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.
Original language | English |
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Pages (from-to) | 357-372 |
Number of pages | 16 |
Journal | Acta Crystallographica Section A: Foundations and Advances |
Volume | 74 |
Issue number | 4 |
DOIs | |
Publication status | Published - 1 Jul 2018 |
Funding
We thank the Institut Laue–Langevin for funding A. G. Hart’s internship. Further, we thank Joellen Preece for advice on Markov theory and probability, as well as Henry Fischer for guidance on PDF analysis. Further thanks are owed to the anonymous reviewers for their knowledgeable and detailed suggestions which helped to improve the manuscript considerably. We extend our gratitude to Chris Cook, Michael Green, Matthew Hill, Daniel Hoare and Lucy Roche for offering corrections and criticism.
Keywords
- Aperiodic crystals
- Chaotic crystallography
- Markov chains
- Markov models
ASJC Scopus subject areas
- Structural Biology
- Biochemistry
- General Materials Science
- Condensed Matter Physics
- Physical and Theoretical Chemistry
- Inorganic Chemistry