Tangible symmetry elements and space-group models to guide from molecular to solid-state composition

The ability to imagine symmetry and the spatial arrangement of atoms and molecules is crucial in chemistry in general. Teaching and understanding crystallography and the composition of the solid state therefore require understanding of symmetry elements and their relationships. To foster the student's spatial imagination, models representing a range of concepts from individual rotation axes to complete space groups have been designed and built. These models are robust and large enough to be presented and operated in a lecture hall, and they enable students to translate conventional 2D notations into 3D objects and vice versa. Tackling them hands-on means understanding them.

The Euler characteristic as a basis for teaching topology concepts to crystallographers

The simple Euler polyhedral formula, expressed as an alternating count of the bounding faces, edges and vertices of any polyhedron, V − E + F = 2, is a fundamental concept in several branches of mathematics. Obviously, it is important in geometry, but it is also well known in topology, where a similar telescoping sum is known as the Euler characteristic χ of any finite space. The value of χ can also be computed for the unit polyhedra (such as the unit cell, the asymmetric unit or Dirichlet domain) which build, in a symmetric fashion, the infinite crystal lattices in all space groups. In this application χ has a modified form (χm) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), χm has an elegant topological interpretation through the concept of orbifolds. Alternatively, χm can be illustrated using the theorems of Harriot and Descartes, which predate the discovery made by Euler. Those historical theorems, which focus on angular defects of polyhedra, are beautifully expressed in the formula of de Gua de Malves. In a still more general interpretation, the theorem of Gauss–Bonnet links the Euler characteristic with the general curvature of any closed space. This article presents an overview of these interesting aspects of mathematics with Euler's formula as the leitmotif. Finally, a game is designed, allowing readers to absorb the concept of the Euler characteristic in an entertaining way.

Electron Diffraction Study of the Space Group Variation in the Al–Mn–Pt T-Phase

Binary high temperature “Al3Mn” (T-phase) and its extensions in ternary systems were the subjects of numerous crystallographic investigations. The results were ambiguous regarding the existence or lack of the center of symmetry: both Pna21 and Pnam space groups were reported. Our research on the Al–Mn–Pt T-phase allowed concluding that inside a continuous homogeneity region, the structure of the Al-rich T-phase (e.g., Al78Mn17.5Pt4.5) belongs to the non-centrosymmetric Pna21 space group, while the structure of the Al-poor T-phase (such as Al71.3Mn25.1Pt3.6) is centrosymmetric, i.e., Pnam. Following metallurgical and crystallographic considerations, the change in the symmetry was explained.

On Designing, Composing and Performing Networked Collective Interactions

In this article, we discuss some of our research with Local Area Networks (LAN) in the context of sound installations or musical performances. Our systems, built on top of Web technologies, enable novel possibilities of collective and collaborative interaction, in particular by simplifying public access to the artwork by presenting the work through the web browser of their smartphone/tablet. Additionally, such a technical framework can be extended with so-called nano-computers, microprocessors and sensors. The infrastructure is completely agnostic as to how many clients are attached, or how they connect, which means that if the work is available in a public space, groups of friends, or even informally organised flash mobs, may engage with the work and perform the contents of the work at any time, and if available over the Internet, at any place. More than the technical details, the specific artistic directions or the supposed autonomy of the agents of our systems, this article focuses on how such ‘networks of devices’ interleave with the ‘network of humans’ composed of the people visiting the installation or participating in the concert. Indeed, we postulate that an important point in understanding and describing such proposals is to consider the relation between these two networks, the way they co-exist and entangle themselves through perception and action. To exemplify these ideas, we present a number of case studies, sound installations and concert works, very different in scope and artistic goal, and examine how this interaction is materialised from several standpoints.

Glide symmetry protected higher-order topological insulators from semimetals with butterfly-like nodal lines

Most topological insulators (TIs) discovered today in spinful systems can be transformed from topological semimetals (TSMs) with vanishing bulk gap via introducing the spin-orbit coupling (SOC), which manifests the intrinsic links between the gapped topological insulator phases and the gapless TSMs. Recently, we have discovered a family of TSMs in time-reversal invariant spinless systems, which host butterfly-like nodal-lines (NLs) consisting of a pair of identical concentric intersecting coplanar ellipses (CICE). In this Communication, we unveil the intrinsic link between this exotic class of nodal-line semimetals (NLSMs) and a $${{\mathbb{Z}}}_{4}$$ Z 4 = 2 topological crystalline insulator (TCI), by including substantial SOC. We demonstrate that in three space groups (i.e., Pbam (No.55), P4/mbm (No.127), and P42/mbc (No.135)), the TCI supports a fourfold Dirac fermion on the (001) surface protected by two glide symmetries, which originates from the intertwined drumhead surface states of the CICE NLs. The higher order topology is further demonstrated by the emergence of one-dimensional helical hinge states, indicating the discovery of a higher order topological insulator protected by a glide symmetry.

