Inferring planar disorder in close-packed structures via [epsilon]-machine spectral reconstruction theory: examples from simulated diffraction patterns

A previous paper detailed a novel algorithm, [epsilon]-machine spectral reconstruction theory ([epsilon]MSR), that infers pattern and disorder in planar-faulted, close-packed structures directly from X-ray diffraction patterns [Varn et al. (2013). Acta Cryst. A69, 197–206]. Here [epsilon]MSR is applied to simulated diffraction patterns from four close-packed crystals. It is found that, for stacking structures with a memory length of three or less, [epsilon]MSR reproduces the statistics of the stacking structure; the result being in the form of a directed graph called an [epsilon]-machine. For stacking structures with a memory length larger than three, [epsilon]MSR returns a model that captures many important features of the original stacking structure. These include multiple stacking faults and multiple crystal structures. Further, it is found that [epsilon]MSR is able to discover stacking structure in even highly disordered crystals. In order to address issues concerning the long-range order observed in many classes of layered materials, several length parameters are defined, calculable from the [epsilon]-machine, and their relevance is discussed.