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

Abstract

We present a mathematical framework 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 form 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 tools of Fourier analysis and functional analysis, we develop expressions for the scattering cross sections of a broad class of disordered materials.