Quasicrystals and instabilities of the two-dimensional Wigner crystal.

摘要

This work concerns two unrelated topics: quasicrystals and the two-dimensional Wigner crystal.; Despite a great deal of effort since the discovery of quasicrystals in 1984 determine their atomic structures, it remains an only partially solved problem. One approach to solving the quasicrystal structure problem is to determine the structure of a finite-unit cell crystalline approximant whose structure is related to that of the quasicrystal. The structure of the approximant can be refined with respect to experimental diffraction data. This approach is developed here and then applied to the case of icosahedral AlCuFe, where large single-grain X-ray and neutron diffraction data sets are used in comparing several different models for the structure.; A geometry problem that is also of interest in quasicrystallography is the problem: given a sphere radius and a desired quasiperiodic point group, how can one pack these spheres without overlap so as to achieve the highest packing fraction? A technique is, the "pinwheel construction" is given in detail that can increase the packing fraction of certain dodecagonal, octagonal and icosahedral sphere packings over those based on the "cut and project" method that use simpler shapes for the projection window.; The ground state of the two-dimensional electron system is a fundamental and difficult problem in physics. Of interest is the transition between the "Wigner crystal" and fluid states and the magnetic state of the crystal. This system is first investigated in the semiclassical limit where the energies of point defects are calculated. These point defects could play a role in the melting transition, although the critical density calculated here is not in agreement with published estimates of the critical density. Then the quantum electron system is considered. Trial wavefunctions for the ground state are used that are linear superpositions of "multigaussians", where a multigaussian is a Slater determinant of gaussian one-electron wavefunctions. Particular wavefunctions are chosen to evaluate various aspects of the 2D electron system. Evidence is found that the Wigner crystal may undergo a transition from an unpolarized state at low density to a ferromagnetic state at higher densities.