Tetrahedral Symmetry Research Articles

SALEM NUMBERS, SPECTRAL RADII AND GROWTH RATES OF HYPERBOLIC COXETER GROUPS

We show that not every Salem number appears as the growth rate of a cocompact hyperbolic Coxeter group. We also give a new proof of the fact that the growth rates of planar hyperbolic Coxeter groups are spectral radii of Coxeter transformations, and show that this need not be the case for growth rates of hyperbolic tetrahedral Coxeter groups.

The Study on Eu3+ Doped Lithium Bismuth Aluminum Borate Glass: New Red Luminescence Medium

The lithium bismuth aluminum borate glasses doped with Eu3+ ion were studied on the physical, structural network, optical and luminescence properties. The glass density fluctuates by following the Eu2O3 content due to the variation of BO4 group in glass network. The FTIR spectra indicate the main glass structure units which occupies by the trigonal BO3 and tetrahedral BO4 borate groups mixing with AlO6 complex. The Eu3+ acts as the glass modifier in glass as the Eu2O3 concentration is higher than 1.00 mol%. The glasses more absorb photons in visible light – near infrared region with addition of Eu2O3 content. Under 394 nm excitation, glasses represented the strongest emission at 613 nm by the 5D0→7F0 transition of Eu3+ and overall color of this emission is red. The optimum concentration of Eu2O3 for this developed glass is 3.00 mol% that owns the highest emission intensity. The Eu3+ doped lithium bismuth aluminum borate glass performs the high potential for using as a red luminescence medium in light sources and display devices.

Structural and crystallization behavior studies on unfamiliar LiF–B2O3 glasses

Glasses from the binary high LiF–B2O3 system with variable LiF contents (10–55 mol%) were prepared by normal melting-annealing technique. Optical, FTIR spectral properties were measured for the glasses in addition to thermal expansion parameters were derived to be able to convert glassy samples to their corresponding glass-ceramic derivatives. FTIR, X-ray diffraction and SEM were carried out for the prepared heat-treated glass-ceramics to derive their structural building groups and types of crystalline phases formed by controlled heat treatment regime. Optical spectra of the prepared glasses show only distinct UV absorption peaks which are related to unavoidable contaminated traces of impurities of Fe3+ ions. FTIR spectra reveal distinct extended vibrational bands within the mid-region 400-1600 cm−1 comprising both of triangular and tetrahedral borate groups (BO3, BO3F) beside the possibility of sharing of (LiF4) groups at high LiF content. Thermal expansion measurements indicate the changes of the dilatometric softening temperatures with the LiF content but with some derived variations depending on the constituents housing in the network structure. X-ray diffraction patterns data reveal the appearance of lithium borate crystalline phase at low LiF content and at high content both two crystalline phases of lithium borate Li2B4O7 (Diomignite) and LiB3O5 beside crystalline lithium fluoride (LiF) are identified. This result refers to the distinct action of both lithium ions (Li+) and fluoride ions (F−) as initiator to produce phase separation, ease of nucleation and followed by crystallization, similar to the behavior of Li2O within borate and silicate glasses. SEM images support the X-ray diffraction data revealing different crystalline phases with variable texture at high LiF content.

Peabodies of constant width

We describe a new family of 3-dimensional bodies of constant width that we have called peabodies, obtained from the Reuleaux tetrahedron by replacing a small neighborhood of all six edges with sections of an envelope of spheres. This family contains, in particular, the two Meissner solids and a body with tetrahedral symmetry that we have called Robert’s body. Behind the construction of this family lies the classical notion of confocal quadrics discussed, for example, by Hilbert and Cohn-Vossen in their famous book. We will study confocal quadrics and prove that the distances of an alternating sequence of four points in two confocal quadrics always satisfies a simple equation and we will use this equation to prove that our bodies have constant width.

