# Structure Factor for Generalized Penrose Tiling

__M. Chodyń__^{1}, P. Kuczera^{1,2} and J. Wolny^{1}

^{1}Faculty of Physics and
Applied Computer Science, AGH University of Science and Technology, Mickiewicz
Avenue 30, 30-059 Krakow, Poland

^{2}Laboratory of
Crystallography, ETH Zurich, Wolfgang-Pauli-Strasse 10, Zurich, CH-8093,
Switzerland

maciej.chodyn@gmail.com

The Generalized Penrose Tiling (GPT)[1,2] can
be considered a promising alternative for Penrose Tiling (PT) as a model for
decagonal quasicrystal refinement procedure, particularly in the statistical
approach (also called the Average Unit Cell (AUC) approach) [3]. The
statistical method using PT has been successfully applied to the structure optimization
of various decagonal phases [4]. The application of the AUC concept to the GPT
will be presented.

In the higher dimensional (*n*D)
approach, PT is obtained by projecting a 5D hypercubic lattice through a window
consisting of four pentagons, called the atomic surfaces (ASs), in the
perpendicular space. The vertices of these pentagons together with two
additional points form a rhombicosahedron. The GPT is created by projecting the
5D hypercubic lattice through a window consisting of five polygons, generated
by shifting the ASs along the body diagonal of the rhombicosahedron. Three of
the previously pentagonal ASs will become decagon, one will remain pentagonal
and one more pentagon will be created (for PT it is a single point). The
structure of GPT will depend on the shift parameter, its building units are
still thick and thin rhombuses, but the matching rules and the tiling changes.
Diffraction pattern of GPT have peaks in the same positions as regular PT,
however their intensities are different.

Binary decagonal quasicrystal structure with
arbitrary decoration for a given shift value was simulated. Its diffraction
pattern was calculated using AUC method[5,6]. Generated diffraction pattern were
used as "experimental data set" in structure refinement algorithm
made to test the refining of shift parameter.

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#### 3. J. Wolny, *Phil. Mag. *(1998), **77**, 395-414

#### 4. P. Kuczera, J. Wolny, W. Steurer, *Acta Cryst.* (2012), **B68**, 578–589

#### 5. B. Kozakowski, J. Wolny, *Acta **Cryst *(2010),* ***A66**,
489-498

#### 6. M. Chodyn, P. Kuczera, J. Wolny, *Acta Cryst*. (2015), **A71**,
161–168