# Periodically distributed objects with quasicrystalline diffraction pattern

## Abstract

It is possible to construct fully periodically distributed objects with a diffraction pattern identical to the one obtained for quasicrystals. These objects are probability distributions of distances obtained in the statistical approach to aperiodic structures distributed periodically. The diffraction patterns have been derived by using a two-mode Fourier transform—a very powerful method not used in classical crystallography. It is shown that if scaling is present in the structure, this two-mode Fourier transform can be reduced to a regular Fourier transform with appropriately rescaled scattering vectors and added phases. Detailed case studies for model sets 1D Fibonacci chain and 2D Penrose tiling are discussed. Finally, it is shown that crystalline, quasicrystalline, and approximant structures can be treated in the same way.

- Authors:

- Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow (Poland)
- (Switzerland)

- Publication Date:

- OSTI Identifier:
- 22398811

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Applied Physics Letters; Journal Volume: 106; Journal Issue: 13; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; CRYSTALLOGRAPHY; CRYSTALS; DIFFRACTION; DISTANCE; FOURIER TRANSFORMATION; PERIODICITY; PROBABILITY; VECTORS

### Citation Formats

```
Wolny, Janusz, E-mail: wolny@fis.agh.edu.pl, Strzalka, Radoslaw, Kuczera, Pawel, and Laboratory of Crystallography, ETH Zurich, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich.
```*Periodically distributed objects with quasicrystalline diffraction pattern*. United States: N. p., 2015.
Web. doi:10.1063/1.4916830.

```
Wolny, Janusz, E-mail: wolny@fis.agh.edu.pl, Strzalka, Radoslaw, Kuczera, Pawel, & Laboratory of Crystallography, ETH Zurich, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich.
```*Periodically distributed objects with quasicrystalline diffraction pattern*. United States. doi:10.1063/1.4916830.

```
Wolny, Janusz, E-mail: wolny@fis.agh.edu.pl, Strzalka, Radoslaw, Kuczera, Pawel, and Laboratory of Crystallography, ETH Zurich, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich. Mon .
"Periodically distributed objects with quasicrystalline diffraction pattern". United States.
doi:10.1063/1.4916830.
```

```
@article{osti_22398811,
```

title = {Periodically distributed objects with quasicrystalline diffraction pattern},

author = {Wolny, Janusz, E-mail: wolny@fis.agh.edu.pl and Strzalka, Radoslaw and Kuczera, Pawel and Laboratory of Crystallography, ETH Zurich, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich},

abstractNote = {It is possible to construct fully periodically distributed objects with a diffraction pattern identical to the one obtained for quasicrystals. These objects are probability distributions of distances obtained in the statistical approach to aperiodic structures distributed periodically. The diffraction patterns have been derived by using a two-mode Fourier transform—a very powerful method not used in classical crystallography. It is shown that if scaling is present in the structure, this two-mode Fourier transform can be reduced to a regular Fourier transform with appropriately rescaled scattering vectors and added phases. Detailed case studies for model sets 1D Fibonacci chain and 2D Penrose tiling are discussed. Finally, it is shown that crystalline, quasicrystalline, and approximant structures can be treated in the same way.},

doi = {10.1063/1.4916830},

journal = {Applied Physics Letters},

number = 13,

volume = 106,

place = {United States},

year = {Mon Mar 30 00:00:00 EDT 2015},

month = {Mon Mar 30 00:00:00 EDT 2015}

}