Preparation and characterization of glasses from the binary sodium fluoroborate system NaF–B2O3 within the range (10–50 mol% NaF) assessed by structural FTIR, optical and thermal properties and effects of gamma irradiation

Extended glasses from the binary system (NaF-B2O3) with (10→50 mol% NaF) were synthesized by the traditional melting-annealing route. These unfamiliar glasses with high extended NaF contents than Na2O were characterized by multi-techniques including optical and FTIR spectral analysis beside measurements of their thermal expansion parameters. The optical (UV–visible) spectra of the glasses reveal obvious transparency except for the appearance of distinct UV absorption due to unavoidable trace ferric ions impurities present within the laboratory chemicals. Structural FTIR spectra indicate that NaF primarily acts the same as Na2O progressively converting some of the BO3 units to produce tetrahedral (BO3F) units and then at a high percent of NaF is assumed to form additional (NaF4) groups and hence the intensity of the bands within the range 800–1200 cm−1 due to combined tetrahedral groups (BO4 + NaF4) is always higher than that for the peaks at the range 1200–1600 cm−1due to vibrational BO3 groups. The thermal expansion data reveal that the dilatometric softening temperature primarily increases due to the formation of compact tetrahedral (BO3F) groups. It is obvious that this unfamiliar (NaF–B2O3) glass does not show an anomalous behavior as that reported for (Na2O–B2O3) glassy system and other alkali-B2O3 glasses in their thermal expansion properties. Gamma irradiation did not show any distinct variations in the FTIR spectra indicating the stability of the structural building groups in this NaF–B2O3 glass system. On the other hand, the optical spectra show the displacement of one of the two broad bands in the UV spectra which is assumed to be due to a photochemical reaction between some Fe3+ ions present as trace impurities and formed freed electrons during the irradiation process.

The Chemistry of Zirconium/Carboxylate Clustering Process: Acidic Conditions to Promote Carboxylate-Unsaturated Octahedral Hexamers and Pentanuclear Species.

Clustering chemistry is a key point in the design and synthesis of the secondary building units that comprise metal-organic frameworks (MOFs) based on group IV metals. In this work, the first stages of the zirconium-carboxylate clustering process in alcohol/water mixtures are studied in detail using the monocarboxylic benzoic and hydroxybenzoic acids to avoid the polymerization. Mass spectroscopy measurements performed on the reactions revealed the presence of hexa- and pentanuclear species even at low pH values and also evidenced the acid-base nature and pH dependence of the transformation between both species. The control on the chemistry governing the equilibria between these species has allowed us to isolate six new compounds in the solid state. The single-crystal X-ray diffraction analysis revealed that they are closely related to the well-known [Zr6(O)4(OH)4(OOC)12] secondary building unit found in many MOFs by removing carboxylic ligands in the case of the hexameric species ([Zr6(O)4(OH)4(OOC)8(H2O)8]4+) or by additionally removing one of the metal centers in the case of the pentameric entities ([Zr5(O)2(OH)6(OOC)4(H2O)11(alcohol)]6+). Going in detail, the unsaturated hexameric clusters exhibit different dispositions of their eight carboxylate ligands in such a way that the remaining four carboxylate-free positions are arranged according to a square planar or tetrahedral symmetry. It should be highlighted that the pentameric complexes imply an unprecedented core nuclearity in zirconium clusters and thus their isolation provides a novel building block for the design of metal-organic materials.

Common defects in diamond lattices as instances of the general T ⊗ (e+t2) Jahn-Teller effect

Finding the ground-state configuration of point defects in semiconductors is in general a high-dimensional optimization problem. However, their adiabatic potential energy surfaces (APES) can be described with few effective coordinates by exploiting symmetry arguments within the Jahn-Teller (JT) theoretical framework. In this paper, we propose a general framework, that combines group theory and ab initio calculations, to build a full picture of common defects in diamond lattices (vacancies and substitutional impurities) as instances of the general T $\ensuremath{\bigotimes}\phantom{\rule{4pt}{0ex}}(e+{t}_{2})$ APES. Starting from the original tetrahedral symmetry, we consider all possible JT symmetries and we numerically investigate the shape of the APES by targeting the minimum energy path between equivalent JT distortions taking as examples the vacancy (V) and the metallic impurities Pd, Pt, and Au in crystalline silicon.

