The Equivalence Between Unit-Cell Twinning and Tiling in Icosahedral Quasicrystals

Albert Prodan,Marion A Van Midden,Ram Dušić Hren,Herman J P Van Midden,Erik Zupanič

doi:10.1038/s41598-017-12669-w

Abstract

It is shown that tiling in icosahedral quasicrystals can also be properly described by cyclic twinning at the unit cell level. The twinning operation is applied on the primitive prolate golden rhombohedra, which can be considered a result of a distorted face-centered cubic parent structure. The shape of the rhombohedra is determined by an exact space filling, resembling the forbidden five-fold rotational symmetry. Stacking of clusters, formed around multiply twinned rhombic hexecontahedra, keeps the rhombohedra of adjacent clusters in discrete relationships. Thus periodicities, interrelated as members of a Fibonacci series, are formed. The intergrown twins form no obvious twin boundaries and fill the space in combination with the oblate golden rhombohedra, formed between clusters in contact. Simulated diffraction patterns of the multiply twinned rhombohedra and the Fourier transform of an extended model structure are in full accord with the experimental diffraction patterns and can be indexed by means of three-dimensional crystallography. The alternative approach is fully compatible to the rather complicated descriptions in a hyper-space.

Highlights

Pauling was convinced that none of the existing crystallographic rules was violated in the newly discovered materials[5,6,7,8,9,10]
It is shown in the present work that tiling in the icosahedral QC structure can be properly explained by cyclic unit cell twinning[20,21], applied on primitive golden rhombohedra, forming intergrown twins without explicit twin-boundaries
Instead of describing a QC structure in a hyper-space, the present description is based on twinning of the basic building units

Summary

Introduction

Pauling was convinced that none of the existing crystallographic rules was violated in the newly discovered materials[5,6,7,8,9,10]. Contrary to Pauling, a number of researchers[12,13,14,15,16,17,18] considered QCs an exception to the known solid state structures, which required a novel approach Their explanation was based on the so-called Amman tiling[19], the three-dimensional equivalent of the two-dimensional Penrose tiling. Likewise to two Penrose rhombic tiles filling a plane, their three-dimensional equivalents, the prolate and the oblate golden rhombohedra, will fill the space and form the QC structure It is shown in the present work that tiling in the icosahedral QC structure can be properly explained by cyclic unit cell twinning[20,21], applied on primitive golden rhombohedra, forming intergrown twins without explicit twin-boundaries

Results

Discussion

Conclusion