Ein neues Syntheseverfahren f�r die Wolframbronze Cs0,29WO3

Theoretical and computational improvements to the algebraic method for discovering cooperative rigid-unit modes

A linear-algebraic algorithm for identifying rigid-unit modes in networks of interconnected rigid units has recently been demonstrated. This article presents a series of enhancements to the original algorithm, which greatly improve its conceptual simplicity, numerical robustness, computational efficiency and interpretability. The improvements include the efficient isolation of constraints, the observation of variable-block separability, the use of singular value decomposition and a quantitative measure of solution inexactness.

An algebraic approach to cooperative rotations in networks of interconnected rigid units

Crystalline solids consisting of three-dimensional networks of interconnected rigid units are ubiquitous amongst functional materials. In many cases, application-critical properties are sensitive to rigid-unit rotations at low temperature, high pressure or specific stoichiometry. The shared atoms that connect rigid units impose severe constraints on any rotational degrees of freedom, which must then be cooperative throughout the entire network. Successful efforts to identify cooperative-rotational rigid-unit modes (RUMs) in crystals have employed split-atom harmonic potentials, exhaustive testing of the rotational symmetry modes allowed by group representation theory, and even simple geometric considerations. This article presents a purely algebraic approach to RUM identification wherein the conditions of connectedness are used to construct a linear system of equations in the rotational symmetry-mode amplitudes.

Octahedral tilting in the tungsten bronzes

Possibilities for `simple' octahedral tilting in the hexagonal and tetragonal tungsten bronzes (HTB and TTB) have been examined, making use of group theory as implemented in the computer programISOTROPY. For HTB, there is one obvious tilting pattern, leading to a structure in space groupP63/mmc. This differs from the space groupP63/mcmfrequently quoted from X-ray studies – these studies have in effect detected only displacements of the W cations from the centres of the WO6octahedra. The correct space group, taking account of both W ion displacement and the octahedral tilting, isP6322 – structures in this space group and matching this description have been reported. A second acceptable tilting pattern has been found, leading to a structure inP6/mmmbut on a larger `2 × 2 × 2' unit cell – however, no observations of this structure have been reported. For TTB, a search at the special points of the Brillouin zones revealed only one comparable tilting pattern, in a structure with space-group symmetryI4/mon a `21/2 × 21/2by 2' unit cell. Given several literature reports of larger unit cells for TTB, we conducted a limited search along the lines of symmetry and found structures with acceptable tilt patterns inBbmmon a `21/22 × 21/2 × 2' unit cell. A non-centrosymmetric version has been reported in niobates, inBbm2 on the same unit cell.