Influence of lanthanum doping on the structural and optical properties of cerium tungstate nanocrystals

In this study, the effect of lanthanum (La) doping on the structural and optical properties of cerium tungstate nanocrystals synthesized by the chemical precipitation method is investigated. The X-ray diffraction studies confirmed the crystalline phase formation and observed no phase change with doping. It is interesting to note that the crystallite size is decreased with La doping. The characteristic vibrational peaks of W-O bonds are confirmed from the FTIR and Raman spectroscopic studies and found no appreciable changes in the spectral positions with doping. The optical properties are studied using UV-Vis spectroscopy and photoluminescence spectroscopy. The characteristic broad blue PL emission from the tetrahedral tungstate group is observed for the pure and doped samples, and the appreciable variation in PL intensity can be associated with the formation of defects that occurred while doping.

CsY(SO4)2·4H2O: A Deep-Ultraviolet Birefringent Crystal Induced by an Edge-Sharing Connection.

SO4 tetrahedral groups have weak polarization anisotropy, which thus results in the small birefringence of sulfates. Here, we report new sulfate CsY(SO4)2·4H2O with unprecedented birefringence among deep-ultraviolet (deep-UV) sulfates. Its single crystal (10 mm × 3.5 mm × 1.5 mm) was simply grown by an aqueous solution evaporation technique, and it features a rare layered structure composed of YO9 polyhedra, SO4 tetrahedra, and H2O molecules. Interestingly, each SO4 group donates two oxygen atoms to edge-share with one adjacent YO9 polyhedron and thus causes severe distortion of these groups. The characteristic edge-sharing mode gives CsY(SO4)2·4H2O a large birefringence of ∼0.045@546 nm, which is the maximum among deep-UV sulfates and phosphates with similar non-π-conjugated anionic groups. The ultraviolet-visible-near-infrared diffuse reflection and transmission spectra, infrared spectrum, thermal stability, and theoretical calculations are also presented. The fascinating results will improve our understanding of sulfates and may provide useful insights into the exploration of deep-UV sulfates with large birefringence.

Dielectric relaxation and conduction mechanism of NaMgPO4 compound

The polycrystalline NaMgPO4 ceramic, synthesized via a high-temperature solid-state reaction route, has been characterized by using different experimental procedures. The X-ray powder diffraction confirmed the phase formation of the synthesized compound in the orthorhombic phase. It assumed an olivine-type structure made up of corners linked between tetrahedral PO4 and octahedral NaO6 and MgO6 groups. Infrared and Raman spectroscopies confirmed the presence of PO[Formula: see text] groups. Local structure and chemical bonding between MgO6 octahedral and PO[Formula: see text] tetrahedral groups investigated by diffusion Raman is the feature in the phase transition at [Formula: see text] 693 K. The temperature dependences of the real [Formula: see text] and imaginary [Formula: see text] parts of dielectric permittivity show a distribution of relaxation times. From Nyquist plots, the presence of grain and grain boundary effect in the material is noticed. The impedance spectroscopy measurement showed a non-Debye-type process. From the impedance data, the determined grain resistance reduces with increment of temperature showing negative temperature coefficient of resistance (NTCR)-type nature of the material which also confirmed from conductivity analysis. The temperature dependence of [Formula: see text] reveals an Arrhenius-type behavior with two activation energies, 0.98 eV in region I and 0.67 eV in region II. Studied sample’s conduction is assured by Na[Formula: see text] ions’ hopping in tunnels and its mechanism was discussed.

Gamma irradiation effectuality on the antibacterial and bioactivity behavior of multicomponent borate glasses against methicillin-resistant Staphylococcus aureus (MRSA).

Recently some borate bioactive glasses have been discovered to have an antibacterial effect when interacting with pathogenic bacteria. In this study, borate bioactive glasses (BG) doped with metal oxide (MO) ZnO, TiO2, TeO2, and CeO2 (encoded BG-Zn, BG-Ti, BG-Te, and BG-Ce, respectively) were prepared using the melt-quench method and have been characterized before and after gamma irradiation at 25.0kGy. X-ray diffraction was performed to recognize the amorphous phases of all samples. Infrared absorption of glasses confirms vibrational bands in their wave number according to mixed main triangular and tetrahedral borate groups. After immersion in the simulated body fluid (SBF) solution, two characteristic peaks are generated indicating the bioactivity of the studied glasses through the formation of hydroxyapatite. SEM micrographs of glass after immersion display that the crystalline phases are identified to be more distinct in different shapes because of the multi-composition involved. The antibacterial activity of borate glasses was assessed against methicillin-resistant Staphylococcus aureus (MRSA) ATCC 6538. The antibacterial results showed that BG-Te was the most efficient against S. aureus ATCC 6538. Furthermore, BG-Te reduced biofilm production (79.23%) at the concentration of 20.0mg/mL. (BG-Te) at 20.0mg/mL significantly decreased the viability percent, cell count, protein content, and protease activity of S. aureus cells. BG-Te presents powerful activity against bacterial infections. It was necessary to equilibrate the antibacterial efficiency with the biocompatibility, so the MTT assay confirmed that BG-Te has low cytotoxicity on the human fibroblast cells (WI-38). It is expected that borate bioglass contained TeO2could be a promising biomaterial for bone tissue engineering.

