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Point Symmetry (point + symmetry)
Selected AbstractsAnalysis of a class of potential Korteweg-de Vries-like equationsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2010R. M. Edelstein Abstract We analyze a class of third-order evolution equations, i.e. ut = f(x, ux, uxx) uxxx+g(x, ux, uxx) via the method of preliminary group classification. This method is a systematic means of analyzing the equation for symmetries. We find explicit forms of f and g, which allow for a larger dimensional Lie algebra of point symmetries. Copyright © 2009 John Wiley & Sons, Ltd. [source] Theoretical investigation of the optical and EPR spectra for ytterbium(III) in strontium tungstate crystalPHYSICA STATUS SOLIDI (B) BASIC SOLID STATE PHYSICS, Issue 2 2010Y. C. Zhong Abstract A theoretical method for studying the optical spectra and electron paramagnetic resonance (EPR) parameters of Yb3+ ions in SrWO4 crystal is presented. Using the crystal-field theory, the formulas relating the optical spectra and EPR parameters are derived. Based on the theoretical formulas, the 14×14 full energy matrices for 4f13 configuration ions in a D2d point symmetry are constructed. By diagonalizing the full energy matrices, the optical spectra and EPR data of Yb3+ in SrWO4 crystal have been investigated. The theoretical results are in good agreement with the experimental values. Furthermore, the crystal-field parameters for SrWO4 have been determined in this work. [source] The maximal axial parameters in equivalent parametrizations of high symmetry crystal-field HamiltoniansPHYSICA STATUS SOLIDI (B) BASIC SOLID STATE PHYSICS, Issue 2 2007Jacek Mulak The cover picture represents a fingerprint of crystal-field parametrization, that is a map of the normalized crystal-field parameter B40 for the Pr3+ ion in Pr2CuO4 of C4v point symmetry according to the reference frame orientation (based on the data by Riou et al. [2]). This map is equivalent to the angle dependence of the k = 4 component of the crystal-field Hamiltonian. The contour lines are the equipotential lines. The picture relates to the work by Jacek Mulak, Maciej Mulak, and Ryszard Gonczarek [1]. The first author is a graduate in Chemistry (Wroclaw University of Technology, 1962) and Mathematics (University of Wroclaw, 1968), now holding the position of a professor at the Institute of Low Temperature and Structure Research, Polish Academy of Sciences in Wroclaw, Poland. He specializes in the crystal-field theory and its application to magnetism and spectroscopy. He embarked on his scientific career as an assistant professor to Professor W,odzimierz Trzebiatowski (1906,1982). The paper is then a tribute to Professor W,odzimierz Trzebiatowski, the founder of the physico-chemical school at Wroclaw's Academic Center. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Structures of the ZrZn22 family: suprapolyhedral nanoclusters, methods of self-assembly and superstructural orderingACTA CRYSTALLOGRAPHICA SECTION B, Issue 3 2009G. D. Ilyushin A combinatorial topological analysis is carried out by means of the program package TOPOS4.0 [Blatov (2006), IUCr Comput. Commun. Newsl.7, 4,38] and the matrix self-assembly is modeled for crystal structures of the ZrZn22 family (space group , Pearson code cF184), including the compounds with superstructural ordering. A number of strict rules are proposed to model the crystal structures of intermetallics as a network of cluster precursors. According to these rules the self-assembly of the ZrZn22 -like structures was considered within the hierarchical scheme: primary polyhedral cluster , zero-dimensional nanocluster precursor , one-dimensional primary chain , two-dimensional microlayer , three-dimensional microframework (three-dimensional supraprecursor). The suprapolyhedral cluster precursor AB2X37 of diameter ,,12,Å and volume ,,350,Å3 consists of three polyhedra (one AX16 of the point symmetry and two regular icosahedra BX12 of the point symmetry); the packing of the clusters determines the translations in the resulting crystal structure. A novel topological type of the two-dimensional crystal-forming 4,4-coordinated binodal net AB2, with the Schläfli symbols 3636 