This thesis describes two theoretical studies which are concerned with the formation and stability of quasicrystals. We propose a new "cluster picture" to explain the structure and formation of quasicrystals. Quasicrystal ordering is explained in terms of a small set of low-energy atomic clusters. The temperature-dependent elastic behavior of quasicrystals is also studied. Numerical evidence for a novel transition in elasticity is obtained from an extensive series of Monte Carlo simulations. The cluster picture is based on the notion that the ground state of a solid system is likely to have a structure that maximizes the density of clusters with low-energy atomic configurations. We show that the quasicrystalline order can arise naturally if we allow the overlapping of the low-energy clusters so that their densities are increased. The nature of the quasicrystals in the cluster picture depends on the choice of low-energy clusters. The ground state, as T goes to zero, can range from a perfect-tiling-like to a random-tiling-like quasicrystalline state. Quasicrystals can be stabilized either for energetic reasons (as in the perfect-tiling picture) or for entropic reasons (as in the random-tiling-picture) depending on the low-energy clusters. The second study concerns the phason elasticity which is thought to be critical in determining the thermodynamic stability of quasicrystals. We present a series of studies on the temperature-dependent elastic behavior of energetically stabilized quasicrystals and show that there is a novel phase transition from a locked to an unlocked phason elasticity. Numerical evidence for a finite-temperature phase transition in the three-dimensional decagonal quasicrystals is presented. In the low-temperature locked phase the quasicrystals remain in a highly ordered configuration. In the high-temperature unlocked phase clusters of atoms are rearranging themselves continuously as in a random-tiling-like phase. It is also demonstrated that this unlocking transition in quasicrystals may be analogous to a disordering transition in the Ising model.
Supervisor: Paul J. Steinhardt. Thesis (Ph.D. in Physics) -- Graduate School of Arts and Sciences, University of Pennsylvania, 1994. Includes bibliographical references.