Electrons of <Emphasis Type="Italic">l</Emphasis> shells of free atoms as a regular system of points on a sphere: III. Factors of self-organization of electrons in the periodic table

The factors determining the self-organization of the electron system of an atom at different levels of the periodic table are considered. Specifically, these factors are the isotropy and three-dimensional nature of space and the indistinguishability of electrons. The concept of a simplex is used, whose vertices correspond to a regular system of particles (minimum in number for a given space) in the state of the global minimum of the system’s potential. These factors implement the principle of simplicity (small number of particles) and hierarchy in the periodic table of elements. The global minimum of the potential of s, p, d, and f shells is reached in odd-dimensional spaces. In a three-dimensional space, such a minimum is reached for d and f shells, in contrast to s and p shells, through shell mixing.     

Synthesis and structure of (Rb<Subscript>0.50</Subscript>Ba<Subscript>0.25</Subscript>)[UO<Subscript>2</Subscript>(CH<Subscript>3</Subscript>COO)<Subscript>3</Subscript>]

A new compound (Rb_0.50Ba_0.25)[UO₂(CH₃COO)₃] is synthesized and its crystal structure is studied by X-ray diffraction. The compound crystallizes in the form of yellow plates belonging to the cubic crystal system. The unit cell parameter a = 17.0367(1) Å, V = 4944.89(5) ų, space group I \(\bar 4\)3d, Z = 16, and R = 0.0182. The coordination polyhedron of the uranium atom is a hexagonal bipyramid with oxygen atoms of three acetate groups and the uranyl group in the vertices. The crystal chemical formula of the uranium-containing group is AB ₃ ⁰¹ (A = UO ₂ ²⁺ , B ⁰¹ = CH₃COO^?). The oxygen atoms of the acetate groups that enter the coordination polyhedron of uranium are bound to barium and rubidium atoms.     

Electrons of <Emphasis Type="Italic">l</Emphasis> shells of free atoms as a regular system of points on a sphere: II. Some manifestations of the symmetry of electron shells of ions of transition metals in ordered media

Manifestations of the spatial symmetry of d and f shells formed by equivalent electrons of transition-metal ions in crystals and other ordered media are considered. It is shown that sizes of crystals and the rare coordination number of ions can be explained in terms of the noncrystallographic symmetry of these shells. The symmetry of the d shell may cause manifestations of elements of fivefold symmetry in the crystal faceting and the structure of molecules and affect the properties dependent on matching of the intrinsic symmetry of the shell and the translational symmetry of a crystal.     

Specific stereochemical features of metallochelate complexes with tridentate <Emphasis Type="Italic">N,N,N</Emphasis>(<Emphasis Type="Italic">N,N</Emphasis>,O)-donating ligands

Earlier X-ray diffraction studies of a series of 12 adducts (I–XII) between metallochelate complexes [M = Co(II), Ni(II), and Cu(II)] with tridentate N,N,N(N,N,O)-donating Schiff bases (L) and monodentate or bidentate ligands (L′) revealed a similarity in the stereochemistry of these compounds. The coordination polyhedron of metal atoms in compounds I–XII is a tetragonal pyramid (bipyramid) with two vertices occupied competitively. The L ligand occupies three coordination sites in the base of the pyramid. The L′ ligand approaches the metal atom, as a rule, in a direction perpendicular to the basal plane. The fourth site in the base of the pyramid and the apical vertex are occupied competitively. Different patterns of occupation of these positions are observed: they include the donor atoms of both the L and L′ ligands.     

Cluster self-organization of crystal-forming systems: Suprapolyhedral cluster precursors and self-assembly of the icosahedral structure of ZrZn<Subscript>22</Subscript>(<Emphasis Type="Italic">c</Emphasis>F184)

