Discovery Of Quasicrystals Research Articles

Nucleation and growth of quasicrystal-related precipitates within the Al-matrix of AlEr(Fe) alloys

Since the discovery of quasicrystals, their applications to industry have been interesting issues, among which attempts of introducing quasicrystal-related precipitates into the Al-matrix to enhance properties of Al alloys have been made without adequate understandings. Here, we report a formation pathway of quasicrystal-related precipitates in the Al-matrix of an AlEr(Fe) alloy. It is shown that upon thermal aging of the alloy, Al3Er precipitates form firstly in the Al-matrix, then quasicrystal-related (AlFe)-precipitates may nucleate heterogeneously within the Al3Er precipitates and grow large. As the quasicrystal-related precipitate grows large, the Al3Er-precipitate reform as the thin “skin” of the formed composite precipitate, keeping the quasicrystal-related portion well separated from the matrix. Atomic-resolution electron microscopy and spectroscopy reveal that these composite precipitates are the mixtures of the Al13Fe4 quasicrystal approximant structure and/or quasicrystal-related aperiodic structures, as well as their surrounding skin with Al3Er-structure. As such, Fe atoms as vexing impurity in Al alloys can be absorbed continuously into such composite precipitates. Our findings provide insights into the feasibility to form quasicrystal-related precipitates in the Al-matrix.

25 Years of Quasiperiodic Crystallography in Physical Space using the Average Unit Cell Approach

AbstractSince the discovery of quasicrystals 40 years ago, many new paradigms and methods have been introduced to crystallography. 25 years ago, a statistical method of structure and diffraction analysis of aperiodic materials was proposed and, over these years, developed to describe model and real systems. This short review paper briefly invokes the basic concepts of the method: a reference lattice and an average unit cell, but also gives an overview of its application to atomic structure and diffraction analysis of various systems. Results are briefly discussed for mathematical sequences (Fibonacci and Thue‐Morse), model quasilattices in 2D and 3D (Penrose and Ammann tiling), refinements of real decagonal and icosahedral quasicrystals, analysis of structure disorder in quasicrystals, description of modulated systems, including macromolecular biological systems, and beyond usual application in crystallography.

Aspects of Aperiodic Order

The theory of aperiodic order expanded and developed significantly since the discovery of quasicrystals, and continues to bring many mathematical disciplines together. The focus of this workshop was on harmonic analysis and spectral theory, dynamical systems and group actions, Schrödinger operators, and their roles in aperiodic order – with links into a full range of problems from number theory to operator theory.

Superconducting Quasicrystals

AbstractDan Shechtman's discovery of quasicrystals in 1982 introduced the scientific world to aperiodic crystals with unique rotational symmetries, redefining traditional crystallography. Although superconductivity in related periodic approximants has since been observed, true bulk superconductivity in quasicrystals was confirmed only in 2018. This recent discovery opens a new horizon not only for the study of correlated quasicrystals but more generally for the study of superconductivity with nontrivial spatial order. The theoretical understanding of superconducting quasicrystals poses challenges due to their lack of periodicity. Notably, they exhibit non‐BCS type superconductivity and distinct electromagnetic responses, reminiscent of the so‐called FFLO state. In this review, we provide an overview of superconducting quasicrystals, along with some “behind‐the‐scenes” information.

Derived from the traditional principles of Islamic geometry, a methodology for generating non-periodic long-range sequences in one-dimension for 8-fold, 10-fold, and 12-fold rotational symmetries

This paper explores the use of traditional principles of Islamic geometry to generate the long-range order of different non-period sequences in one-dimension. The discovery of quasicrystals with forbidden long-range order has uncovered a range of puzzling non-periodic symmetries, triggering a variety of investigations into understanding their complicated structural logic. The discovery of historical patterns in Islamic architecture with similar quasi-periodic formations is providing new insights into understanding the mathematical properties of these complicated formations. This paper argues that the hierarchical self-similar logic underlying the construction process of the historical patterns can serve as the key ingredient for resolving the long-range order of the one-dimensional non-periodic sequences. A case study approach is employed to test the hypotheses. Three historical patterns with non-period symmetries were tested: 10-fold rotational symmetry, 8-fold rotational symmetry and 12-fold rotational symmetry. The results show that the methodology is successful in generating three different non-periodic sequences in one-dimension.

