Refinement of the crystal structure of lithium-bearing uvite

The crystal structure of a natural calcium tourmaline, i.e., uvite with a high lithium content (0.51 au per formula (aupf) at the Y site, is refined to R = 0.019, R _w = 0.020, and S = 1.11. It is shown that, in nature, there exist uvites in which the charge balance in the case where the Z site is occupied by trivalent cations is provided by the replacement of part of the divalent magnesium cations at the Y site by univalent cations, divalent calcium cations at the X site by sodium cations, and univalent anions at the W site by oxygen anions. The W site is found to be split into two sites, namely, the W1 and W11 sites (the W1–W11 distance is 0.14 Å), which are partially occupied by the fluorine and oxygen anions, respectively. An analysis of the results obtained in this study and the data available in the literature on the crystal structure of uvites allows the conclusion that uvite can be considered a superspecies and that the nomenclature of this mineral group needs refinement with the use of structural data.     

Refinement of the crystal structures of three fluorine-bearing elbaites

The crystal structures of three Li-Al natural tourmalines (elbaites) containing 0.88–1.39 wt % F are refined to R= 0.0294, 0.0308, and 0.0417. It is revealed that the W threefold anion site is split into two sites, namely, the W1 threefold site and W2 ninefold site (W1–W2 ～ 0.4 Å, Y-W₁ ≥ 1.94 Å, Y-W2 ≥ 1.75 Å). The following hypothesis is proposed and justified: the W1 and W2 sites are partially occupied by OH groups and fluorine anions, respectively. The ratio of the [YO₄(OH)₂] octahedra to the [YO₄(OH)F] octahedra depends on the fluorine content and varies from structure to structure. The fact that the W site is more than 50% occupied by fluorine in the structures of two tourmalines under investigation allows the conclusion that fluor-elbaite with the ideal formula Na(Li_1.5A1_1.5)A1₆(Si₆O₁₈)(BO₃)₃(OH)₃F is a new mineral species and that elbaite can be considered a superspecies.     

Comparative analysis of two methods of calculation of the orientation of domain walls in ferroelastics

Two types of domain-wall equations are analyzed: the equations derived by the Sapriel method and the equations obtained by interface matching of the thermal-expansion tensor. It is shown that, for W-type domain walls, these methods yield the same equations. For W′-type domain walls, the equations obtained by different methods coincide for proper ferroelastics and differ for improper ferroelastics.     

Structural investigation of cooperite (PtS) crystals

The single-crystal structure of cooperite, a natural platinum sulfide PtS, is studied by X-ray diffraction supported by high-resolution scanning transmission electron microscopy and X-ray spectrum microanalysis. It is found that, in addition to the main reflections corresponding to the known tetragonal cell (a = 3.47 and c = 6.11 Å; space group P4₂/mmc), many weak reflections with intensities I ≤ 60σ(I) are clearly observed. These reflections fit the tetragonal cell (space group I4/mmm) with doubled parameters. In structures with small (P4₂/mmc) and large (I4/mmm) cells, the S atoms occupy statistically two special positions. It is shown that the chemical composition of the cooperite crystals deviates from the stoichiometric composition: sulfur-deficient specimens predominate.     

Growth and structure of La<Subscript>3</Subscript>Zr<Subscript>0.5</Subscript>Ga<Subscript>5</Subscript>Si<Subscript>0.5</Subscript>O<Subscript>14</Subscript> crystals

The complete X-ray structure determination of Czochralski grown La₃Zr_0.5Ga₅Si_0.5O₁₄ single crystals with the Ca₃Ga₂Ge₄O₁₄ structure is performed (sp. gr. P321, a = 8.226(1) Å, c = 5.1374(6) Å, Z = 1, Mo K_α1 radiation, 1920 crystallographically independent reflections, R = 0.0166, R_w = 0.0192). The absolute structure is determined. It is shown that possible transition of some of La atoms (～1.2%) from the 3e to 6g position may give rise to the formation of structural defects.     

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.     