The mechanism of self-assembly of symmetrically and topologically different chains and microlayers (in the form of planar nets) from cyclic three-node clusters A ₃ is considered in the model system. The obtained nets correspond to the uninodal Shubnikov nets N 3 12 12 and N 3 6 3 6 and the new binodal net N1 3 6 3 6 + N2 3 3 6 6 (1: 2). A complete three-dimensional reconstruction of the self-assembly of the icosahedral structure of the ZrZn₂₂ compound (cF184) is performed using computer methods (with the TOPOS program package) according to the following scheme: cluster precursor → primary chain → microlayer → microframework (supraprecursor) → ... → framework. It is revealed that the suprapolyhedral cluster precursor (nanocluster ～12 Å in size) of the AB ₂ composition is formed by three polyhedra shared by vertices in a cyclic manner: the A-ZrZn₁₆ polyhedron (sixteen-vertex polyhedron with the point symmetry \(\bar 4\)3m) and two B-ZrZn₁₂ polyhedra (icosahedra with the point symmetry \(\bar 3\) m). The AB ₂ cluster precursor in the structure retains the symmetry m. It is established that the structural mechanism of self-assembly of the two-dimensional layer in the ZrZn₂₂ structure is described by the binodal net N1 3 6 3 6 + N2 3 3 6 6 (1: 2) constructed in the modeling.     

Gosset helicoids: I. 8D crystallographic lattice <Emphasis Type="Italic">E</Emphasis><Subscript>8</Subscript> and crystallographic,quasi-crystallographic,and fractional helicoidal axes determined by this lattice

It is shown that mapping of substructures of a semiregular Gosset polytope, whose 240 vertices form the first coordination sphere of a 8D lattice E ₈, determines the orders of p/d axes of helicoids that are set only by invariants of (sub)algebras. Axes of such (Gosset) helicoids are derived. These axes perform rotation by an angle (360°/p(d; can be crystallographic, quasi-crystallographic (p = 2, 3, 4, 5, 6, …; d = 1), or fractional (1 < d < p/2); and may belong to regular polytopes (conventional, starlike, etc.). Formation of ordered structures is considered as a formation of Gosset helicoids (rods) with their subsequent assembly. Helicoids with axes 15/4 and 15/7 are considered as examples. They correspond to crystallographic approximants—helicoids with axes 4₁ and 2₁ composed of deformed icosahedra (dodecahedra)—in a β-Mn crystal (clathrate IX).     

The crystal structure of vyuntspakhite: A redetermination

The crystal structure of the mineral vyuntspakhite (Y, TR)₆{Al₂(OH)₃[H_1.48Si_1.88O₇][SiO₄][SiO₃(OH)]}₂(a = 5.7551(11) Å, b = 14.752(3) Å, c = 15.906(4) Å, β = 96.046(4)°, sp. gr. P2₁/n, Z = 2), which had been established earlier in the pseudo-unit cell, is redetermined by X-ray diffraction (R = 0.040, T = 100 K). The redetermination of the structure shows that pronounced pseudotranslation along the axis c′ = c/3 is associated with the fact that Y(TR) atoms are related by a 1/3 translation along the [001] direction. Most of the hydrogen atoms are located. The crystal-chemical function of hydrogen bonds is analyzed. In the unit cell of vyuntspakhite, the cationic layers consisting of edge-sharing (Y,TR) eight-vertex polyhedra alternate along the b axis with mixed anionic layers composed of isolated Si tetrahedra (orthotetrahedra), Si₂O₇ double-tetrahedra (diortho) groups, Al five-vertex polyhedra, and Al₂O₈ double-tetrahedra groups linked by shared vertices and through hydrogen bonding.     

Structural chemistry of peroxo compounds of group VI transition metals: II. Peroxo complexes of molybdenum and tungsten: A review

The specific features revealed in the structure of the molybdenum and tungsten peroxo complexes with the ratios M: O₂ = 1: 1, 1: 2, and 1: 4 are considered. It is demonstrated that the geometry of the coordination polyhedron of the metal atom is primarily determined by the “metal: peroxo ligand” ratio. Formally, the pentagonal bipyramidal coordination polyhedra of the Mo(VI) and W(VI) oxo monoperoxo and oxo diperoxo complexes (the coordination numbers of the metal atoms are equal to seven) have different geometries, namely, the MO(O₂)A ₄ pseudooctahedral and MO(O₂)₂ A ₂ pseudotrigonal bipyramidal configurations.     

Structural chemistry of peroxo compounds of group VI transition metals: I. Peroxo complexes of chromium (a review)

The specific features revealed in the structure of the d ³ Cr(III), d ² Cr(IV), d ¹Cr(V), and d ⁰ Cr(VI) peroxo complexes with the ratios M:O₂ = 1:1, 1:2, and 1:4 are considered. It is noted that, in eleven compounds of the general formula Cr(O₂)_nO_m A _p (n = 1, 2, 4; m = 0, 1; p = 0–4), the metal atoms can be in four oxidations states: +3 (d ³), +4 (d ⁴), +5 (d ¹), and +6 (d ⁰). This property distinguishes chromium peroxo compounds from molybdenum and tungsten dioxygen complexes, which, with one exception, are represented by the d ⁰ M(VI) compounds.     

