(Ga,Ge)-Analogue of Tourmaline: Crystal Structure and Composition

Crystallography Reports - Crystals of a (Ga,Ge)-analogue of tourmaline have been synthesized by hydrothermal method in the temperature range of 600–650°C under a pressure of 100 MPa. The...     

Crystal structure of holtite I

The crystal structure of As-containing holtite I is refined (Ital Structures diffractometer, 939 crystallographically independent reflections, anisotropic approximation, R = 0.047). The parameters of the orthorhombic unit cell are a = 4.695(1) Å, b = 11.906(3) Å, c = 20.38(3) Å, sp. gr. Pnma, Z = 4. On the whole, the structural formula obtained, (Si_2.43Sb_0.36As_0.21)BO₃[(Al_0.62Ta_0.26□)Al₂(Al_0.98□)₂(Al_0.94□)₂O₁₂](O,OH,□)_2.65, corresponds to the electron-probe analysis data. The statistical replacement of (Si,As)O₄ tetrahedra by pyramidal [SbO₃] groups is confirmed. The X-ray diffraction spectra of holtite I are compared with those of holtite II.     

Crystal structures of K-and Cs-exchanged forms of zorite

The crystal structures of K-and Cs-exchanged forms of zorite were studied by X-ray diffraction and IR spectroscopy: K_4.75Na_1.82[Ti(Ti_0.79Nb_0.20)₄Si₁₂O₃₄(O,OH)_5.2] × 10.62 H₂O (sp. gr. Cmmm, R= 0.0481 for 516 independent reflections) and Cs_4.34Na_1.90[Ti(Ti_0.80Nb_0.18)₄Si₁₂O₃₄(O,OH)₅] × 5.37 H₂O (sp. gr. Cmmm, R = 0.0285 for 621 independent reflections). Both structures retain the mixed polyhedral framework of zorite: Na₆Ti(Ti,Nb)₄(Si₆O₁₇)₂(O,OH)₅ × nH₂O, where n ～ 11. It is shown that the positions of the atoms located in the cavities of the frameworks of these compounds differ from those in the structures of zorite and its synthetic analogs.     

Erratum to:Mineralogical Crystallography: Look in the Past,New Trends,and Highlights

Crystallography Reports - An Erratum to this paper has been published: https://doi.org/10.1134/S1063774521340071     

Crystal Chemistry of Cation-Exchanged Forms of Hilairite: New Experimental Data and Chemical Composition—Structure—Genesis Relations

Crystallography Reports - The crystal structures of cation-exchanged forms of hilairite obtained at 90°C (Pb- and Rb- exchanged samples (Pb0.60Na0.40H0.40ZrSi3O9 ? 3H2O [R32, a =...     

Crystal structure of new decaborate Na<Subscript>2</Subscript>Ba<Subscript>2</Subscript>[B<Subscript>10</Subscript>O<Subscript>17</Subscript>(OH)<Subscript>2</Subscript>]

The crystal structure of a newly synthesized compound Na₂Ba₂[B₁₀O₁₇(OH)₂] has been determined (Syntex \(P\bar 1\) diffractometer, MoK_α radiation, 1784 crystallographically nonequivalent reflections, anisotropic approximation, R = 1.7%). The parameters of the monoclinic unit cell are a = 11.455(7), b = 6.675(4), c = 9.360(7) Å, β = 93.68(5)°, Z = 2, sp. gr. C2. The structure consists of double pseudohexagonal layers built by BO₄-tetrahedra and BO₃-triangles forming three-membered rings of two mutually orthogonal orientations. The neighboring layers along the [001] direction are bound by Na-polyhedra and hydrogen bonds with participation of OH groups. The interlayer tunnels along the [100] direction are filled with columns of Ba-polyhedra. The crystallochemical characteristics of a number of synthetic Ba-borates (to which the structure of new decaborate is related) are considered in terms of borate building blocks singled out in the structure.     

Duality, triality and complementary extremum principles in non-convex parametric variational problems with applications

This paper presents a reasonably complete duality theory anda nonlinear dual transformation method for solving the fullynonlinear, non-convex parametric variational problem inf{W(u- µ) - F(u)}, and associated nonlinear boundary valueproblems, where is a nonlinear operator, W is either convexor concave functional of p = u, and µ is a given parameter.Detailed mathematical proofs are provided for the complementaryextremum principles proposed recently in finite deformationtheory. A method for obtaining truly dual variational principles(without a dual gap and involving the dual variable p* of uonly) in n-dimensional problems is proposed. It is proved thatfor convex W(p), the critical point of the associated LagrangianL_µ(u, p*) is a saddle point if and only if the so-calledcomplementary gap function is positive. In this case, the systemhas only one dual problem. However, if this gap function isnegative, the critical point of the Lagrangian is a so-calledsuper-critical point, which is equivalent to the Auchmuty'sanomalous critical point in geometrically linear systems. Wediscover that, in this case, the system may have more than oneprimal-dual set of problems. The critical point of the Lagrangianeither minimizes or maximizes both primal and dual problems.An interesting triality theorem in non-convex systems is proved,which contains a minimax complementary principle and a pairof minimum and maximum complementary principles. Applicationsin finite deformation theory are illustrated. An open problemleft by Hellinger and Reissner is solved completely and a purecomplementary energy principle is constructed. It is provedthat the dual Euler-Lagrange equation is an algebraic equation,and hence, a general analytic solution for non-convex variational-boundaryvalue problems is obtained. The connection between nonlineardifferential equations and algebraic geometry is revealed.     

Crystal structure of Pb-exchanged form of zorite

The crystal structure of a Pb-exchanged form of zorite is studied by X-ray diffraction: Pb_3.95(Ca_0.1Sr_0.05)[Ti(Ti_0.80Nb_0.20)₄Si₁₂O₃₈(OH)] · 9.52H₂O (sp. gr. Cmmm, R = 0.0530 for 680 independent reflections). The structure retains the mixed polyhedral framework of zorite, Na₆[Ti(Ti,Nb)₄(Si₆O₁₇)₂(O,OH)₅] · 11H₂O. This framework is composed of xonotlite-like [Si₆O₁₇] ribbons linked to each other by columns of vertex-sharing (Ti,Nb)O₆ octahedra and isolated TiO₅ half-octahedra. Lead atoms in the Pb-exchanged form occupy one site, unlike Cs cations in the Cs-exchanged form of zorite, which are strongly disordered and partially occupy eight positions. The position of Pb²⁺ cations corresponds to the Na(2) position in the zorite structure, the Sr position in the Sr-exchanged form of ETS-4, and the K position in the K-exchanged form and is similar to the position of the water molecule W(3) in the structure of the Cs-exchanged form of zorite.     

Crystal structure of zdenekite NaPbCu5(AsO4)4Cl · 5H2O

The crystal structure of the mineral zdenekite NaPbCu₅(AsO₄)₄Cl · 5H₂O was established (Bruker SMART CCD diffractometer, synchrotron radiation, λ = 0.6843Å, R = 0.096 for 1356 reflections). Single-crystal X-ray diffraction study demonstrated that zdenekite belongs to the monoclinic system with the unit-cell parameters a = 10.023(7) Å, b = 19.55(1) Å, c = 10.023(6) Å, β = 90.02(1)°, sp. gr. P2₁/n, Z = 4. The structure consists of polyhedral layers parallel to the (010) plane. These layers are formed by Cu? polyhedra φ = O, Cl, H₂O) and AsO₄ tetrahedra. Distorted Na octahedra and Pb 7-vertex polyhedra and H₂O molecules coordinated to these metal atoms are located between the layers.