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Published on: August 22, 2017
Multiresolution of quasicrystal diffraction spectra
Avi Elkharrat1, Jean Pierre Gazeau, Françoise Dénoyer
1Boite 7020, APC, CNRS UMR 7164, Université Paris Diderot Paris 7, 75205 Paris Cedex 13, France.
Acta Crystallographica. Section A, Foundations of Crystallography
|October 22, 2009
Summary
This study introduces a novel multiresolution analysis for classifying diffraction spectra of self-similar structures like quasicrystals. The method generates a unique
Area of Science:
- Materials Science
- Crystallography
- Mathematical Physics
Background:
- Analyzing diffraction patterns is crucial for understanding material structures.
- Pure point diffraction spectra from self-similar structures exhibit complex scaling properties.
- Existing methods may not fully capture the multiscale nature of these spectra.
Purpose of the Study:
- To develop a new method for analyzing and classifying two-dimensional pure point diffraction spectra.
- To characterize self-similar structures, including quasicrystals, using their diffraction patterns.
- To create a 'fingerprint' for diffraction spectra based on geometry, scale, and intensity.
Main Methods:
- Viewing the two-dimensional pure point diffraction spectrum as a point set in the complex plane with assigned Bragg intensities.
- Employing a nested sequence of self-similar subsets, termed beta-lattices, for multiresolution analysis.
- Implementing a partitioning of the spectrum based on geometry, scale, and intensity.
Main Results:
- The multiresolution analysis successfully partitions the diffraction spectrum.
- A unique 'fingerprint' is generated for the spectrum, reflecting its inherent properties.
- Numerical tests on mathematical structures and a quasicrystal model validate the method.
Conclusions:
- The developed method provides a robust framework for analyzing complex diffraction spectra.
- The 'fingerprint' offers a powerful tool for classifying and understanding self-similar structures.
- This approach enhances the characterization of materials like quasicrystals through their diffraction signatures.
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