New multicell model for describing the atomic structure of La<Subscript>3</Subscript>Ga<Subscript>5</Subscript>SiO<Subscript>14</Subscript> piezoelectric crystal: Unit cells of different compositions in the same single crystal

Accurate X-ray diffraction study of langasite (La₃Ga₅SiO₁₄) single crystal has been performed using the data obtained on a diffractometer equipped with a CCD area detector at 295 and 90.5 K. Within the known La₃Ga₅SiO₁₄ model, Ga and Si cations jointly occupy the 2d site. A new model of a “multicell” consisting of two different unit cells is proposed. Gallium atoms occupy the 2d site in one of these cells, and silicon atoms occupy this site in the other cell; all other atoms correspondingly coordinate these cations. This structure implements various physical properties exhibited by langasite family crystals. The conclusions are based on processing four data sets obtained with a high resolution (sin θ/λ ≤ 1.35 Å^–1), the results reproduced in repeated experiments, and the high relative precision of the study (sp. gr. P321, Z = 1; at 295 K, a = 8.1652(6) Å, c = 5.0958(5) Å, R/wR = 0.68/0.68%, 3927 independent reflections; at 90.5 K, a = 8.1559(4) Å, c = 5.0913(6) Å, R/wR = 0.92/0.93%, 3928 reflections).     

Crystal structures of copper(II) nitrate,copper(II) chloride,and copper(II) perchlorate complexes with 2-formylpyridine semicarbazone

Compounds dinitrato(2-formylpyridinesemicarbazone)copper (I), dichloro(2-formylpyridinesemicarbazone) copper hemihydrate (II), and bis(2-formylpyridinesemicarbazone)copper(2+) perchlorate hydrate (III) are synthesized and their crystal structures are determined. In compounds I–III, the neutral 2-formylpyridine semicarbazone molecule (L) is tridentately attached to the copper atom via the N,N,O set of donor atoms. In compounds I and II, the Cu: L ratio is equal to 1: 1, whereas, in III, it is 1: 2. In complex I, the coordination sphere of the copper atom includes two nitrate ions with different structural functions in addition to the L ligand. The structure is built as a one-dimensional polymer in which the NO₃ bidentate group fulfills a bridging function. The coordination polyhedron of the copper(2+) atom can be considered a distorted tetragonal bipyramid (4 + 1 + 1). Compound II has an ionic structure in which the main element is the [CuLCl₂ · Cu(H₂O)LCl]⁺ dimer. In the dimer, the copper atoms are linked via one of the μ₂-bridging chlorine atoms. The coordination polyhedra of the central atoms of the Cu(H₂)LCl and CuLCl₂ complex fragments are tetragonal bipyramid and tetragonal pyramid, respectively. In compound III, the copper atom is octahedrally surrounded by two L ligands in the mer configuration.     

Multicell model of La<Subscript>3</Subscript>Ga<Subscript>5</Subscript>GeO<Subscript>14</Subscript> crystal: A new approach to the description of the short-range order of atoms

The multicell model alternative to the model of mixed atomic sites used now is proposed for a single crystal of La₃Ga₅GeO₁₄ belonging to the langasite family. The multicell consists of four unit cells. In three identical cells of the structure, atoms adapt to the Ge atom occupying one of the two 2d positions on the threefold symmetry axis. In the fourth cell, atoms surround the Ge atom located at the 1a position. The multicell model allows one to study the short-range order of atoms by the methods of classical structure analysis based on Bragg scattering. Four high-resolution data sets measured at 295 and 111.5 K are used in the study. The results are obtained with high relative precision (space group P321, Z = 1; at 295 K a = 8.2020(6) Å and c = 5.1065(6) Å, R/wR = 0.81/0.73% for 3829 unique ref lections; at 111.5 K a = 8.1939(1) Å and c = 5.1022(4) Å, R/wR = 0.85/0.76% for 3880 reflections).     