Stabilization of lattice period in quinary solid solutions of the <Emphasis Type="Italic">A</Emphasis><Superscript>3</Superscript><Emphasis Type="Italic">B</Emphasis><Superscript>5</Superscript> compounds

The stabilizing effect of elastic strains on the lattice period of a quinary solid solution is considered. The expression for the stabilization factor for quinary solid solutions of the A_xB_1?xC_yD_zE_1?y?z type is derived. It is shown that the stabilizing influence of the substrate sharply increases in the vicinity of the region of the chemical spinodal. The stabilization factors are calculated for the Ga_xIn_1?xP_yAs_zSb_1?y?z and A1_xGa_yIn_1?x?yP_zAs_1?z quinary solid solutions isoperiodic to InAs, GaSb, and GaAs. It is shown that in the region of thermodynamic instability the stabilization factor has negative values. The changes in the composition of the above elastically strained quinary solid solutions are analyzed with respect to the equilibrium composition. It is also shown that stabilization of the lattice period does not signify the stabilization of its composition.     

Dimorphism of <Emphasis Type="Italic">R</Emphasis>F<Subscript>3</Subscript> (<Emphasis Type="Italic">R</Emphasis> = La–Nd) crystals based on the data of X-ray diffraction studies

The ratio of two forms of tysonite in nominally pure single crystals of RF₃ (R = La–Nd) obtained from melts under identical conditions (the as-grown state) is studied for the first time by X-ray diffraction. Crystals of RF₃ with R = La–Nd belong to the β-LaF₃ structural type (space group \(P\bar 3c1\), Z = 6) and form twins. Samples 0.2–0.4 in diameter contain inclusions of the high-temperature a form (space group P6₃/mmc, Z = 2). It is shown that twinning and dimorphism of the RF₃ crystals (R = La–Nd) have a common structural basis.     

Electrons of <Emphasis Type="Italic">l</Emphasis> shells of free atoms as a regular system of points on a sphere: I. Modeling by polyhedra

Electron l shells of free atoms have been modeled on the basis of regular and semiregular polyhedra by determining stable systems of identical Coulomb particles with an increase in their number. It is shown that such stable systems correspond to the location of the maxima of electron density at the vertices of inversion-center polyhedra forming a sequence in which the first polyhedron has two vertices and the number of vertices in each subsequent polyhedron exceeds the number of vertices in the previous one by four. The electron s, p, d, and f shells are modeled by a dumbbell and trigonal, pentagonal, and heptagonal antiprisms, respectively. Thus, in addition to the quantum-mechanical properties, l shells should have the symmetry properties of these antiprisms. It is believed that consideration of the noncrystallographic (for the three-dimensional Euclidean space) fivefold and seven-fold symmetries of d and f shells of free atoms will make it possible to obtain a unified explanation for a number of phenomena in crystal structures and other ordered media.     

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.     

Domain structure in a nematic layer under oscillatory Couette flow

The formation of a spatially modulated structure (domains) in a homeotropically aligned layer of a nematic liquid crystal (NLC) under an oscillatory Couette flow are described theoretically by using the hydrodynamic equations for an NLC which are linearized with respect to perturbations. The shear frequencies at which the phenomenon is determined by the inertial properties of the medium are considered. It is shown that the domain size is equal to the thickness h of an NLC layer and that the dependence of the threshold value of the shear amplitude u _th on the frequency ω and the layer thickness h is described by the expression u _th = const/(ωh²). The calculated results are compared with the experimental data.     

Phase equilibria of <Emphasis Type="Italic">A</Emphasis><Superscript>3</Superscript><Emphasis Type="Italic">B</Emphasis><Superscript>5</Superscript> quinary systems with an elastically strained solid phase

The contribution of the elastic component of the excess energy of mixing is determined for A³B⁵ quinary solid solutions. A relationship is derived for the activities of the components in an elastically strained solid phase of the quinary solid solutions. The correctness of the possible assumptions used in the calculation of the activity coefficients of the solid phase of a quinary solution is analyzed. The contact supercooling in the vicinity of the binary substrate isoperiods is calculated for a number of A³B⁵ quinary systems. It is shown that the negative contact supercoolings correspond to the thermodynamic instability boundaries of the quinary solid solutions.     

Theory of the formation of P4<Subscript>1</Subscript>32(P4<Subscript>3</Subscript>32)-phase spinels

A group-theoretical, thermodynamic, and structural study of the formation of P4₁32(P4₃32) spinel modification has been performed. In particular, the occurrence of unique hyper-kagome atomic order is analyzed. The critical order parameter inducing a phase transition is established. It is shown that the calculated structure of the low-symmetry P4₁32(P4₃32) phase is formed as a result of displacements of atoms of all types and due to the cation and anion ordering. The problem of the occurrence of unique hyper-kagome atomic order in the structures of P4₁32(P4₃32) spinel modifications is considered theoretically. It is proven within the Landau theory of phase transitions that the P4₁32(P4₃32) phase can be formed from the high-symmetry Fd3m phase with an ideal spinel structure only as a result of first-order phase transition. Therefore, the formation of hyper-kagome sublattice in the P4₁32(P4₃32) phase is accompanied by a significant transformation of the spinel structure.     