and 3366 for nodes A and B, is discovered. It is shown that the ZrZn22 superstructures are formed by substituting some atoms in the cluster precursors. Computer analysis of the CRYSTMET and ICSD databases shows that the cluster AB2X37 occurs in 111 intermetallics belonging to 28 structure types. [source] Quantitative description of the tilt of distorted octahedra in ABX3 structuresACTA CRYSTALLOGRAPHICA SECTION B, Issue 2 2007Rafael Tamazyan A description of the tilt of octahedra in ABX3 perovskite-related structures is proposed that can be used to extract the unique values for the tilt parameters ,, , and , of ABX3 structures with regular and distorted octahedra up to the point symmetry , from atomic coordinates and lattice parameters. The geometry of the BX6 octahedron is described by three B,X bond lengths (r1, r2, r3) and three X,B,X bond angles (,12, ,13 and ,23) or alternatively by a local strain tensor together with an average B,X bond length. Connections between the proposed method and Glazer's tilt system are discussed. The method is used to analyze structural transformations of I2/c, Pbnm and Immm structures. The proposed description allows the analysis of group,subgroup relations for the ABX3 structures with distorted octahedra, in terms of octahedral deformations and tilting. The method might also be of interest in the study of the phase transitions in the family of ABX3 structures. [source] Trinitratobis{N,-[1-(2-pyridyl)ethylidene]isonicotinohydrazide}cerium(III): a three-dimensional cerium,organic supramolecular structureACTA CRYSTALLOGRAPHICA SECTION C, Issue 7 2009Ying-Ying Zhang The CeIII ion in the title complex, [Ce(NO3)3(C13H12N4O)2], is 12-coordinated by six chelating nitrate O atoms and six donors (2 O and 4 N atoms) of two N,-[1-(2-pyridyl)ethylidene]isonicotinohydrazide ligands, exhibiting a bicapped pentagonal-antiprism-type coordination geometry. The title complex possesses C2 point symmetry and is located on a twofold crystallographic axis. Each molecule is linked with four surrounding molecules by four N,H...N hydrogen bonds, resulting in an extended two-dimensional layer parallel to the ab plane, while ,,, interactions between pyridine rings from neighboring complex molecules connect the two-dimensional layers into a three-dimensional cerium,organic supramolecular structure. [source] Li12Cu16+xAl26,x (x = 3.2): a new intermetallic structure typeACTA CRYSTALLOGRAPHICA SECTION C, Issue 8 2008Volodymyr Pavlyuk The new ternary lithium copper aluminide, Li12Cu16+xAl26,x (x = 3.2), dodecalithium nonadecacopper tricosaaluminide, crystallizes in a new structure type with space group P4/mbm. Among nine independent atomic positions, two Al (one of which is statistically disordered with Cu) and three Li atoms have point symmetry m.2m, two statistically disordered Al/Cu atoms are in m.. sites, one Al atom is in a 4/m.. site and one Cu atom occupies a general site. The framework of Li12Cu16+xAl26,x consists of pseudo-Frank,Kasper polyhedra enclosing channels of hexagonal prisms occupied by Li atoms. The crystallochemical peculiarity of this new structure type is discussed in relation to the derivatives from Laves phases (LiCuAl2 and Li8Cu12+xAl6,x) and to the well known CaCu5 structure. [source] The new pentaborate Na3SrB5O10ACTA CRYSTALLOGRAPHICA SECTION C, Issue 7 2008Li Wu Sodium strontium pentaborate, Na3SrB5O10, maintains the same, previously unobserved, structure type at 200, 250 and 293,K. The fundamental building units are anionic [B5O10]5, groups distorted from mm2 point symmetry. The Sr atoms are eightfold coordinated by O atoms, forming trigonal dodecahedra. The Na atoms appear in three crystallographically different environments. The present single-crystal results correct a previous report in which a monoclinic cell was deduced for this compound on the basis of powder diffraction data. The structure of the title compound is discussed in the crystalochemical context of other borates with the same formula type. Although the unit cell of the present compound is similar to that determined in a previous study of the analogous Ca-containing compound, this study demonstrates that the structures of the two are different. These novel alkali,alkaline