High‐Performance Sulfate Optical Materials Exhibiting Giant Second Harmonic Generation and Large Birefringence

AbstractDiscovering novel sulfate optical materials with strong second‐harmonic generation (SHG) and large birefringence is confronted by a great challenge attributed to the intrinsically weak polarizability and optical anisotropy of tetrahedral SO4 groups. Herein, two superior‐performing sulfate optical materials, namely, noncentrosymmetric Hg3O2SO4 and centrosymmetric CsHgClSO4 ⋅ H2O, have been successfully synthesized through the introduction of a highly polarizable d10 metal cation, Hg2+. The unique component layers in the reported compounds, [Hg3O2SO4]∞ layers in Hg3O2SO4 and [HgClSO4(H2O)] layers in CsHgClSO4 ⋅ H2O, induce enlarged birefringence in each sulfate. Remarkably, Hg3O2SO4 exhibits a very large SHG response (14 times that of KH2PO4), which is the strongest efficiency among all the reported nonlinear optical sulfates. Detailed theoretical calculations confirm that the employment of highly polarizable Hg2+ is an effective strategy to design superior optical materials with large birefringence and strong SHG response.

Oxygen Deficiencies in Titanium Oxide Clusters as Models for Bulk Defects.

TD-DFT calculations were performed on neutral TinO2n, TinO2n-1, and TinO2n-2 clusters, where n ≤ 7. Calculations show the TinO2n clusters are closed shell systems containing empty d orbitals and that the partially filled d orbitals of the suboxide clusters have a profound effect on their structural, electronic, and topological properties. The low energy photoexcitations of TinO2n clusters are all O-2p to Ti-3d transitions, while the open-shell suboxide clusters are all characterized by d-d transitions that occur at a much smaller optical gap. Upon low energy photoabsorption, the localization of the hole is accompanied by a local bond elongation, i.e., polaron formation, whereas d-electrons are generally delocalized around the cluster. The properties of the clusters, including the oxygen binding energies and structures, were calculated to account for the variation in relative populations found in experimental cluster distributions. Several TinO2n-2 clusters contain higher symmetry which is reflected in their relative stability. In particular, the tetrahedral symmetry of Ti4O6 inhibits charge carrier localization and therefore exhibits higher stability.

Classification of Fullerene Isomers from C40 to C48

We have designed possible structures of the isomers of midi-fullerenes, namely C40, C42, C44, C46, and C48; three the most natural mechanisms of their formation being used: fusion of carbon cupolas having the same symmetry; fusion of fullerenes having compatible symmetry and embedding carbon dimers into initial fullerenes. The energies of the fullerenes calculated through the use of molecular mechanics are presented together with their graphs. It is found that in the majority of cases the minimum-energy fullerenes are those, which have tetrahedral symmetries. The maximum-energy fullerenes refer to the three-fold T-symmetry.

Variable dimensionality of the anion framework in four new borophosphates and fluoroborophosphates with short cutoff edges.

