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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 611-612   | 1 | 2 |

Section 4.6.3.3.2.3. Structure factor

W. Steurera* and T. Haibacha

aLaboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich, Switzerland
Correspondence e-mail:  walter.steurer@mat.ethz.ch

4.6.3.3.2.3. Structure factor

| top | pdf |

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)[link], 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[link]).

References

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








































to end of page
to top of page