Optical properties of symmetry breaking of the self-similar order in triadic Cantor photonic crystals.

In general, the photonic crystal (PC) is a periodical optical structure, but there are some studies considering aperiodic structures. If we insert a defect layer into a one-dimensional periodic PC to break its translational symmetry order (TSO), some peaks, called defect modes, appear in the transmittance spectrum. The defect layer thickness governs the frequencies of these defect modes but almost does not affect the other part of the spectrum. The discovery of quasi-crystals tells us that not only the TSO but also other orders can produce Bragg diffraction. It is well known that triadic Cantor set (TCS) PCs, which lack TSO but have a self-similar symmetry order (SSO), still exhibit narrow transmission peaks. In this work, we try to break the SSO in TCS PCs and find the resulting optical phenomena, where single-negative materials and dielectrics are chosen as the constituents of PCs. The study method is the transfer matrix method, and the calculation results show that the background intensity of the transmittance spectrum rather than the frequency of peaks obviously periodically changes with the break of SSO. It follows that the SSO does have physical meaning, and not only the transmission peaks but also the background should be treated as a significant optical property.

Quasi-Crystal, Not Quasi-Scientist

Materials science investigates the structure and properties of different materials. One of these materials is the crystal. Crystals are solid materials with building blocks (atoms, ions, or molecules) that are arranged in a highly organized manner. Salt, quartz, and diamonds are examples of crystals. In ordinary crystals, these building blocks are organized in a repeating pattern in all directions. In contrast, in special crystals called quasi-crystals, the building blocks are organized in a non-repeating manner. The discovery of quasi-crystals created a revolution in the science of crystallography and changed our most basic definition of a crystal. Since their discovery, many hundreds of quasi-crystals have been found. Some of these quasi-crystals have unique physical properties and are useful for a variety of different applications.

Emergent Quasicrystalline Symmetry in Light-Induced Quantum Phase Transitions

The discovery of quasicrystals with crystallographically forbidden rotational symmetries has changed the notion of the ordering in materials, yet little is known about the dynamical emergence of such exotic forms of order. Here we theoretically study a nonequilibrium cavity-QED setup realizing a zero-temperature quantum phase transition from a homogeneous Bose-Einstein condensate to a quasicrystalline phase via collective superradiant light scattering. Across the superradiant phase transition, collective light scattering creates a dynamical, quasicrystalline optical potential for the atoms. Remarkably, the quasicrystalline potential is "emergent" as its eightfold rotational symmetry is not present in the Hamiltonian of the system, rather appears solely in the low-energy states. For sufficiently strong two-body contact interactions between atoms, a quasicrystalline order is stabilized in the system, while for weakly interacting atoms the condensate is localized in one or few of the deepest minima of the quasicrystalline potential.

Limit cycles and their period detection via numeric and symbolic hybrid computations

Computing the attracting cycles via iterations near bifurcation parameters for the logistic map could be misleading for the beginner. The matter is the recognition of numerical convergence when it is not ultimately monotone, thereby the true length of the limiting cycle as opposed to the impression of a double length. Among the tools used are Gröbner basis, Mathematica calculations, and symbolic dynamics (word-lifting). We compute all the 209 superstable points of period length ≤ 11. We also obtain the degree of these algebraic integers. Such computer-assisted proofs are of philosophical interest too. The realization of a 5-periodicity as covered in this paper may resemble the discovery of quasicrystals, a topic we briefly mention. Both share the symbolic dynamics aspect of self-similarity. Finally, for the antisymmetric cubic map we calculate some singly and some doubly superstable parameters.

Electronic band structure of a two-dimensional oxide quasicrystal

Since the discovery of quasicrystals, the study of their electronic structure has been a challenging topic due to the absence of translational symmetry. Here, the valence bands of a novel two-dimensional ultrathin BaTiO${}_{3}$-derived oxide quasicrystal are determined by momentum-resolved photoelectron spectroscopy. The dispersion of the oxygen 2$p$ bands and the occupation of the Ti 3$d$ states are identified. They bridge the knowledge deriving from free-electron-like wave functions in bulk quasicrystals to that of the more localized electrons in confined dimensions.