Crystallography of Mendeleev’s periodic table

Electrons in each atomic orbital (s, p, d, f, and so on) form regular systems on a sphere, which can be easily revealed when constructing Mendeleev’s periodic table. Electrons in orbitals (including hybrid orbitals) can be located only at the vertices of isogons: Platonic bodies; Archimedean bodies; and two infinite series of prisms and antiprisms and their affine transforms, which retain the vertex transitivity. Electrons in crystal structures are distributed over isogonal three-dimensional regular systems that are vertices of partitions composed of isogons.     

Crystal structure of a new synthetic Ca,Li pentaborate: CaLi<Subscript>4</Subscript>[B<Subscript>5</Subscript>O<Subscript>8</Subscript>(OH)<Subscript>2</Subscript>]<Subscript>2</Subscript>

When studying the phase formation in the CaO-Li₂O-B₂O₃-H₂O system, a new Ca,Li pentaborate was synthesized under hydrothermal conditions. The structure of a new compound with the crystallochemical formula CaLi₄[B₅O₈(OH)₂]₂ (sp. gr. Pb2n, a = 8.807(7), b = 9.372(7), c = 8.265(6) Å, V = 682.2(9) ų, Z = 2, d_calcd = 2.43 g/cm³, automated SYNTEX-\(P\bar 1\) diffractometer, 2690 reflections, 2θ/θ scan, λMo) is refined up to R_hkl = 0.0557 in the anisotropic approximation of atomic thermal vibrations with allowance for the localized H atoms. The structure of the Ca,Li pentaborate is formed by (010) open boron—oxygen layers formed by two independent [B₅O₈(OH)₂]^3? pentagroups, with each of them being formed by three B tetrahedra and two B triangles. The structure framework consists of the above boron—oxygen layers bound by isolated Li tetrahedra. The Ca cations are localized in the centers of eight-vertex polyhedra located in the [001] channels of the Li,B,O framework. Comparative crystallochemical analysis of the new Ca,Li pentaborate and Li pentaborate of the composition Li₃[B₅O₈(OH)₂]-II showed that the anionic matrices of both compounds are completely identical, whereas some of the cationic positions are different.     

Birefringence of (NH<Subscript>4</Subscript>)<Subscript>2</Subscript>SO<Subscript>4</Subscript> crystals under the action of uniaxial pressures

The influence of uniaxial mechanical pressure σ _m ≤ 150 bar on the spectral (300–800 nm) dependence of the birefringence Δn _i of (NH₄)₂SO₄ crystals is studied. The dispersion Δn _i (λ) is shown to be normal and greatly increases when approaching the absorption edge. Uniaxial pressure changes the value of dispersion dΔn _i /dλ but not its character. It is found that the simultaneous action of pressures σ _x ～ σ y ～ 560 bar results in the occurrence of a uniaxial isotropic state. Piezoelectric constants of the crystals are estimated.     

Electron diffraction study of lizardite 1<Emphasis Type="Italic">T</Emphasis> with stacking faults

The stacking faults observed in the structure of the mineral lizardite 1T belonging to the polytype group A are investigated using the digital oblique-texture electron diffraction patterns, difference Fourier-potential maps, and model diffraction patterns obtained for this compound. Numerical simulation of the diffraction profiles along the first (the 02l and 11l reflections) and second (the 20l and 13l reflections) ellipses in the oblique-texture electron diffraction patterns is performed for finite sequences of ten layers with the use of the Markovian statistical model in the quasi-homogeneous approximation. The specific features of the intensity distributions along the first and second ellipses are associated with the manifestation of translational (displacements of the layers by ±b/3) and orientational (rotations of the layers through an angle of 180°) defects of the layer stacking, respectively. For both ellipses, the experimentally observed intensity distributions are in the best agreement with the diffraction profiles calculated for stacking faults at a content of approximately 25%, the short-range order parameter S = 1, and the maximum degree of ordering in the layer alternation. It is demonstrated that the irregularities revealed in the layer alternation in the structure of lizardite 1T (which is characterized by an identical orientation of the adjacent layers) arise from layer displacements by ±b/3 and, to a considerable extent, from the formation of sequences with opposite orientations of the adjacent layers. As a result, the structure of lizardite 1T nanocrystals involves a combination of layer sequences that are typical of structures belonging to the polytype groups A and D.     

