Diffraction pattern of modulated structures described by Bessel functions

We performed detailed analysis of 1D modulated structure (MS) with harmonic modulation within the statistical approach. By applying two-mode Fourier transform, we were able to derive analytically the structure factor for MS with single harmonic modulation component. We confirmed in a very smooth way...

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Bibliographic Details
Published in:Philosophical magazine (Abingdon, England) Vol. 96; no. 13; pp. 1344 - 1359
Main Authors: Wolny, Janusz, Buganski, Ireneusz, Strzalka, Radoslaw
Format: Journal Article
Language:English
Published: Taylor & Francis 02.05.2016
Subjects:
Asymptotic properties
Bessel functions
Diffraction patterns
harmonic modulation
Harmonics
Mathematical models
Modulated structures
Modulation
quasicrystal
Quasicrystals
statistical approach
Tuning
two-mode Fourier transform
ISSN:1478-6435, 1478-6443
Online Access:Get full text
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Summary:We performed detailed analysis of 1D modulated structure (MS) with harmonic modulation within the statistical approach. By applying two-mode Fourier transform, we were able to derive analytically the structure factor for MS with single harmonic modulation component. We confirmed in a very smooth way that ordinary Bessel functions of the first kind define envelopes tuning the intensities of the diffraction peaks. This applies not only to main reflections of the diffraction pattern but also to all satellites. In the second part, we discussed in details the similarities between harmonically modulated structures with multiple modulations and 1D model quasicrystal. The Fourier expansion of the nodes' positions in the Fibonacci chain gives direct numerical definition of the atomic arrangement in MS. In that sense, we can define 1D quasicrystal as a MS with infinite number of harmonic modulations. We prove that characteristic measures (like v(u) relation typical for statistical approach and diffraction pattern) calculated for MS asymptotically approach their counterparts for 1D quasicrystal as large enough number of modulation terms is taken into account.
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ISSN:1478-6435
1478-6443
DOI:10.1080/14786435.2016.1161859