QUASILATTICES AND QUASICRYSTALS

Quasicrystals are atomic structures that exhibit long-range quasiperiodic translational order and an orientational symmetry that is incompatible with spatial periodicity.(\u271) In this thesis, several geometrical and structural properties of quasicrystals are analyzed and it is argued that the observed icosahedral phases(\u272) are examples of quasicrystals. The emphasis in this work is on the unit-cell description of quasicrystal structure, which provides insight into the microscopic structure of quasicrystals. After a brief introduction to four techniques for constructing quasilattices (deflation, projection, the generalized dual method, and matching rules), criteria are discussed for the separation of quasilattices into classes with similar properties. Quasilattices consisting of the same unit cells with the same orientational symmetry are classified with regard to their local isomorphism properties. (The degenerate ground states of a quasicrystal correspond to quasilattices of a single local isomorphism class.) Techniques are discussed for determining the local isomorphism classes of quasilattices of a given symmetry. Tilings of 8-fold, 12- fold, and icosahedral symmetry that share several elegant and useful properties with the original Penrose tilings(\u273) are derived and exhibited. The elastic and hydrodynamic properties of quasicrystals are treated from the point of view of the unit-cell picture. Spatial varia- tions in the phonon and phason fields are shown to correspond to distortions and rearrangements of the unit cells. Linear phason strains with different transformation properties are discussed for icosahedral and pentagonal quasicrystals. Dislocations are described as topologically stable configurations of the phonon and phason fields. The energy of static phason strain is discussed and the mechanism for the relaxation of phason strain is described. Finally, the quasicrystal model is shown to provide a coherent and natural explanation of the experimental observations of icosahedral phases, and it is argued that other models currently under investiga- tion are either incorrect or incomplete. An explanation is given for the distortions observed in diffraction experiments, including the peak shapes, the shifts of peaks from ideal icosahedral positions, and the scaling of the peak widths. (\u271)D. Levine and P. J. Steinhardt, Phys. Rev. Lett 53, 2477 (1984). (\u272)See, for example, D. S. Shechtman et al., Phys. Rev. Lett. 53, 1951 (1984); P. Sainfort et al., C. R. Acad. Sci. (Paris), 301, 689 (1986). (\u273)R. Penrose, Bull. Inst. Math. and Its Appl. 10, 266 (1974)