Two new borophosphates, Cs3B3P4O16 and Li(NH4)2B3P4O16, and two new fluoroborophosphates, K4B2P2O9F2 and Rb3B2PO5F4, have been obtained via the high-temperature solution method. Single-crystal X-ray diffraction indicates that all of them exhibit various anion frameworks although they crystallize in the same space group, P21/c. Two-dimensional (2D) [B3P4O16]∞ layers and a 3D [B3P4O16]∞ network can be found in Cs3B3P4O16 and Li(NH4)2B3P4O16, respectively, while isolated [B2P2O9F2] and [B2PO5F4] exist in K4B2P2O9F2 and Rb3B2PO5F4, respectively. The effect of the alkali metal cation size on the framework structures of Cs3B3P4O16 and Li(NH4)2B3P4O16 has been discussed in detail. The IR spectra confirm their structural validity. UV-Vis-NIR diffuse reflectance spectroscopy indicates that the new compounds exhibit short cutoff edges. In addition, theoretical calculations were carried out to understand their electronic structures and optical properties.

Mechanism of Cluster Formation on Cerium Borosilicate Glasses Based on TEM-EDP and SEM-EDEX Investigations

Cerium oxide has a great capacity to remove nonbridging oxygen atoms (NBO) from the glass network and serves as glass former units. The well formed CeO4 units played the role of decreasing NBO from the silicate network and cause a reduction in the concentration of tetrahedral boron groups (N4). The highest content of NBO in glass of lower CeO2 (1 mol%) has a dominant role in constructing crystalline clusters in the glass. Higher CeO2 concentration leads to formation of an amorphous glass network as documented by XRD and TEM-EDP spectra. Coordination of cerium with oxygen atoms gives uniform units of spherical morphology in the pure CeO2 as well as in cerium rich glass. Clustered species has a great benefit in the field of application, used as a shielding material for ionized radiations.

Achieving Short‐Wavelength Phase‐Matching Second Harmonic Generation in Boron‐Rich Borosulfate with Planar [BO3] Units

AbstractUltraviolet (UV) nonlinear optical (NLO) crystals which can produce short‐wavelength lasers via a direct second harmonic generation (SHG) process are of great importance in modern laser technology. Currently, the exploration of UV NLO crystals in borosulfates is nearly stagnant since the non‐phase matching (PM) property lies on the small birefringence induced by the intrinsically small optical anisotropy of the tetrahedral groups. Herein, for the first time, the planar [BO3] units were introduced into borosulfates leading to a boron‐rich borosulfate (NH4)2B4SO10 with unprecedented [B4SO10]∞ layers and evidently enhanced birefringence. To the best of our knowledge, it achieves the shortest SHG PM wavelength of 252 nm in all reported borosulfates with deep UV cutoff edge (184 nm), large SHG response (1.1×KDP at 1064 nm and 0.15×β‐BBO at 532 nm) and large birefringence (0.053 at 1064 nm) and is easy to grow single crystals via simple chemical vapor deposition method. These results confirm the feasibility of utilizing planar [BO3] units to optimize birefringence of borosulfates, and also open up broad prospects for UV NLO crystals in boron‐rich borosulfates.

Achieving Short-Wavelength Phase-Matching Second Harmonic Generation in Boron-Rich Borosulfate with Planar [BO3 ] Units.

Ultraviolet (UV) nonlinear optical (NLO) crystals which can produce short-wavelength lasers via a direct second harmonic generation (SHG) process are of great importance in modern laser technology. Currently, the exploration of UV NLO crystals in borosulfates is nearly stagnant since the non-phase matching (PM) property lies on the small birefringence induced by the intrinsically small optical anisotropy of the tetrahedral groups. Herein, for the first time, the planar [BO3 ] units were introduced into borosulfates leading to a boron-rich borosulfate (NH4 )2 B4 SO10 with unprecedented [B4 SO10 ]∞ layers and evidently enhanced birefringence. To the best of our knowledge, it achieves the shortest SHG PM wavelength of 252 nm in all reported borosulfates with deep UV cutoff edge (184 nm), large SHG response (1.1×KDP at 1064 nm and 0.15×β-BBO at 532 nm) and large birefringence (0.053 at 1064 nm) and is easy to grow single crystals via simple chemical vapor deposition method. These results confirm the feasibility of utilizing planar [BO3 ] units to optimize birefringence of borosulfates, and also open up broad prospects for UV NLO crystals in boron-rich borosulfates.

