Spatially localized quasicrystalline structures

Subramanian, Priya; Archer, Andrew J.; Knobloch, Edgar; Rucklidge, Alastair M.

doi:10.1088/1367-2630/aaf3bd

Cited by 26 publications

(25 citation statements)

References 54 publications

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“…So far, only the face-centred icosahedral (FCI) and primitive icosahedral (PI) phases have been reported experimentally. The only observation of the body-centred icosahedral phase (I-type) comes from numerical simulations (Engel et al, 2015;Subramanian et al, 2018). iQCs are divided into three groups depending on the local geometry of atomic sites.…”

Section: General Informationmentioning

confidence: 99%

The atomic structure of the Bergman-type icosahedral quasicrystal based on the Ammann–Kramer–Neri tiling

2020

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In this study, the atomic structure of the ternary icosahedral ZnMgTm quasicrystal (QC) is investigated by means of single‐crystal X‐ray diffraction. The structure is found to be a member of the Bergman QC family, frequently found in Zn–Mg–rare‐earth systems. The ab initio structure solution was obtained by the use of the Superflip software. The infinite structure model was founded on the atomic decoration of two golden rhombohedra, with an edge length of 21.7 Å, constituting the Ammann–Kramer–Neri tiling. The refined structure converged well with the experimental diffraction diagram, with the crystallographic R factor equal to 9.8%. The Bergman clusters were found to be bonded by four possible linkages. Only two linkages, b and c, are detected in approximant crystals and are employed to model the icosahedral QCs in the cluster approach known for the CdYb Tsai‐type QC. Additional short b and a linkages are found in this study. Short interatomic distances are not generated by those linkages due to the systematic absence of atoms and the formation of split atomic positions. The presence of four linkages allows the structure to be pictured as a complete covering by rhombic triacontahedral clusters and consequently there is no need to define the interstitial part of the structure (i.e. that outside the cluster). The 6D embedding of the solved structure is discussed for the final verification of the model.

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Section: General Informationmentioning

confidence: 99%

The atomic structure of the Bergman-type icosahedral quasicrystal based on the Ammann–Kramer–Neri tiling

2020

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“…It would be interesting to explore these complex patterns in more detail, in the context of coupled reaction-diffusion systems, in the context of amplitude equations, especially in the case q < 1 2 , as initiated in [49], and in the context of simpler model PDEs, such as the ones proposed by [27,38,39,46] as models for Faraday waves. A related PDE is known to produce three-dimensional icosahedral quasipatterns [50] and localised quasipatterns [51], and such structures may also be possible in Turing systems. We plan to undertake further investigations in the future.…”

Section: Summary and Discussionmentioning

confidence: 99%

Spatiotemporal chaos and quasipatterns in coupled reaction–diffusion systems

Castelino

Ratliff

Rucklidge

et al. 2020

Physica D: Nonlinear Phenomena

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In coupled reaction-diffusion systems, modes with two different length scales can interact to produce a wide variety of spatiotemporal patterns. Three-wave interactions between these modes can explain the occurrence of spatially complex steady patterns and time-varying states including spatiotemporal chaos. The interactions can take the form of two short waves with different orientations interacting with one long wave, or vice verse. We investigate the role of such three-wave interactions in a coupled Brusselator system. As well as finding simple steady patterns when the waves reinforce each other, we can also find spatially complex but steady patterns, including quasipatterns. When the waves compete with each other, time varying states such as spatiotemporal chaos are also possible. The signs of the quadratic coefficients in three-wave interaction equations distinguish between these two cases. By manipulating parameters of the chemical model, the formation of these various states can be encouraged, as we confirm through extensive numerical simulation. Our arguments allow us to predict when spatiotemporal chaos might be found: standard nonlinear methods fail in this case. The arguments are quite general and apply to a wide class of pattern-forming systems, including the Faraday wave experiment.

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“…Some of the most documented examples are localized structures which resemble a patterned or activated state inside of a compact spatial region and a second homogeneous state outside of this compact region. Localized structures can be found in many applications, including as crime hotspots [5,24,38], vegetation patterns [6,27,32] and soft matter quasicrystals [34]. They have further been observed in chemical reactions [39], supported elastic struts [28], semiconductors [36] and ferrofluids [16].…”

Section: Introductionmentioning

confidence: 99%

Isolas of multi-pulse solutions to lattice dynamical systems

Bramburger¹

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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This work investigates the existence and bifurcation structure of multi-pulse steady-state solutions to bistable lattice dynamical systems. Such solutions are characterized by multiple compact disconnected regions where the solution resembles one of the bistable states and resembles another trivial bistable state outside of these compact sets. It is shown that the bifurcation curves of these multi-pulse solutions lie along closed and bounded curves (isolas), even when single-pulse solutions lie along unbounded curves. These results are applied to a discrete Nagumo differential equation and we show that the hypotheses of this work can be confirmed analytically near the anti-continuum limit. Results are demonstrated with a number of numerical investigations.

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Spatially localized quasicrystalline structures

Cited by 26 publications

References 54 publications

The atomic structure of the Bergman-type icosahedral quasicrystal based on the Ammann–Kramer–Neri tiling

The atomic structure of the Bergman-type icosahedral quasicrystal based on the Ammann–Kramer–Neri tiling

Spatiotemporal chaos and quasipatterns in coupled reaction–diffusion systems

Isolas of multi-pulse solutions to lattice dynamical systems

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