Metallic-mean quasicrystals as aperiodic approximants of periodic crystals

Ever since the discovery of quasicrystals, periodic approximants of these aperiodic structures constitute a very useful experimental and theoretical device. Characterized by packing motifs typical for quasicrystals arranged in large unit cells, these approximants bridge the gap between periodic and aperiodic positional order. Here we propose a class of sequences of 2-D quasicrystals that consist of increasingly larger periodic domains and are marked by an ever more pronounced periodicity, thereby representing aperiodic approximants of a periodic crystal. Consisting of small and large triangles and rectangles, these tilings are based on the metallic means of multiples of 3, have a 6-fold rotational symmetry, and can be viewed as a projection of a non-cubic 4-D superspace lattice. Together with the non-metallic-mean three-tile hexagonal tilings, they provide a comprehensive theoretical framework for the complex structures seen, e.g., in some binary nanoparticles, oxide films, and intermetallic alloys.

Equilibrium Configurations for Generalized Frenkel–Kontorova Models on Quasicrystals

I study classes of generalized Frenkel–Kontorova models whose potentials are given by almost-periodic functions which are closely related to aperiodic Delone sets of finite local complexity. Since such Delone sets serve as models for quasicrystals, this setup presents Frenkel–Kontorova models for the type of aperiodic crystals which have been discovered since Shechtman’s discovery of quasicrystals. Here I consider models with configurations $$u:{\mathbb {Z}}^r \rightarrow {\mathbb {R}}^d$$ , where d is the dimension of the quasicrystal, for any r and d. The almost-periodic functions used for potentials are called pattern-equivariant and I show that if the interactions of the model satisfies a mild $$C^2$$ requirement, and if the potential satisfies a mild non-degeneracy assumption, then there exist equilibrium configurations of any prescribed rotation rotation number/vector/plane. The assumptions are general enough to satisfy the classical Frenkel–Kontorova models and its multidimensional analoges. The proof uses the idea of the anti-integrable limit.

Discovery of quasicrystals: The early days

Abstract This paper is a survey of the initial developments of the research on quasicrystals starting from their discovery by Daniel Shechtman (Nobel Prize in Chemistry in 2011) in 1982 at the National Bureau of Standards (now National Institute for Standard and Technology) in Gaithersburg (Maryland, USA) up to the beginning of the early 1990s, a time when the crystallographic methods were well developed and mastered enough to decipher the ultimate atomic structures of quasicrystals. These early works have enlarged our understanding of spatial order in solids through a strong multidisciplinary effort between mathematicians, physicists, chemists and material scientists.

Matter-Wave Diffraction from a Quasicrystalline Optical Lattice.

Quasicrystals are long-range ordered and yet nonperiodic. This interplay results in a wealth of intriguing physical phenomena, such as the inheritance of topological properties from higher dimensions, and the presence of nontrivial structure on all scales. Here, we report on the first experimental demonstration of an eightfold rotationally symmetric optical lattice, realizing a two-dimensional quasicrystalline potential for ultracold atoms. Using matter-wave diffraction we observe the self-similarity of this quasicrystalline structure, in close analogy to the very first discovery of quasicrystals using electron diffraction. The diffraction dynamics on short timescales constitutes a continuous-time quantum walk on a homogeneous four-dimensional tight-binding lattice. These measurements pave the way for quantum simulations in fractal structures and higher dimensions.

Wave Harmonization in Hierarchic Quasicrystals by the Analytic Metric

Thirty seven years after the discovery of quasicrystals, their diffraction is completely described by harmonization between the sine wave probe with hierarchic translational symmetry in a structure that is often called quasiperiodic. The diffraction occurs in geometric series that is a special case of the Fibonacci sequence. Its members are irrational. When substitution is made for the golden section τ by the semi-integral value 1.5, a coherent set of rational numbers maps the sequence. Then the square of corresponding ratios is a metric that harmonizes the sine wave probe with the hierarchic structure, and the quasi-Bragg angle adjusts accordingly. From this fact follows a consistent description of structure, diffraction and measurement.

