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Periodic diffraction from an aperiodic monohedral tiling.

Craig S Kaplan1, Michael O'Keeffe2, Michael M J Treacy3

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Acta Crystallographica. Section A, Foundations and Advances
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PubMed
Summary

Analysis of a chiral hat tiling reveals sixfold periodicity and unique diffraction patterns. Discrete satellite peaks, not diffuse scattering, break mirror symmetry, confirming the tiling

Keywords:
hat monotilingquasicrystalline tilingquasiperiodic diffraction

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Area of Science:

  • Crystallography
  • Materials Science
  • Mathematical Physics

Background:

  • A novel aperiodic monohedral tiling, known as the 'einstein' or 'hat' tiling, has been recently reported.
  • This tiling is based on the hexagonal mta net, a kite tiling, characterized by aperiodic vertex deletions.

Purpose of the Study:

  • To analyze the diffraction pattern generated by the aperiodic 'hat' tiling.
  • To investigate the structural properties and symmetry of the tiling through its diffraction characteristics.

Main Methods:

  • Analysis of the diffraction pattern from a large model of the 'hat' tiling.
  • Examination of the spatial distribution and nature of diffracted intensity, including Bragg peaks and features between them.

Main Results:

  • The diffraction pattern exhibits robust sixfold periodicity, consistent with plane group p6.
  • A distinct, repeating triangular motif of diffused intensity is observed between strong Bragg peaks.
  • This motif comprises high-density regions of discrete satellite peaks, indicative of broken mirror symmetry.

Conclusions:

  • The observed diffraction pattern, particularly the satellite peaks and broken mirror symmetry, is consistent with the chiral nature of the hat tiling.
  • The findings validate the structural model and provide experimental evidence for the unique properties of this aperiodic tiling.