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.

QSalignWeb: A Server to Predict and Analyze Protein Quaternary Structure

The identification of physiologically relevant quaternary structures (QSs) in crystal lattices is challenging. To predict the physiological relevance of a particular QS, QSalign searches for homologous structures in which subunits interact in the same geometry. This approach proved accurate but was limited to structures already present in the Protein Data Bank (PDB). Here, we introduce a webserver (www.QSalign.org) allowing users to submit homo-oligomeric structures of their choice to the QSalign pipeline. Given a user-uploaded structure, the sequence is extracted and used to search homologs based on sequence similarity and PFAM domain architecture. If structural conservation is detected between a homolog and the user-uploaded QS, physiological relevance is inferred. The web server also generates alternative QSs with PISA and processes them the same way as the query submitted to widen the predictions. The result page also shows representative QSs in the protein family of the query, which is informative if no QS conservation was detected or if the protein appears monomeric. These representative QSs can also serve as a starting point for homology modeling.

Микроструктура и фазовый состав сплава дисилицидов железа и хрома

The phase composition, microstructure, and interphase interfaces of the disordered CrSi2-FeSi2 solid solution obtained by spontaneous crystallization (before and after annealing) have been investigated by scanning, transmission electron microscopy, electron diffraction, and X-ray energy dispersive spectrometry. The as-grown samples contained the phases of CrSi2 with the P6422 hexagonal structure and FeSi2 with the P4/mmm tetragonal structure. Annealing of the samples led to the phase transformation of tetragonal FeSi2 into the orthorhombic modification Cmca. Precipitates of cubic iron monosilicide FeSi with space group P213, nano-precipitates of Si and silicon silicide Cr5Si3 with a tetragonal structure I4/mcm were observed inside the FeSi2 grains. Impurities of interstitial Cr atoms with a concentration up to 2.0 at% are found in iron (di)silicides grains in all samples. The structure of the CrSi2 phase remains unchanged after annealing; the concentration of impurity iron atoms is about 0.7 at%. Orientation relationships between the crystal lattices of the phases are established and strains due to the mismatch of the crystal lattices are determined.

Vibrational dynamics in 2D crystal lattices of borophene

Vibration dynamics of crystalline borophene is considered in the framework of the Born–von Karman model. The vibrations perpendicular to the plane of 2D borophen lattice (flexural modes) are studied.

Giant single-crystal-to-single-crystal transformations associated with chiral interconversion induced by elimination of chelating ligands

Numerous single crystals that exhibit single-crystal-to-single-crystal (SCSC) transformations have been reported, and some of them show great promise for application to advanced adsorption materials, magnetic switches, and smart actuators. However, the development of single crystals with super-adaptive crystal lattices capable of huge and reversible structural change remains a great challenge. In this study, we report a ZnII complex that undergoes giant SCSC transformation induced by a two-step thermal elimination of ethylene glycol chelating ligands. Although the structural change is exceptionally large (50% volume shrinkage and 36% weight loss), the single-crystal nature of the complex persists because of the multiple strong hydrogen bonds between the constituent molecules. This allows the reversible zero-dimensional to one-dimension and further to three-dimensional structural changes to be fully characterized by single-crystal X-ray diffraction analyses. The elimination of chelating ligands induces a chiral interconversion in the molecules that manifests as a centric-chiral-polar symmetric variation of the single crystal. The study not only presents a unique material, featuring both a periodic crystal lattice and gel-like super-ductility, but also reveals a possible solid-state reaction method for preparing chiral compounds via the elimination of chelating ligands.

Puzzling Low-Temperature Behavior of the Van Der Waals Friction Force between Metallic Plates in Relative Motion

This paper presents the results of calculating the van der Waals friction force (dissipative fluctuation-electromagnetic force) between metallic (Au) plates in relative motion at temperatures close to 1 K. The stopping tangential force arises between moving plates along with the usual Casimir force of attraction, which has been routinely measured with high precision over the past two decades. At room temperatures, the former force is 10 orders of magnitude less than the latter, but at temperatures T<50 K, friction increases sharply. The calculations have been carried out in the framework of the Levin-Polevoi-Rytov fluctuation electromagnetic theory. For metallic plates with perfect crystal lattices and without defects, van der Waals friction force is shown to increase with decreasing temperature as T-4. In the presence of residual resistance ρ0 of the metal, a plateau is formed on the temperature dependence of the friction force at T→0 with a height proportional to ρ0-0.8. Another important finding is the weak force-distance dependence ~a-q (with q<1). The absolute values of the friction forces are achievable for measurements in AFM-based experiments.

Effect of the lattice mismatch on the efficiency of the GaAs nanowire/Si substrate solar cell

The effect of the crystal lattice mismatch between single p-GaAs nanowire grown on p-Si substrate on the solar cell efficiency is studied. The study is performed by measuring the I-V curves under red (wavelength=650 nm) laser illumination. The measurement of the single nanowire was done by conductive atomic force microscopy (C-AFM). The measured curve was reproduced by numerical simulations accounting piezoresistance and piezoelectric effects. The analysis demonstrated the presence of the tensile (2%) zinc blend insert at the interface between nanowire and substrate induced by crystal lattices mismatch. Strained insert at the interface changes the polarity of the photogenerated current and increases the efficiency by 2 times.

A Detailed Insight into the Detrital and Diagenetic Mineralogy of Metal(oid)s: Their Origin, Distribution and Associations within Hypersaline Sediments

Hypersaline environments are among the most vulnerable coastal ecosystems and are extremely noteworthy for a variety of ecological reasons. Comprehensive assessment of metal(oid) contamination in hypersaline sediments from Sečovlje (Northern Adriatic, Slovenia) was addressed by introducing the detrital and diagenetic mineralogy and geochemical properties within the solid sediment material. Close associations between Fe/Mn oxides and oxyhydroxides with As, Cr, Ni, Pb and Zn, and between organic matter with Cu, Pb and Zn were confirmed using X-ray powder diffraction, SEM-EDS and ICP emission spectrometry analysis. Possible incorporation or adsorption on the crystal lattices of clay minerals (As, Cr, Pb, Sn and Zn), halite (As) and aragonite/calcite (Cd, Cu, Pb, Sr and Zn) were also detected. All presented correlations were highlighted by various statistical analyses. The enrichment factor (EF) values showed a low degree of anthropogenic burden for As, Bi, Hg and Zn, while Cd, Cr, Cu, Ni, Pb, Sn and Sr originated from the geological background. These results emphasise that a detailed mineralogical and geochemical characterisation of solid (especially detrital and diagenetic) sediment particles is crucial in further understanding the metal(oid) translocation within the hypersaline ecosystems.

The Mechanism of Destruction Under the Influence of Shock Waves

The spectra of the plasma emitted from the studied samples consist of several dozens of narrow bands superimposed on each other. Tables of spectral lines were used to interpret the spectra. It turned out that the largest number of bands corresponds to the radiation of positively charged ions and atoms of elements that make up the crystal lattices of minerals that make up the studied rocks. Thus, the spectra of the plasma emitted from quartz corresponded to the radiation of atoms and positively charged silicon ions, the charge of which varied from 1 to 4, as well as atoms and positively charged oxygen ions, the charge of which varied from 1 to 3. Positively charged ions and atoms of Si, O, K, Ca, Al and Na, which are part of the crystal lattices of quartz and feldspar, flew out of granites. Positively charged ions and Ca, C and O atoms flew out of the calcite.