Generating Islamic Quasi-Periodic Patterns

The discovery of quasi-crystals has led to a great debate about their unusual structure. The big surprise is that these structures were found in Islamic art several centuries ago. This latest discovery drew the attention of scientists to propose several approaches for the comprehension of these structures by analyzing several quasi-periodic patterns spread around the Islamic world. In this article, we propose a systematic method for generating new quasi-periodic patterns inspired by existing Islamic historical patterns. The method builds Islamic quasi-periodic patterns based on a quasi-periodic tiling and a few intuitive parameters. Given a quasi-periodic tiling, the method divides its tiles (rhombs) into symmetric right triangles and constructs their template motifs. The construction of these template motifs is achieved by a systematic and well-organized process. The content of the tiles is obtained by applying mirror reflections to the constructed template motifs. Finally, the pattern is drawn by putting the content of the constructed tiles in the tiling. To show the effectiveness of this generative method, examples of new quasi-periodic patterns will be presented.

Quasicrystals: What do we know? What do we want to know? What can we know?

More than 35 years and 11 000 publications after the discovery of quasicrystals by Dan Shechtman, quite a bit is known about their occurrence, formation, stability, structures and physical properties. It has also been discovered that quasiperiodic self-assembly is not restricted to intermetallics, but can take place in systems on the meso- and macroscales. However, there are some blank areas, even in the centre of the big picture. For instance, it has still not been fully clarified whether quasicrystals are just entropy-stabilized high-temperature phases or whether they can be thermodynamically stable at 0 K as well. More studies are needed for developing a generally accepted model of quasicrystal growth. The state of the art of quasicrystal research is briefly reviewed and the main as-yet unanswered questions are addressed, as well as the experimental limitations to finding answers to them. The focus of this discussion is on quasicrystal structure analysis as well as on quasicrystal stability and growth mechanisms.

Mackay clusters and beyond in icosahedral quasicrystals seen from 6D space

It is a great pleasure and honour for us to participate to the celebration of the 90th birthday of Alan Mackay, one of the most inspired crystallographers of our time who has been the authentic predecessor of the quasicrystal discovery. We discuss here several ways to construct Mackay-type atomic clusters and others for describing quasicrystalline structures from the standard 6D framework. We show that they are several simple solutions for both the 6D natural cluster and the original Mackay derivation that are consistent with special points of the basic icosahedral 6D lattice and the actually determined clusters in usual cubic 1/1 approximants of the icosahedral phases. This technique works as well for describing the two first shells of the so-called Bergman clusters but the situation is far more complicated for the so-called Tsai cluster that cannot be directly obtained from the icosahedral cut and projection of the simple 6D lattice special points without significantly large differences in the radii of the various orbits with respect to their actual positions in the YbCd icosahedral-type alloys. This shows that the 6D approach using special points as locations of the mean atomic surfaces—although very efficient for constructing initial simple models of the icosahedral phases—requires subsequent refinement techniques, especially in the actual locations and sizes of the various atomic orbits of the implied clusters, for leading to final acceptable structural models.

Superior room-temperature ductility of typically brittle quasicrystals at small sizes.

The discovery of quasicrystals three decades ago unveiled a class of matter that exhibits long-range order but lacks translational periodicity. Owing to their unique structures, quasicrystals possess many unusual properties. However, a well-known bottleneck that impedes their widespread application is their intrinsic brittleness: plastic deformation has been found to only be possible at high temperatures or under hydrostatic pressures, and their deformation mechanism at low temperatures is still unclear. Here, we report that typically brittle quasicrystals can exhibit remarkable ductility of over 50% strains and high strengths of ∼4.5 GPa at room temperature and sub-micrometer scales. In contrast to the generally accepted dominant deformation mechanism in quasicrystals—dislocation climb, our observation suggests that dislocation glide may govern plasticity under high-stress and low-temperature conditions. The ability to plastically deform quasicrystals at room temperature should lead to an improved understanding of their deformation mechanism and application in small-scale devices.

Numerical methods for crack loading analyses in quasicrystals

AbstractSince the first discovery of quasicrystals in a man made Al‐Mn alloy about thirty years ago, people made great effort to investigate this kind of outstanding material. The materials scientists are constantly trying to produce stable quasicrystalline particles or even a complete single quasicrystalline specimen. On the other hand, the fracture behaviour of quasicrystals is started to be investigated because the coupling effect between phonon and phason fields can rebuild the conventional fracture criteria. This work develops a numerical tool for simulating in‐plane problems of 1D QC and extends the fracture quantities i.e. stress intensity factors (SIF) and strain energy release rate to phason fields. Finally, numerical results are given to reveal what difference the phason field can bring into conventional fracture quantities. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)