An Exact Method to Determine the Photonic Resonances of Quasicrystals Based on Discrete Fourier Harmonics of Higher-Dimensional Atomic Surfaces
A rigorous method for obtaining the diffraction patterns of quasicrystals is presented. Diffraction patterns are an essential analytical tool in the study of quasicrystals, since they can be used to determine their photonic resonances. Previous methods for approximating the diffraction patterns of q...
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| Format: | Article |
| Language: | English |
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MDPI AG
2016-08-01
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| Series: | Crystals |
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| Online Access: | http://www.mdpi.com/2073-4352/6/8/93 |
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| author | Farhad A. Namin Douglas H. Werner |
| author_facet | Farhad A. Namin Douglas H. Werner |
| author_sort | Farhad A. Namin |
| collection | DOAJ |
| description | A rigorous method for obtaining the diffraction patterns of quasicrystals is presented. Diffraction patterns are an essential analytical tool in the study of quasicrystals, since they can be used to determine their photonic resonances. Previous methods for approximating the diffraction patterns of quasicrystals have relied on evaluating the Fourier transform of finite-sized super-lattices. Our approach, on the other hand, is exact in the sense that it is based on a technique that embeds quasicrystals into higher dimensional periodic hyper-lattices, thereby completely capturing the properties of the infinite structure. The periodicity of the unit cell in the higher dimensional space can be exploited to obtain the Fourier series expansion in closed-form of the corresponding atomic surfaces. The utility of the method is demonstrated by applying it to one-dimensional Fibonacci and two-dimensional Penrose quasicrystals. The results are verified by comparing them to those obtained by using the conventional super-lattice method. It is shown that the conventional super-cell approach can lead to inaccurate results due to the continuous nature of the Fourier transform, since quasicrystals have a discrete spectrum, whereas the approach introduced in this paper generates discrete Fourier harmonics. Furthermore, the conventional approach requires very large super-cells and high-resolution sampling of the reciprocal space in order to produce accurate results leading to a very large computational burden, whereas the proposed method generates accurate results with a relatively small number of terms. Finally, we propose how this approach can be generalized from the vertex model, which assumes identical particles at all vertices, to a more realistic case where the quasicrystal is composed of different atoms. |
| first_indexed | 2024-04-13T01:19:57Z |
| format | Article |
| id | doaj.art-6922bda292504ca495d5c8f0b744b439 |
| institution | Directory Open Access Journal |
| issn | 2073-4352 |
| language | English |
| last_indexed | 2024-04-13T01:19:57Z |
| publishDate | 2016-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Crystals |
| spelling | doaj.art-6922bda292504ca495d5c8f0b744b4392022-12-22T03:08:48ZengMDPI AGCrystals2073-43522016-08-01689310.3390/cryst6080093cryst6080093An Exact Method to Determine the Photonic Resonances of Quasicrystals Based on Discrete Fourier Harmonics of Higher-Dimensional Atomic SurfacesFarhad A. Namin0Douglas H. Werner1Department of Electrical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran 15875, IranDepartment of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802, USAA rigorous method for obtaining the diffraction patterns of quasicrystals is presented. Diffraction patterns are an essential analytical tool in the study of quasicrystals, since they can be used to determine their photonic resonances. Previous methods for approximating the diffraction patterns of quasicrystals have relied on evaluating the Fourier transform of finite-sized super-lattices. Our approach, on the other hand, is exact in the sense that it is based on a technique that embeds quasicrystals into higher dimensional periodic hyper-lattices, thereby completely capturing the properties of the infinite structure. The periodicity of the unit cell in the higher dimensional space can be exploited to obtain the Fourier series expansion in closed-form of the corresponding atomic surfaces. The utility of the method is demonstrated by applying it to one-dimensional Fibonacci and two-dimensional Penrose quasicrystals. The results are verified by comparing them to those obtained by using the conventional super-lattice method. It is shown that the conventional super-cell approach can lead to inaccurate results due to the continuous nature of the Fourier transform, since quasicrystals have a discrete spectrum, whereas the approach introduced in this paper generates discrete Fourier harmonics. Furthermore, the conventional approach requires very large super-cells and high-resolution sampling of the reciprocal space in order to produce accurate results leading to a very large computational burden, whereas the proposed method generates accurate results with a relatively small number of terms. Finally, we propose how this approach can be generalized from the vertex model, which assumes identical particles at all vertices, to a more realistic case where the quasicrystal is composed of different atoms.http://www.mdpi.com/2073-4352/6/8/93quasicrystalsdiffraction patternhigher dimensional approachcut-and-project |
| spellingShingle | Farhad A. Namin Douglas H. Werner An Exact Method to Determine the Photonic Resonances of Quasicrystals Based on Discrete Fourier Harmonics of Higher-Dimensional Atomic Surfaces Crystals quasicrystals diffraction pattern higher dimensional approach cut-and-project |
| title | An Exact Method to Determine the Photonic Resonances of Quasicrystals Based on Discrete Fourier Harmonics of Higher-Dimensional Atomic Surfaces |
| title_full | An Exact Method to Determine the Photonic Resonances of Quasicrystals Based on Discrete Fourier Harmonics of Higher-Dimensional Atomic Surfaces |
| title_fullStr | An Exact Method to Determine the Photonic Resonances of Quasicrystals Based on Discrete Fourier Harmonics of Higher-Dimensional Atomic Surfaces |
| title_full_unstemmed | An Exact Method to Determine the Photonic Resonances of Quasicrystals Based on Discrete Fourier Harmonics of Higher-Dimensional Atomic Surfaces |
| title_short | An Exact Method to Determine the Photonic Resonances of Quasicrystals Based on Discrete Fourier Harmonics of Higher-Dimensional Atomic Surfaces |
| title_sort | exact method to determine the photonic resonances of quasicrystals based on discrete fourier harmonics of higher dimensional atomic surfaces |
| topic | quasicrystals diffraction pattern higher dimensional approach cut-and-project |
| url | http://www.mdpi.com/2073-4352/6/8/93 |
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