A Markov theoretic description of stacking-disordered aperiodic crystals including ice and opaline silica

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.

Analysis of stacking disorder in ice I using pair distribution functions

“…First steps towards this goal were the use of local distortion modes for the analysis of pair distribution data of BaTiO 3 (Senn et al, 2016) and the implementation of rigid bodies and symmetry modes in the TOPAS software to reduce the degrees of freedom in the structural models (Coelho et al, 2015). The description of stacking disorder with the help of Markov chains has very recently been reviewed (Hart et al, 2018).…”

A hidden Markov model for describing turbostratic disorder applied to carbon blacks and graphene

“…The theory presented by Riechers et al (2015) and Hart et al (2018) assumes A is finite, and therefore that the probability distribution over symbols from the hidden state s 2 S is a vector v s 2 V. Further, the probability of emitting a symbol x 2 A is one of the entries of the vector v s , denoted v s (x). Every hidden state emits a symbol from the alphabet according to some distribution that depends on the hidden state.…”

“…When the state space S is finite, Hart et al (2018) argue that the Monte Carlo approach is much slower than computing the cross section explicitly using matrix operations. When the state space S is finite, Hart et al (2018) argue that the Monte Carlo approach is much slower than computing the cross section explicitly using matrix operations.…”

“…The significance of these reversible crystals is treated extensively by Ellison et al (2009) and discussed in the context of ice and opal by Hart et al (2018). These conditions hold for a crystal where the probability density of a state y following state x is equal to the probability density of state x following y, which is to say k(x, y) = k(y, x).…”

“…Carbon blacks therefore comprise a Markov chain of layers with infinite state space, and are therefore not fully understood by the analysis of chains with finite state space explored by Riechers et al (2015), Varn & Crutchfield (2016) and Hart et al (2018). However, since Shi's model allows carbon layers to be shifted both parallel and perpendicular to the basal plane by any magnitude, there are an uncountable infinity of positions that a carbon layer may find itself in.…”