earth borates are considered as potential host materials for optical applications (fluorescence materials or phosphors). [source] rac -9-Ethyl-12a-hydroxytetradecahydrotriphenylene-1,5(2H,4bH)-dione: stabilization of a new isomer of a functionalized perhydrotriphenylene through a tandem Michael addition,aldol reactionACTA CRYSTALLOGRAPHICA SECTION C, Issue 6 2008Luis Arturo García The title compound, C20H30O3, is a new functionalized perhydrotriphenylene derivative formed via a tandem Michael addition,aldol reaction. The structural study reveals that the system of fused rings approximates a C2 point symmetry, with trans,cis,cis ring junctions, while highly symmetric all- trans perhydrotriphenylene, previously characterized, approximates a D3 symmetry. The perhydrotriphenylene nucleus of the title compound corresponds to the third stable stereoisomer isolated for this polycyclic system. Considering that the Cs isomer was obtained recently through a similar tandem reaction, a general strategy is proposed which may help to obtain other stable stereoisomers of perhydrotriphenylene. [source] A new whitlockite, Ca8.42Na1.16V(PO4)7ACTA CRYSTALLOGRAPHICA SECTION C, Issue 2 2006Alexander A. Tsirlin Single crystals of a new complex phosphate, calcium sodium vanadium phosphate, Ca8.42Na1.16V(PO4)7, have been grown from a melt under an inert atmosphere. The crystal structure has rhombohedral (R3c) symmetry and belongs to the whitlockite structure type. Vanadium(III) ions occupy nearly regular octahedral sites (M5 with 3 point symmetry), which share corners with six PO4 tetrahedra to form isolated units. The calcium ions occupy eight- and nine-coordinated sites. The sodium ions partially occupy one octahedral position and share one nine-coordinated position with a Ca atom. [source] In1.06Ho0.94Ge2O7: a thortveitite-type compoundACTA CRYSTALLOGRAPHICA SECTION C, Issue 2 2004Erick A. Juarez-Arellano A new indium holmium digermanate, In1.06Ho0.94Ge2O7, with a thortveitite-type structure, has been prepared as a polycrystalline powder material by high-temperature solid-state reaction. This new compound crystallizes in the monoclinic system (space group C2/c, No. 15). The structure was characterized by Rietveld refinement of powder laboratory X-ray diffraction data. The In3+ and Ho3+ cations occupy the same octahedral site, forming a hexagonal arrangement on the ab plane. In their turn, the hexagonal arrangements of (In/Ho)O6 octahedral layers are held together by sheets of isolated diortho groups comprised of double tetrahedra sharing a common vertex. In this compound, the Ge2O7 diortho groups lose the ideal D3d point symmetry and also the C2h point symmetry present in the thortveitite diortho groups. The Ge,O,Ge angle bridging the diortho groups is 160.2,(3)°, compared with 180.0° for Si,O,Si in thortveitite (Sc2Si2O7). The characteristic mirror plane in the thortveitite space group (C2/m, No. 12) is not present in this new thortveitite-type compound and the diortho groups lose the C2h point symmetry, reducing to C2. [source] In1.08Gd0.92Ge2O7: a new member of the thortveitite familyACTA CRYSTALLOGRAPHICA SECTION C, Issue 10 2002Erick-Adrian Juarez-Arellano Indium gadolinium digermanium heptaoxide, In1.08Gd0.92Ge2O7, with a thortveitite-type structure, has been prepared as a polycrystalline powder material by a high-temperature solid-state reaction. As in the mineral thortveitite, the crystal structure belongs to the monoclinic system, with space group C2/m (No. 12). The precise structural parameters were obtained by applying the Rietveld method of refinement to the X-ray powder diffraction data. This layered structure presents, on one side, a honeycomb-like arrangement of the unique octahedral site, which is occupied randomly by In and Gd atoms, and, on the other side, sheets of isolated Ge2O7 diortho-groups made up of double tetrahedra sharing a common vertex and displaying C2h point symmetry. This compound showed a remarkable photoluminescence effect when it was irradiated with the X-ray beam during the X-ray diffraction measurements, and with the , beam during the Rutherford back-scattering spectrometry experiments employed to analyze the chemical stoichiometry. [source] | ||||||||||