Synthesis and Characterization of High Temperature Properties of YBa2Cu3O6+δ Superconductor as Potential Cathode for Intermediate Temperature Solid Oxide Fuel Cells

YBa2Cu3O6+δ (YBC) oxygen deficient perovskite was synthesized by an auto-combustion method and was studied as potential cathode for Intermediate Temperature Solid Oxide Fuel Cell (IT-SOFC). Synchrotron X-ray thermodiffraction in air shows a phase transition from orthorhombic Pmmm to tetragonal P4/mmm space groups at ~ 425 °C. The chemical compatibility with Ce0.9Gd0.1O1.95 (GDC) electrolyte was investigated in air where certain reactivity was observed above 800 °C. However, the main phase is Ba(Ce1-xYx)O3, a good ionic conductor. The catalytic performance in air was obtained by electrochemical impedance spectroscopy (EIS) measurements on YBC/GDC/YBC symmetrical cells. The area specific resistance (ASR) values change from 13.66 to 0.14 Ω cm2 between 500 and 800 °C, with activation energy (Ea) of 0.41 eV. The results suggest potential applications of YBC as IT-SOFC cathode.

Synthesis and Crystal Structures of Novel Glycoluryl Carboxylic Acids Conglomerates

The two novel conglomerats were obtained by crystallization of racemic (S)-2-((3aS,6aR)- and (R)-2-((3aR,6aS)-2,5-dioxohexahydroimidazo[4,5-d]imidazole-1(2H)-yl)-3-methylbutanoic acids (racemate I), (R)-2-(3aR,6aR)- and (S)-2-((3aS,6aS)-4,6-dimethyl-2,5-dioxohexahydroimidazo[4,5-d]imidazole-1(2H)-yl)pentanoic acids (racemate II), which were synthesized by highly diastereoselective condensation of 4,5-dihydroxyimidazolidine-2-ones with racemic ureido acids for the first time. The differences in the molecular geometry of I and II are studied by X-ray diffraction that showed them to crystallize as conglomerates in non-centrosymmetric space groups Pna21 and P212121, respectively.

Structural and Biochemical Analysis of the Dual Inhibition of MG-132 against SARS-CoV-2 Main Protease (Mpro/3CLpro) and Human Cathepsin-L

After almost two years from its first evidence, the COVID-19 pandemic continues to afflict people worldwide, highlighting the need for multiple antiviral strategies. SARS-CoV-2 main protease (Mpro/3CLpro) is a recognized promising target for the development of effective drugs. Because single target inhibition might not be sufficient to block SARS-CoV-2 infection and replication, multi enzymatic-based therapies may provide a better strategy. Here we present a structural and biochemical characterization of the binding mode of MG-132 to both the main protease of SARS-CoV-2, and to the human Cathepsin-L, suggesting thus an interesting scaffold for the development of double-inhibitors. X-ray diffraction data show that MG-132 well fits into the Mpro active site, forming a covalent bond with Cys145 independently from reducing agents and crystallization conditions. Docking of MG-132 into Cathepsin-L well-matches with a covalent binding to the catalytic cysteine. Accordingly, MG-132 inhibits Cathepsin-L with nanomolar potency and reversibly inhibits Mpro with micromolar potency, but with a prolonged residency time. We compared the apo and MG-132-inhibited structures of Mpro solved in different space groups and we identified a new apo structure that features several similarities with the inhibited ones, offering interesting perspectives for future drug design and in silico efforts.

Crystal Structure of the Intermediate Na2V2(PO4)3 Phase and Electrochemical Reaction Mechanisms in NaxV2(PO4)3 (1 ≤ x ≤ 4) System

The Na-superionic-conductor (NASICON) Na3V2(PO4)3 is an important positive electrode material for Na-ion batteries. Here, we investigate the mechanisms of phase transition in NaxV2(PO4)3 (1 ≤ x ≤ 4) upon a non-equilibrium battery cycling. Unlike the widely believed two-phase reaction in Na3V2(PO4)3 – Na1V2(PO4)3 system, we determine a new intermediate Na2V2(PO4)3 phase using operando synchrotron X-ray diffraction. Density functional theory calculations further support the existence of the Na2V2(PO4)3 phase. We propose for the first time two possible crystal structures of Na2V2(PO4)3 analyzed by Rietveld refinement. The two structure models with the space groups P21/c or P2/c for the new intermediate Na2V2(PO4)3 phase show similar unit cell parameters but different atomic arrangements, including a vanadium charge ordering. As the appearance of the intermediate Na2V2(PO4)3 phase is accompanied by symmetry reduction, Na(1) and Na(2) sites split into several positions in Na2V2(PO4)3, in which one of the splitting Na(2) position is found to be a vacancy whereas the Na(1) positions are almost fully filled. The intermediate Na2V2(PO4)3 phase reduces the lattice mismatch between Na3V2(PO4)3 and Na1V2(PO4)3 phases facilitating a fast phase transition. This work paves the way for a better understanding of great rate capabilities of Na3V2(PO4)3.