Numerical dependence of the unit-cell parameters of minerals of the lamprophyllite group on the cationic composition in the interlayer position

Analysis of the data on the unit-cell parameters of the chemically characterized minerals of the lamprophyllite group showed that the parameter c and the angle β are independent of the cationic composition in the interlayer position. The value of the parameter b correlates with the Ba content. It is shown that the parameter a linearly depends on the average ionic radius of cations in the interlayer position. The Ba and K contents can be estimated from the equations K + Ba = 2.1431a-41.091 and Ba = 17.764b-125.36.     

Cationic transport mechanism in α-Ag<Subscript>1 − <Emphasis Type="Italic">x</Emphasis></Subscript>Cu<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>I(0 < <Emphasis Type="Italic">x</Emphasis> < 0.25): Molecular-dynamics simulation

The structural and transport characteristics of an Ag_{1 ? x} Cu _x I(0 < x < 0.25) solid solution have been simulated by the molecular-dynamics method. It is found that the cation diffusion coefficient decreases with increasing copper concentration; this correlation is in agreement with the experimentally observed decrease in ionic conductivity. It is shown that the cationic transport in disordered Ag_{1 ? x} Cu _x I phases is mainly due to the migration of silver cations, whereas the mobility of copper cations is much lower. Cu⁺ cations are found to reside in the 8c positions in a bcc cell; this finding suggests the existence of nanoscale α-CuI regions.     

<Emphasis Type="Italic">P</Emphasis>-wave ray velocities and the inverse acoustic problem for anisotropic media

The specific features of the calculation of ray velocities of quasi-longitudinal waves in anisotropic media have been considered. A technique for calculating elastic constants using P-wave ray velocities measured in an ultrasonic experiment on spherical samples is presented. It is shown by an example of tabular data that elastic constants С₁₁, С₂₂, and С₃₃ and combinations of constants (С₁₂ + 2С₆₆), (С₁₃ + 2С₅₅), (С₂₃ + 2С₄₄), (С₁₄ + 2С₅₆), (С₂₅ + 2С₄₆), and (С₃₆ + 2С₄₅) can be calculated most accurately for the general case of anisotropic media with elastic properties of arbitrary symmetry. Since the determining system of equations is illconditioned, the values of elastic constants entering these combinations depend on the choosed initial approximation.     

Relationship between the temperature dependence of the induced helical pitch and the anisometry of molecules of chiral dopants

The regularities of changes in the temperature dependence of the helical pitch P and the parameter dP/dt are analyzed for induced cholesteric liquid-crystal systems based on nematic 4-pentyl-4′-cyanobiphenyl, including chiral arylidene derivatives of (S)-1-phenylethylamine and (1R),(4R)-isomenthone. In the molecules of these compounds, the length of the π-electron arylidene fragment, the nature of the bridge group between its benzene rings and of the lateral substitution in the cyclohexanone ring, and the length of the terminal alkyl or oxyalkyl substituent are varied. The statistical molecular anisometry of the chiral dopants is calculated using the density functional method and the semiempirical AM1 method. It is shown that the temperature dependence of the induced helical pitch and, correspondingly, its quantitative characteristic dP/dt and the parameter dt _t /dC, characterizing the effect of a chiral dopant on the thermal stability of the mesophase, are determined by the molecular anisometry of the chiral component.     

Strain of interatomic bonds and multiple-well potential in crystals with perovskite structure

A method for determining the strain characteristics of interatomic bonds in crystals of ABX₃ compounds with perovskite structure is developed. The bond strain energy ΔU _s is estimated. This energy is responsible for the formation of ordered lattice distortions related to the rotation of octahedra and/or the cooperative displacement of cations. It is shown that ΔU _s correlates with the characteristics of ordered distortions and the temperature of the phase transition of compounds to the cubic structure. It is ascertained that the interatomicinteraction potential in crystals with strained bonds is a local multiple-well potential, the form of which depends on the nature of interacting atoms. It is shown that the occurrence of the ferroelectric state in the noted crystals may be due to the presence of bond strains and the anisotropy of the covalent component of the cation-anion interaction.     

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).