σ‐Hole Interactions of Tetrahedral Group IV–VIII Lewis Acid Centers with Lewis Bases: A Comparative Study

AbstractIn the current study, σ‐hole interactions of group IV–VIII element‐containing molecules in tetrahedral geometry with Lewis bases were comparatively scrutinized. The FSiF3, OPF3, NSF3, FClO3, and OArO3 molecules were devoted as Lewis acid centers to interact with NH3 and NCH Lewis bases. According to the results, the most significant σ‐hole interactions were ascribed to the FSiF3⋅⋅⋅ and FClO3⋅⋅⋅Lewis base complexes with interaction energy up to –29.74 kcal mol−1 in the case of the FSiF3⋅⋅⋅NH3 complex. Symmetry‐adapted perturbation theory‐based energy decomposition analysis (SAPT‐EDA) demonstrated that the electrostatic energy (Eelst) was the most prevalent force dominating the explored interactions, while the exchange energies exhibited unfavorable contribution. Quantum theory of atoms in molecules (QTAIM) and noncovalent interaction (NCI) results emphasized the closed‐shell nature for all the investigated complexes except FSiF3⋅⋅⋅NH3 complex that demonstrated a covalent nature. Crucially, the utilization of the positively‐ and negatively‐directed external electric field (EEF) led to amelioration and debilitation of the strength of the studied interactions, respectively. The present findings provide knowledge essential to further expanded applications in the fields pertinent to materials science and crystal engineering.

4-dimensional lattice models for the quantum range

Some finite subspace models L are presented for quantum structures which replace the use of countable infinite Hilbert space H dimensions. A maximal Boolean sublattice, called block, is 24, where its four atoms directly above 0εL, base vectors of H in 24 are drawn as four points on an interval. Blocks can overlap in one or two atoms. Different kinds of operators can map one block onto another and interpretations are given such that subspaces can carry on their base vector tuple real, complex or quaternionic numbers, energies, symmetries and generate coordinate lines. Describing states of physical systems is done using L and its applications for dynamical modelling. They don‘t need the infinte dimensional vectors of H. L has in the first model 11 blocks and 24 atoms (figure 1). They correspond to the 24 elements of the tetrahedral S4 symmetry. S4 arises from a spin-line rgb-graviton whirl operator with center at the tip of a tetrahedron and a nucleon triangle base with three quarks as vertices. The triangles factor group D3 of S4 is due to the CPT Klein normal subgroup Z2 x Z2 of S4 . It has a strong interaction SI rotor for the nucleons inner dynamics which is used for integrating functions, exchanging energies of nucleon with its environment and setting barycentrical coordinates in the triangle. At their intersection B as barycenter sets a Higgs boson or field the rescaled quark mass of a nucleon. Each factor class of one element from D3 assigns to it a color charge, a coordinate, an energy vector and a symmetry. Symmetries attached can be different according to interactions involved. Every atom of L has then a specific character with different properties.Three characters are added to octonian base vectors, listed by their indices as n = 0,1,…,7, and named for the atoms of L as na, nb, nc. The structure and element attributes of the finite subspace lattices L are desribed in many examples and models which technical constructed run macroscopically. Several models are described below. Example, the color charge whirl as rgb-graviton projection operator maps the block 2c3b5a6a to 0a1a2a3a. The symmetries change dimension from 3x3- to 2x2-matrices. From SU(3) are λ1 on 3b mapped to the SU(2) x-coordinate Pauli matrix σ1, from λ2 on 5a to σ2 y-coordinate and from λ3 on 6a to σ3 z-coordinate of real Euclidean space R³. The SU(3) matrices have complex w3 = z +ict, w2 = (iy,f), w1 = (x,m) coordinates. In figure 3 is shown how a rotation of two proton tetrahedrons for fusion changes the two linearly independent wj vectors to the 1-dimensional x,y,z base vectors. In deuteron then on one coordinate line sit with Cooper paire u-d-quarks at the ends the Heisenberg coupled energy or space vector rays 15 (x,m), m mass measured in kg, x in meter, 23 (iy,E(rot)), E(rot) rotational energy measured in Joule J, y in meter, 46 (ict,f), t time measured in seconds, f = 1/∆t frequency s inverse time interval measured in Hz. The six color charges are red r on +x as octonian coordinate 1, green g on +y as 2 , blue b on -z as 6, turquoise on -x as 5, magenta on -y as 3, yellow on +z as 4..