Quasi crystals: Studies of stability and phason relaxation

This dissertation is in two distinct parts. In chapter I the author considers a simple model of solidification based on Landau theory and investigates whether this model can have stable or metastable quasicrystalline solutions. The model is that proposed by Kalugin, Kitaev, and Levitov with an additional local quartic term in the free energy. In this case, the body-centered cubic (bcc) crystal is the global minimum. He assesses the stability of the quasicrystalline solutions and shows that they are not even metastable, being unstable against a collapse to the bcc crystal. In chapter II he proposes a simple model for phason dynamics in quasicrystals. Phason shifts in the Penrose tiling model of quasicrystals appear as flips of rows of tiles, known as worms. When worms cross one another a hierarchy is established in which some of the worms cannot flip until others have. A complex set of constraints on worm flips is thereby introduced by the intricate pattern of worm crossings in quasicrystalline tilings. He introduces a simple model of interacting sets of one-dimensional Ising chains that mimics this set of constraints and study the possible consequences of these constraints for phason dynamics and the relaxation of phason strain in quasicrystals