Crystals

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13 pages, 1903 KiB

Open AccessArticle

Twisted Graphene Bilayers and Quasicrystals: A Cut and Projection Approach

by José L. Aragón, Gerardo G. Naumis and Alfredo Gómez-Rodríguez

Crystals 2019, 9(10), 519; https://doi.org/10.3390/cryst9100519 - 10 Oct 2019

Cited by 6 | Viewed by 5552

In this work, a modified version of the cut and projection approach is proposed to describe the structure of graphene bilayers with twist angles. With this method, the rotation between two graphene layers is viewed as a rotation of the projection space and [...] Read more.

In this work, a modified version of the cut and projection approach is proposed to describe the structure of graphene bilayers with twist angles. With this method, the rotation between two graphene layers is viewed as a rotation of the projection space and the resulting projected structure is interpreted as the set of points of best fit between the two rotated structures. Additionally, focus is given to the pertinence of the many algebraic and geometric tools used in grain boundaries and in quasicrystals to graphene bilayer system (or any other bilayer system, for that matter) case. Full article

(This article belongs to the Special Issue Structure and Properties of Quasicrystals)

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28 pages, 19617 KiB

Open AccessArticle

Aperiodic Photonics of Elliptic Curves

by Luca Dal Negro, Yuyao Chen and Fabrizio Sgrignuoli

Crystals 2019, 9(9), 482; https://doi.org/10.3390/cryst9090482 - 14 Sep 2019

Cited by 9 | Viewed by 4322

Abstract

In this paper we propose a novel approach to aperiodic order in optical science and technology that leverages the intrinsic structural complexity of certain non-polynomial (hard) problems in number theory and cryptography for the engineering of optical media with novel transport and wave [...] Read more.

In this paper we propose a novel approach to aperiodic order in optical science and technology that leverages the intrinsic structural complexity of certain non-polynomial (hard) problems in number theory and cryptography for the engineering of optical media with novel transport and wave localization properties. In particular, we address structure-property relationships in a large number (900) of light scattering systems that physically manifest the distinctive aperiodic order of elliptic curves and the associated discrete logarithm problem over finite fields. Besides defining an extremely rich subject with profound connections to diverse mathematical areas, elliptic curves offer unprecedented opportunities to engineer light scattering phenomena in aperiodic environments beyond the limitations of traditional random media. Our theoretical analysis combines the interdisciplinary methods of point patterns spatial statistics with the rigorous Green’s matrix solution of the multiple wave scattering problem for electric and magnetic dipoles and provides access to the spectral and light scattering properties of novel deterministic aperiodic structures with enhanced light-matter coupling for nanophotonics and metamaterials applications to imaging and spectroscopy. Full article

(This article belongs to the Special Issue Structure and Properties of Quasicrystals)

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11 pages, 594 KiB

Open AccessArticle

Analytic Solutions to Two-Dimensional Decagonal Quasicrystals with Defects Using Complex Potential Theory

by Haobai Cao, Yiqing Shi and Wu Li

Crystals 2019, 9(4), 209; https://doi.org/10.3390/cryst9040209 - 17 Apr 2019

Cited by 1 | Viewed by 2757

Abstract

An analytical treatment for two-dimensional point group 10 mm decagonal quasicrystals with defects was suggested based on the complex potential method. On the basis of the assumption of linear elasticity, two new conformal maps were applied to two examples: the first was an [...] Read more.

An analytical treatment for two-dimensional point group 10 mm decagonal quasicrystals with defects was suggested based on the complex potential method. On the basis of the assumption of linear elasticity, two new conformal maps were applied to two examples: the first was an arc with an elliptic notch inner surface in a decagonal quasicrystal, where the complex potentials could be exactly obtained; and the second was concerned with a decagonal point group 10 mm quasicrystalline strip weakened by a Griffith crack, which was subjected to a pair of uniform static pressures. Using the basic idea underlying crack theory, the extent of the stress intensity factors was analytically estimated. If the height was allowed to approach infinity, these results can be turned into the known results of an “ordinary” crystal with only phonon elastic parameters when the phason and phonon-phason elastic constants are eliminated. Full article

(This article belongs to the Special Issue Structure and Properties of Quasicrystals)

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10 pages, 1245 KiB

Open AccessArticle

Atomic Structure of Decagonal Al-Cu-Rh Quasicrystal–Revisited: New Correction for Phonons

by Radoslaw Strzalka, Ireneusz Buganski, Pawel Kuczera, Lucjan Pytlik and Janusz Wolny

Crystals 2019, 9(2), 78; https://doi.org/10.3390/cryst9020078 - 1 Feb 2019

Cited by 6 | Viewed by 3305

Abstract

The standard approach applies the Gaussian distribution function to estimate atomic displacements due to thermal vibrations in periodic and aperiodic systems, which is used in a form of the Debye–Waller factor during the structure refinement. Acoustic phonons provide the largest contribution to the [...] Read more.

