∊-Machine spectral reconstruction theory: a direct method for inferring planar disorder and structure from X-ray diffraction studies

Abstract

In previous publications [Varn et al. (2002). Phys. Rev. B, 66, 174110; Varn et al. (2007). Acta Cryst. B63, 169-182] we introduced and applied a new technique for discovering and describing planar disorder in close-packed structures directly from their diffraction patterns. Here, we provide the theoretical development behind those results, adapting computational mechanics to describe one-dimensional structure in materials. We show that the resulting statistical model of the stacking structure - called the ε-machine - allows the calculation of measures of memory, structural complexity and configurational entropy. The methods developed here can be adapted to a wide range of experimental systems in which power spectra data are available.

Pairwise correlations in layered close-packed structures

“…While simulation studies (Varn & Crutchfield, 2004) and "-machine spectral reconstruction (Varn et al, 2002(Varn et al, , 2007(Varn et al, , 2013a express stacking structure in terms of the Hä ggmachine, our theoretical methods developed in x4 require that it be re-expressed in terms of the ABC-machine. This section is devoted to detailing techniques to do just that.…”

“…An SSC is defined analogous to a causal state cycle (Varn et al, 2013a) on an "-machine as a 'finite, closed, nonself-intersecting, symbol-specific path' along the graph. The winding number W for an SSC on a Hä ggmachine is similar to the parameter Á previously defined by Yi & Canright (1996) and the cyclicity (C) (Dornberger-Schiff & Schmittler, 1971) for a polytype of a CPS.…”

“…In the following, we demonstrate how pairwise correlation functions can be calculated either analytically or to a high degree of numerical certainty for an arbitrary HMM and, thus, for an arbitrary "-machine. Previous researchers often calculated pairwise correlation functions for particular realizations of stacking configurations (Berliner & Werner, 1986;Kabra & Pandey, 1988;Shrestha & Pandey, 1997;Estevez-Rams, Martinez et al, 2001;Varn et al, 2013a) or from analytic expressions constructed for particular models (Tiwary & Pandey, 2007;Estevez-Rams et al, 2008;Varn & Crutchfield, 2004). The techniques developed here, however, are the first generally applicable methods that do not rely on samples of a stacking sequence.…”

“…Disordered layered materials are often studied via the pairwise correlations between layers, as these correlations are experimentally accessible from the Fourier analysis of a diffraction pattern (Estevez-Rams, Martinez et al, 2001;Estevez-Rams, Penton-Madrigual et al, 2001;Varn et al, 2002Varn et al, , 2013a, or directly from simulation studies (Kabra & Pandey, 1988Shrestha & Pandey, 1996a,b, 1997Shrestha et al, 1996;Varn & Crutchfield, 2004). Such studies yield important insights into the structural organization of materials.…”

“…Chaotic crystallography provides a unified platform not only to calculate physical quantities of interest but also to give insight into their physical structure. For layered materials, where there is but one axis of interest, namely the organization along the stacking direction, the appropriate mathematical framework for this formalism has been identified (Varn et al, 2002(Varn et al, , 2007(Varn et al, , 2013a as computational mechanics (Crutchfield & Young, 1989;Crutchfield, 2012). The mathematical entity that gives a compact, statistical description of the disordered material (along its stacking direction) is its "-machine, a kind of hidden Markov model (HMM) (Rabiner, 1989;Elliot et al, 1995).…”