Recent Advances in Quasicrystal Research

The study of quasicrystals – materials that are characterized by their aperiodic yet long-range ordered structures1-5 – continues to evolve, offering new challenges and opportunities in understanding complex-ordered systems that transcend traditional crystallography. They have emerged as a rich interdisciplinary field, encompassing mathematics, physics, chemistry, and materials science. This introduces the additional challenge of bridging varied research communities, each with its distinct language and culture. Central to the investigation of quasicrystals are mathematical tools that describe their intricate geometric and topological properties, physical models that capture their unique electronic and other physical properties, as well as innovative experimental methods that can be employed to analyze their unique and complex nature. This special issue of the Israel Journal of Chemistry is dedicated to the latest advancements in quasicrystal research, presented at ICQ15 – the 15th International Conference on Quasicrystals – held on the campus of Tel Aviv University in June 2023 (see Fig. 1 for a group photo). It offers a valuable snapshot of the current state of quasicrystal research, highlighting the progress made in recent years and the challenges that lie ahead. The collection of Reviews and Research Articles, included here, spans a broad spectrum of topics, reflecting the diverse and interdisciplinary nature of quasicrystal research, providing a good entry point, as well as some deep insight, into the theoretical, experimental and practical underpinnings of aperiodic long-range order. Tremendous progress has been made in recent years in advancing our understanding of the mathematical foundations of quasicrystals and aperiodic long-range order. Yet, much of this has appeared only in technical mathematical literature. This special issue features several Reviews, whose purpose is to explain these new understandings in a language, accessible to non-mathematicians. Baake, Gähler, and Mazáč,6 in their Review, On the Fibonacci Tiling and Its Modern Ramifications, employ the one-dimensional Fibonacci chain as the quintessential model of aperiodic order, in order to introduce and elucidate some new mathematical notions and results. Their Review recalls the mathematics behind methods to generate the Fibonacci tiling, such as substitution and high-dimensional embedding. These are integrated into the description of advancements in the understanding of pair correlations and the spectral theory of diffraction, leading up to the dynamical systems approach, which is instrumental in much of the recent progress in the mathematics of aperiodic order. With some examples and generalizations into two dimensions, this Review is a good starting point for studying the nature of long-range aperiodic order. Mañibo’s Review, Substitutions and Their Generalisations,7 focuses on substitution tilings, a cornerstone of quasicrystal modeling. Mañibo examines the spectral and structural properties of generalized substitution rules, namely, with infinite (yet compact) alphabets – or tilings with infinitely many different tiles – while highlighting the role of their hierarchical structure in obtaining concrete new results. Tenenbaum complements this work in his Research Article, Approximations of Symbolic Substitution Systems in One Dimension,8 by introducing a method for approximating so-called substitution subshifts with periodic systems, providing quantitative tools to analyze their convergence and spectral properties. Together, these studies emphasize the versatility and recent important contribution of the substitution method to the understanding of quasicrystals. Kellendonk’s Review,9 Topological Quantum Numbers in Quasicrystals, explores the role of topology in understanding the nature of physical excitations in quasicrystals. Using K-theory and non-commutative topology, Kellendonk examines, among other things, how the quantum properties of quasicrystals can be characterized through gap labeling theorems, which assign topological invariants to spectral gaps in the electronic spectra. The review bridges abstract theoretical constructs with measurable observables, shedding light on bulk-boundary correspondence and elucidating the features that distinguish quasicrystals from periodic crystals through their unique topological classification.1 Conference group photo, outside the Zion Gate entrance to the Old City of Jerusalem. Understanding the diffraction properties of quasicrystals is central to characterizing their unique nature. This issue includes contributions addressing the mathematical description and analysis, as well as the experimental methods, for studying diffraction and inferring the structure of quasicrystals, including the effects of atomic disorder. Lenz, Spindeler and Strungaru,10 in their Review Pure Point Diffraction and Almost Periodicity, explore the mathematical foundations underlying pure point diffraction – the defining hallmark of long-range order – and its connection to almost periodicity. The study clarifies distinctions between Bohr, Besicovitch, and Weyl almost periodicity and discusses their implications for the diffraction patterns of aperiodic crystals, analyzing the Fibonacci chain as an example. This work deepens our understanding of the interplay between mathematical definitions of long-range order and experimentally observed diffraction phenomena, and provides an answer to the decades-long question: “what is a crystal?”