The standard approach applies the Gaussian distribution function to estimate atomic displacements due to thermal vibrations in periodic and aperiodic systems, which is used in a form of the Debye–Waller factor during the structure refinement. Acoustic phonons provide the largest contribution to the Gaussian correction although the character of other phonon modes remains relatively unclear. In this paper, we provide an alternative description of localized and dispersionless phonons based on an assumption of the harmonic displacement distribution function, which was recently proposed for model quasicrystals, and apply this approach for a decagonal Al-Cu-Rh quasicrystal that was previously studied by Kuczera et al. in 2012. We used the same X-ray diffraction data and the statistical method of structural analysis of the aperiodic systems. The correction function for phonons takes the form of a Bessel function instead of a conventional (Gaussian) Debye–Waller factor. This allowed us to achieve R-factor of 7.2% compared to 7.9% reported in the original paper. A significant improvement of the calculated atomic composition towards experimentally obtained and minor positional changes is also reported compared to the original paper. The results show the usefulness of investigating different corrective terms for diffraction data during a structure refinement. Full article

(This article belongs to the Special Issue Structure and Properties of Quasicrystals)

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12 pages, 70692 KiB

Open AccessArticle

Non-Local Game of Life in 2D Quasicrystals

by Fang Fang, Sinziana Paduroiu, Dugan Hammock and Klee Irwin

Crystals 2018, 8(11), 416; https://doi.org/10.3390/cryst8110416 - 6 Nov 2018

Cited by 3 | Viewed by 7281

Abstract

On a two-dimensional quasicrystal, a Penrose tiling, we simulate for the first time a game of life dynamics governed by non-local rules. Quasicrystals have inherently non-local order since any local patch, the emperor, forces the existence of a large number of tiles at [...] Read more.

On a two-dimensional quasicrystal, a Penrose tiling, we simulate for the first time a game of life dynamics governed by non-local rules. Quasicrystals have inherently non-local order since any local patch, the emperor, forces the existence of a large number of tiles at all distances, the empires. Considering the emperor and its local patch as a quasiparticle, in this case a glider, its empire represents its field and the interaction between quasiparticles can be modeled as the interaction between their empires. Following a set of rules, we model the walk of life in different setups and we present examples of self-interaction and two-particle interactions in several scenarios. This dynamic is influenced by both higher dimensional representations and local choice of hinge variables. We discuss our results in the broader context of particle physics and quantum field theory, as a first step in building a geometrical model that bridges together higher dimensional representations, quasicrystals and fundamental particles interactions. Full article

(This article belongs to the Special Issue Structure and Properties of Quasicrystals)

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10 pages, 15696 KiB

Open AccessArticle

Dynamic Fracture Mechanism of Quasicrystal-Containing Al–Cr–Fe Consolidated Using Spark Plasma Sintering

by Ruitao Li, Zhiyong Wang, Zhong Li, Khiam Aik Khor and Zhili Dong

Crystals 2018, 8(10), 385; https://doi.org/10.3390/cryst8100385 - 10 Oct 2018

Cited by 1 | Viewed by 3437

Abstract

The potential applications of quasicrystals (QCs) in automotive and aerospace industries requires the investigation of their fracture and failure mechanisms under dynamic loading conditions. In this study, Al–Cr–Fe powders were consolidated into pellets using spark plasma sintering at 800 °C for 30 min. [...] Read more.

The potential applications of quasicrystals (QCs) in automotive and aerospace industries requires the investigation of their fracture and failure mechanisms under dynamic loading conditions. In this study, Al–Cr–Fe powders were consolidated into pellets using spark plasma sintering at 800 °C for 30 min. The microhardness and dynamic failure properties of the samples were determined using nanoindentation and split-Hopkinson pressure bar technique, respectively. Scanning electron microscopy and transmission electron microscopy were employed to analyze fracture particles. The dynamic failure strength obtained from the tests is 653 ± 40 MPa. The dynamic failure process is dominated by transgranular fracture mechanisms. The difficulty in the metadislocation motion in the dynamic loading leads to the high brittleness of the spark plasma sintered (SPSed) Al–Cr–Fe materials. Full article

(This article belongs to the Special Issue Structure and Properties of Quasicrystals)

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16 pages, 29266 KiB

Open AccessArticle

Quasicrystal Tilings in Three Dimensions and Their Empires

by Dugan Hammock, Fang Fang and Klee Irwin

Crystals 2018, 8(10), 370; https://doi.org/10.3390/cryst8100370 - 20 Sep 2018

Cited by 3 | Viewed by 6231

Abstract

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal’s vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and [...] Read more.

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal’s vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the

D_{6}

and

Z^{6}

lattices to

R^{3}

and we present our analyses. We discuss the implications of this new generalization. Full article

(This article belongs to the Special Issue Structure and Properties of Quasicrystals)

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498 KiB

Open AccessArticle

Lempel-Ziv Complexity of Photonic Quasicrystals

by Juan J. Monzón, Angel Felipe and Luis L. Sánchez-Soto

Crystals 2017, 7(7), 183; https://doi.org/10.3390/cryst7070183 - 23 Jun 2017

Cited by 2 | Viewed by 4742

Abstract

The properties of one-dimensional photonic quasicrystals ultimately rely on their nontrivial long-range order, a hallmark that can be quantified in many ways depending on the specific aspects to be studied. Here, we assess the quasicrystal structural features in terms of the Lempel-Ziv complexity. [...] Read more.

The properties of one-dimensional photonic quasicrystals ultimately rely on their nontrivial long-range order, a hallmark that can be quantified in many ways depending on the specific aspects to be studied. Here, we assess the quasicrystal structural features in terms of the Lempel-Ziv complexity. This is an easily calculable quantity that has proven to be useful for describing patterns in a variety of systems. One feature of great practical relevance is that it provides a reliable measure of how hard it is to create the structure. Using the generalized Fibonacci quasicrystals as our thread, we give analytical fitting formulas for the dependence of the optical response with the complexity. Full article

(This article belongs to the Special Issue Structure and Properties of Quasicrystals)

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