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

Abstract

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. Addendum.

“…For HTB we found structures associated with irreps A þ 3 , A þ 6 and L À 2 -for detail see Tables 3 and 4, Figs. 1, 2 and 3, and supporting information in Campbell et al (2018). There were tilting patterns, particularly those associated with irrep A þ 6 , that were missed in our earlier work (Whittle et al, 2015).…”

“…There were tilting patterns, particularly those associated with irrep A þ 6 , that were missed in our earlier work (Whittle et al, 2015). For TTB we found structures associated with all of the irreps Z þ 5 , A À 5 and R 1 -see Tables 4 and 5 As explained by Campbell et al (2018), and emphasized by Phillips (2018), our new analysis rests on a linearization of equations, valid for infinitesimal angles of tilt. What happens at finite angles of tilt?…”

“…Acta Cryst. B71, 342-348] were continued in the context of a more general approach to cooperative rotations of interconnected rigid units [Campbell et al (2018). Acta Cryst.…”

“…It is for this reason that acceptable structures were missed. At this point we began collaboration with a team at Brigham Young University to devise a systematic approach to the tilting problem, the result of which has just been published (Campbell et al, 2018). In this approach we calculated the effects of tilting around any 'pivot' atom on the atoms influenced (displaced) by this tilting, these atoms being called 'passenger' atoms.…”

“…We used this new approach to re-examine the possible tilting patterns in the tungsten bronzes (Campbell et al, 2018). For HTB we found structures associated with irreps A þ 3 , A þ 6 and L À 2 -for detail see Tables 3 and 4, Figs.…”