.4 In their Review, Unravelling Disorder in Aperiodic Crystals – Diffuse Scattering and Atomic Resolution Holography, Stellhorn, Meekel and Minelli11 present experimental advances in the study of local atomic disorder in quasicrystals. The authors highlight diffuse scattering and atomic-resolution holography as powerful techniques for probing structural deviations at the atomic scale. These methods enable the visualization of correlated disorder and offer a pathway to bridging the gap between idealized theoretical models and real-world quasicrystalline materials. Finally, Wolny and collaborators,12 in their Review, 25 Years of Quasiperiodic Crystallography in Physical Space using the Average Unit Cell Approach, summarize the statistical method for structure and diffraction analysis of aperiodic crystals. The method, based on the average unit cell concept, provides a physical-space framework for studying quasicrystals and modulated crystals, including ones with structural defects. This alternative to the so-called superspace approach3 has contributed to the efforts for obtaining high-quality structural refinement, and to the understanding of disorder in quasicrystals, highlighting its relevance as a complementary tool in quasiperiodic crystallography. Theoretical tiling models remain central for understanding quasiperiodic long-range order. In Canonical-Cell Tilings and their Atomic Decorations, Fujita, Mihalkovič, and the late Chris Henley13 review the canonical-cell tiling framework, which models the atomic structure of icosahedral quasicrystals and their periodic approximants. This framework, built on the assembly of canonical polyhedral cells, offers a versatile approach for describing quasiperiodic and periodic structures. Their Review also examines advances in atomistic modeling, providing insights into the geometric and physical constraints that govern quasicrystalline arrangements. Adding to this discussion, Jagannathan and Duneau,14 in their Review, Properties of the Ammann–Beenker Tiling and its Square Periodic Approximants, present a detailed introductory guide, or a user's manual, to constructing and studying the octagonal Ammann–Beenker tiling. They emphasize its geometric simplicity and utility as a paradigm for quasiperiodic systems, describing in simple terms the calculation of its geometric properties, such as the relative densities of local configurations, and its transformation under inflation and deflation. Their analysis emphasizes the attractiveness of the Ammann-Beenker tiling as a model for studying the physics of quasicrystals, while highlighting the notion of phasons, which is unique to quasiperiodic long-range order. In their Review, Hexagonal and Trigonal Quasiperiodic Tilings, Coates et al.15 explore a family of rank-4 quasiperiodic tilings with hexagonal and trigonal symmetries. Using generalized versions of de Bruijn's dual-grid method1 and projections from six-dimensional superspace, they pedagogically construct and analyze tilings analogous to the familiar square Fibonacci tiling,16 yet with 3-fold and 6-fold rotational symmetry. The study provides substitution rules for generating specific tilings and discusses their structural and spectral properties. This work has practical implications for modeling a variety of two-dimensional quasiperiodic systems, including multilayer graphene and the trigonal surfaces of icosahedral quasicrystals. The stability and formation of quasicrystals, and the ability to predict their structure, are still active research questions in materials science. In Quasicrystal Structure Prediction: A Review, Widom and Mihalkovič17 provide a comprehensive survey of methods for predicting quasicrystal structures, with a focus on metallic alloys. The Review discusses first-principles calculations, statistical mechanical approaches, and empirical modeling techniques that have advanced our ability to predict and understand the stability of known and hypothetical quasicrystalline phases. By bridging computational modeling and experimental validation, this work highlights the progress made in unraveling the thermodynamic landscapes of quasicrystals. More specifically, Bugański, Strzałka, and Wolny18 investigate the stability of icosahedral quasicrystal approximants, in their Article, The Tsai vs. Bergman Cluster Stability in ZnMgSc 1/1 Periodic Approximant Crystal. Using Density Functional Theory, they compare the Tsai and Bergman clusters, identifying sp-d hybridization as a critical factor in stability over traditional Hume-Rothery mechanisms. The presence of an additional electron-accepting atom in the Tsai cluster lowers energy levels, explaining differences in structural prevalence between the two cluster types in quasicrystal compositions. Recent years have brought major advances in the study of the physical properties of quasicrystals, a few of which are detailed in this special issue. In her