Diffraction pattern of modulated structures described by Bessel functions

Abstract

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.

Keywords:

• Modulated structures
• harmonic modulation
• Bessel functions
• two-mode Fourier transform
• quasicrystal
• statistical approach

Acknowledgements

Authors are grateful to Lorenzo Sadun for inspiring comments and discussions on the problem of two-mode Fourier transforming. The financial support from Polish National Science Center under grants no. DEC-2013/11/B/ST3/03787 and 2014/13/N/ST3/03776 (R.S.) needs also to be acknowledged.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by Polish National Science Center [grant number DEC-2013/11/B/ST3/03787] and [grant number 2014/13/N/ST3/03776].