Quasicrystals form in alloys of aluminum with transition metals such as iron or copper. They are distinguished by their novel symmetries, such as the rotational symmetry of an icosahedron, that are forbidden in crystalline structures. Although lacking translational periodicity, quasicrystals display translational quasiperiodicity, leading to the dense collection of Bragg peaks in the diffraction pattern shown at the left. Widespread commercialization of quasicrystals awaits discovery of uses for these materials which display unusual hardness and brittleness, low coefficients of friction, and high thermal and electrical resistance. Some proposed applications include wear-resistant surface treatments for other metals in ball bearings and frying pans.

Research in the group of Michael Widom at Carnegie Mellon focusses on understanding the structure and thermodynamics of quasicrystalline alloys. Popular structural models of quasicrystals invoke Penrose tilings like the one shown at the left (taken from the WWW site of Hillman) that covers the plane with fat and thin rhombi according to a set of "matching rules" that force a particular aperiodic structure. Diffraction from such structures is nicely described in a web site maintained by Lifshitz. Widom has shown that randomly tiling the plane with the same set of shapes spontaneously generates structures of appropriate symmetry and quasiperiodicity. The configuational entropy of tiling the plane at random may help explain thermodynamic stability of real quasicrystalline alloys.

Current research in abstract tiling theory examines tilings of high rotational symmetry in collaboration with Remy Mosseri and co-workers. Possibly the limit of high rotational symmetry may prove easier to analyze than specific finite symmetries (10-fold, for example) of direct physical interest. Surprisingly, rhombus tilings are related to algorithms for sorting of lists. Counting the number of distinct tilings enumerates simultaneously the number of equivalance classes of sorting algorithms, a problem previously considered by computer scientist D.E. Knuth. Our random tiling theory implies an upper bound of log(2) for the tiling entropy per vertex, consistent with a conjecture by Knuth. Click here for a preprint on this research. The figure above displays one particular rhombus tiling from an ensemble of 18-fold rotational symmetry. The tiling (left) corresponds (examine color coding of tiles and crossings of lines) to the bubble sort on 9 element lists (right).

In addition to research on tilings, Widom collaborates with others on the problem of atomic modeling of quasicrystal structures. This includes testing interatomic potentials calculated by John Moriarty (Livermore National Lab) for aluminum-rich intermetallic alloys, and performing ab-initio total energy calculations utilizing density functional programs such as VASP and LSMS together with Nassrin Moghadam (Oad Ridge National Lab) and Yang Wang (Pittsburgh Supercomputer Center). These are applied to realistic atomic structures in collaboration with Eric Cockayne (NIST), Chris Henley (Cornell) and Marek Mihalkovic (Chemnitz). Typical atomic configurations of decagonal quasicrystals are generated by Monte Carlo simulations. Structures that resulted inspired the model of decagonal AlCoCu shown at the left. Papers and posters describing this work are available on the web.

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