Review, Superconducting Quasicrystals, Takemori19 surveys experimental and theoretical developments in the study of superconductivity in quasicrystals, highlighting the unique challenges posed by aperiodic order. Takemori underscores the recent discovery in 2018 of bulk superconductivity in an Al-Mg-Zn quasicrystalline alloy,20 while emphasizing theoretical insights that reveal deviations from conventional BCS superconductivity, including non-BCS pairing and unusual electromagnetic responses. These findings emphasize the potential of quasicrystals to challenge and expand the understanding of superconductivity, on the one hand, with implications for the braoder study of correlated-electron behavior in systems with aperiodic long-range order, on the other hand. The Review stresses the need for further experimental and theoretical investigations to fully characterize these novel phenomena. Quasiperiodic tilings are a useful playground for studying the physical properties of quasicrystals, whether using theoretical models, or by realizing them as artificially made metamaterials. Pratyay Ghosh's21 Research Article, Ising Spins on a Frustrated Bronze-Mean Hexagonal Quasicrystal, investigates the Ising model on a geometrically frustrated quasiperiodic tiling with 6-fold symmetry. Monte Carlo simulations reveal six distinct magnetic phases, including exotic spin-glass states with replica symmetry breaking and slow relaxation dynamics. This study provides critical insights into the interplay of frustration and aperiodicity, uncovering unconventional magnetic order and thermodynamic properties. Liam Chandler and collaborators,22 in their Review, Fabricating Quasiperiodic Tilings with Thermal-Scanning Probe Lithography, take a first step toward investigating the magnetic properties of artificially created metamaterials based on quasiperiodic tilings. The authors fabricate nanoscale quasiperiodic structures, such as the Ammann-Beenker14 and the Square Fibonacci16 tilings, using thermal-scanning probe lithography. This work addresses the challenges of nanolithographic design and implementation, and explores potential applications, paving the way for experimental studies of magnetic properties in these artificial quasicrystals. The experimental investigation of quasicrystalline surfaces and thin films, provide important insights into their structural and reactive properties. In Scanning Tunnelling Microscopy Studies of Tsai-Type Quasicrystal Approximants, Coates et al.23 review STM studies of periodic Tsai-type approximants, exploring their surface chemistries and symmetries. The study identifies rare-earth atom stabilization as a common feature, with terrace quality generally lower than that of related quasicrystals. These studies are complemented by Bajoun Mbajoun et al.,24 who employ STEM and HAADF imaging to study Tsai-type and pseudo-Tsai-type clusters, in STEM Structural Investigation of RE-Au-Si 1/1 Approximants. Their findings reveal distinct contrasts between clusters with rare-earth atoms and those with distorted tetrahedrons, providing a detailed understanding of short-range ordering and the structural quality of these approximants. In Surface Reactivity of the Au-Si-Ho Quasicrystalline 1/1 Approximant, Bajoun Mbajoun and collaborators25 investigate oxidation-induced changes in surface structure. Oxygen adsorption leads to segregation of Si and Ho atoms, causing disorder and loss of surface long-range order. Reaction kinetics depend on temperature, with faster changes observed at higher temperatures. These findings highlight the dynamic surface properties of quasicrystalline approximants. Finally, Sharma et al.,26 in Quasicrystalline Antimony Thin Films, study Sb thin films grown on the fivefold surface of Ag-In-Yb quasicrystals. At low coverage, Sb forms pentagonal structures with quasicrystalline long-range order, aligned with the substrate‘s tiling. Higher deposition disrupts this order, resulting in a disordered film. This work establishes a structural relationship between quasicrystalline substrates and thin films, shedding light on pseudomorphic growth mechanisms. The articles featured in this issue underscore the multifaceted nature of quasicrystal research, spanning theoretical insights, experimental advances, and computational breakthroughs. By addressing fundamental questions about structure, diffraction, stability, and physical properties, this collection of studies paves the way for a deeper understanding of quasicrystals. Together, these works demonstrate the ongoing vitality of quasicrystal science as a field that continues to challenge and expand the boundaries of materials research. The author acknowledges the support of the Israel Science Foundation (ISF), through Grant No. 1259/22. ICQ15 – The 15th International Conference on Quasicrystals – was supported by the Israel Science Foundation (ISF) through Grant No. 2888/23; The Israel node of the Centre Européen de Calcul Atomique et Moléculaire (CECAM); The International Union of Crystallography (IUCr); The European Integrated Centre for the Development of New Metallic Alloys and Compounds (C-MetAc); and by Tel Aviv University through its Faculty of Exact Sciences and its School of Physics and Astronomy.