Proposal: DMS-0204823, DMS-0204845 Principal Investigators: Benji Fisher, David Rabson
A B S T R A C T
Fisher and Rabson are classifying the symmetry types of quasicrystals in two and three dimensions and studying the physical consequences of these symmetries. The classification generalizes the work begun in the nineteenth century on space groups of crystals. The investigators follow the Fourier-space approach to crystallography, which has the advantage of avoiding space groups in higher dimensions. Thus the classification starts from the known list of finite subgroups of the orthogonal groups O(2) and O(3). They introduce techniques of group cohomology, some of which have long been used in "direct space," to the Fourier-space formulation. These techniques lead to both theoretical simplifications and efficient computational methods. One important application of these ideas is the description of certain gauge invariants in terms of group homology. The two simplest types of homology class are connected with known physical phenomena: systematic extinctions in diffraction patterns and crossing of electronic bands ("band sticking"). The next simplest type first occurs in a rank-five tetragonal modulated crystal and should be connected with some similar phenomenon. Tiling models are produced, and the ideas are also extended to magnetic and color groups.
Crystallography underlies and informs much of physics, chemistry, and geology. The present research has applications to recent experiments in liquid crystals (related to the popular LCD displays on wristwatches and other electronic equipment), plasmas, and modulated crystals, highly symmetric systems that cannot be described by the classical theory of crystals. Ever since 1784, when the French abbot Rene Just Hauy deduced the microscopic structure of crystals, it had been believed impossible for a crystal to have the symmetries of an icosahedron (or a soccer ball). Precisely 200 years later, such materials, called "quasicrystals," were discovered, and much research has ensued into their properties. Quasicrystals possess strange electronic and physical properties and have already found application as high-quality, non-stick coatings on electrosurgical blades. The present research aims to classify the symmetry types of crystals and to study physical properties associated with these symmetries.
This project is being funded jointly by the Division of Mathematical Sciences and the Division of Materials Research.
