = 0}^{4}} h_{j}a_{j}^{*} \pmatrix{0\cr 0\cr 0\cr \cos (6\pi j/5)\cr \sin (6\pi j/5)\cr}]$ and the vectors $[{\bf e}_{j} = {1 \over a_{j}^{*}} \pmatrix{0\cr 0\cr 0\cr \cos (2\pi j/5)\cr \sin (2\pi j/5)\cr} \hbox{ with } j = 0, \ldots, 4.]$

As shown by Ishihara & Yamamoto (1988), the Penrose tiling can be considered to be a superstructure of a pentagonal tiling with only one type of pentagonal atomic surface in the nD unit cell. Thus, for the Penrose tiling, three special reflection classes can be distinguished: for $[|{\textstyle\sum_{i = 1}^{4}} h_{i}| = m\hbox{ mod }5]$ and [m = 0] the class of strong main reflections is obtained, and for $[m = \pm 1, \pm 2]$ the classes of weaker first- and second-order satellite reflections are obtained (see Fig. 4.6.3.18).

References

Ishihara, K. N. & Yamamoto, A. (1988). Penrose patterns and related structures. I. Superstructure and generalized Penrose patterns. Acta Cryst. A44, 508–516.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.6, pp. 506-507