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Hart AG, Hansen TC, Kuhs WF. A Markov theoretic description of stacking-disordered aperiodic crystals including ice and opaline silica. Acta Crystallogr A Found Adv 2018; 74:357-372. [DOI: 10.1107/s2053273318006083] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2017] [Accepted: 04/20/2018] [Indexed: 11/10/2022]
Abstract
This article reviews the Markov theoretic description of one-dimensional aperiodic crystals, describing the stacking-faulted crystal polytype as a special case of an aperiodic crystal. Under this description the centrosymmetric unit cell underlying a topologically centrosymmetric crystal is generalized to a reversible Markov chain underlying a reversible aperiodic crystal. It is shown that for the close-packed structure almost all stackings are irreversible when the interaction reichweite
s > 4. Moreover, the article presents an analytic expression of the scattering cross section of a large class of stacking-disordered aperiodic crystals, lacking translational symmetry of their layers, including ice and opaline silica (opal CT). The observed stackings and their underlying reichweite are then related to the physics of various nucleation and growth processes of disordered ice. The article discusses how the derived expressions of scattering cross sections could significantly improve implementations of Rietveld's refinement scheme and compares this Q-space approach with the pair-distribution function analysis of stacking-disordered materials.
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Riechers PM, Varn DP, Crutchfield JP. Pairwise correlations in layered close-packed structures. Acta Crystallogr A Found Adv 2015; 71:423-43. [DOI: 10.1107/s2053273315005264] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2014] [Accepted: 03/14/2015] [Indexed: 11/10/2022]
Abstract
Given a description of the stacking statistics of layered close-packed structures in the form of a hidden Markov model, analytical expressions are developed for the pairwise correlation functions between the layers. These may be calculated analytically as explicit functions of model parameters or the expressions may be used as a fast, accurate and efficient way to obtain numerical values. Several examples are presented, finding agreement with previous work as well as deriving new relations.
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Varn DP, Canright GS, Crutchfield JP. ∊-Machine spectral reconstruction theory: a direct method for inferring planar disorder and structure from X-ray diffraction studies. Acta Crystallogr A 2013; 69:197-206. [DOI: 10.1107/s0108767312046582] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2012] [Accepted: 11/11/2012] [Indexed: 11/10/2022] Open
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