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

• Published in: Acta crystallographica. Section A, Foundations and advances Foundations and advances, 2018-09, Vol.74 (5), p.408-424
• Main Authors: Campbell, Branton, Howard, Christopher J., Averett, Tyler B., Whittle, Thomas A., Schmid, Siegbert, Machlus, Shae, Yost, Christopher, Stokes, Harold T.
• Format: Article
• Language: English
• Subjects: Algebra; cooperative rotations; Crystals; group theory; High pressure; Low temperature; perovskites; quartz; rigid‐unit modes; Stoichiometry; Symmetry; symmetry modes; tungsten bronzes
• ISSN: 2053-2733; 0108-7673; 2053-2733
• DOI: 10.1107/S2053273318009713

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. This article presents an algebraic approach to the analysis of cooperative rotations in networks of interconnected rigid units wherein the geometric constraints of connectedness reduce, in the small rotation‐angle limit, to a homogeneous linear system of equations. The approach is illustrated by application to perovskites, to quartz, and to the hexagonal and tetragonal tungsten bronzes.