Application of X-ray diffraction methods in the study of micrometer-sized porous Si layers

An X-ray analysis of porous silicon layers (Sb-doped n ⁺-Si(111)) obtained by anodic oxidation for different times with a current of 50 mA/cm² is performed by the methods of double-crystal rocking curves and total external reflection. A nondestructive method for monitoring the stationary process of the formation of micrometer-sized porous silicon layers and estimating their porosity and thickness is proposed. The parameters obtained for porous silicon layers with a thickness of ～6 μm are confirmed by the joint processing of diffraction curves for the 111 and 333 reflections on the basis of the developed model of dynamic scattering from layers while taking into account the strain profiles Δd(z)/d, the static Debye-Waller factor f(z), and the porosity P(z). The advantages and drawbacks of the proposed method are discussed.     

Crystal structures of bis{4-bromo-2-[(2-hydroxyethylimino)methyl]phenolato}copper and bis{4-chloro-2-[(2-hydroxyethylimino)methyl]phenolato}copper

The crystal structures of bis{4-bromo-2-[(2-hydroxyethylimino)methyl]phenolato}copper (I) and bis{4-chloro-2-[(2-hydroxyethylimino)methyl]phenolato}copper (II) are determined. Crystals I are monoclinic, space group P2₁/c, Z = 2, and R = 0.0732 (for all reflections). Crystals II are likewise monoclinic, space group P2₁/n, Z = 2, and R = 0.1106. In the structures of compounds I and II, the metal atom is situated at the center of symmetry and coordinated by two singly deprotonated bidentate 4-bromo-or 4-chloro-2-[(2-hydroxyethylimino)methylphenol molecules, respectively, through phenol oxygen and azomethine nitrogen atoms, which form a distorted planar square. In the structures of compound II, the coordination polyhedron of the central atom is completed to an elongated tetragonal bipyramid by the amino alcohol oxygen atoms of the adjacent complexes.     

Unstrained bond lengths and corresponding cation radii in crystals with perovskite structure

A method for determining average lengths of unstrained bands A-X (l_0AX) and B-X (l_0BX) and the ratio of the rigidity constants of these bonds for ABX₃ compounds with perovskite structure is proposed. The values of l_0AX and l_0BX correspond to the minimum energies of cation-anion interaction of the crystal sublattices. Values of l_0AX and l_0BX are obtained for several groups of halide and oxide compounds: A⁺B²⁺F₃, Cs⁺B²⁺Cl₃, A⁺B⁵⁺O₃, A²⁺B⁴⁺O₃, and A³⁺B³⁺O₃. It is ascertained that, for most compounds studied, the values of l_0AX and l_0BX are equal or close to the interatomic distances in crystals of binary compounds. The values of l_0AX and l_0BX are compared with the sums of the radii of the corresponding cations (R _A , R _B ) and anions (\(^{VI} R_{O^{2 - } } ,^{VI} R_{F^ - } ,^{VI} R_{Cl^ - }\)). It is found that the differences \(l_{0AO^ - } ^{VI} R_{O^{2 - } } (L_{0AF^ - } ^{VI} R_{F^ - } )\) and \(l_{0BO^ - } ^{VI} R_{O^{2 - } } (l_{0BF^ - } ^{VI} R_{F^ - } )\), regarded as the radii of the A and B cations in unstrained bonds, are close to the Shannon radii for a coordination number of six. It is shown that the rigidity constant for A-X bonds is several times smaller than that for B-X bonds.     

Crystal structure of byelorussite-(Ce) NaMnBa<Subscript>2</Subscript>Ce<Subscript>2</Subscript>(TiO)<Subscript>2</Subscript>[Si<Subscript>4</Subscript>O<Subscript>12</Subscript>]<Subscript>2</Subscript>(F,OH) · H<Subscript>2</Subscript>O

The crystal structure of the mineral byelorussite-(Ce) NaMnBa₂Ce₂Ti₂Si₈O₂₆(F,OH) · H₂O belonging to the joaquinite group was solved and refined to R = 0.033 based on 4813 reflections with I > σ2(I). The parameters of the orthorhombic unit cell are a = 22.301(4) Å, b = 10.514(2) Å, c = 9.669(2) Å, V = 2267.1(8) ų, sp. gr. Ama2, and Z = 4. The structure is composed of three-layer sheets, which consist of dimers of edge-sharing Ti octahedra located between isolated four-membered [Si₄O₂] rings. The sheets are linked to each other by Mn 5-vertex polyhedra to form a heteropolyhedral framework. Large cavities in the framework are occupied by Na 6-vertex polyhedra, Ba 11-vertex polyhedra, and REE 9-vertex polyhedra.