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A mathematical framework is presented to represent turbostratic disorder in materials like carbon blacks, smectites and twisted n-layer graphene. In particular, the set of all possible disordered layers, including rotated, shifted and curved layers, forms a stochastic sequence governed by a hidden Markov model. The probability distribution over the set of layer types is treated as an element of a Hilbert space and, using the tools of Fourier analysis and functional analysis, expressions are developed for the scattering cross sections of a broad class of disordered materials.

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.