Mathematics Colloquia and Seminars

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A significant challenge in condensed matter science is the discovery and characterization of structure in complex, disordered materials directly from their x-ray diffraction spectra. A broad class of layered materials, called polytypes, can exist in a wide range of both ordered and disordered stacking configurations. Examples of polytypes include micas and kaolins, and substances of technological importance, such as the wide band gap semiconductor silicon carbide. While standard crystallographic techniques can identify most ordered stacking structures, understanding the diffuse diffraction spectra arising from disordered specimens has proven more challenging. In this talk, I will briefly discuss the phenomenon of polytypism at a level suitable for a general scientific audience. I will introduce a novel technique for detecting and characterizing disordered stacking structure directly from x-ray diffraction spectra. The resulting expression for the structure is a directed graph. I will demonstrate the technique on x-ray diffraction spectra obtained from zinc sulphide crystals and show how it provides insight into the complex stacking structure of these crystals as well as allows for the calculation of material properties of physical import. The techniques introduced here are quite general, and are applicable to the problem of inferring structure (either spacial or temporal) given an experimental signal in the form of a power spectrum.