Zorka Papadopolos

Coverings of Discrete Quasiperiodic Sets

1.1 Packing, Tiling, and Covering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Aperiodic and Quasiperiodic Systems with Long-Range Order . . . . . . . . 6 1.3 The Quasiperiodic Fibonacci Tiling and its Covering by Delone Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 1.3.1 Fibonacci Tiling and Klotz Construction . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Alternative Fundamental Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 Quasiperiodic functions on a parallel line section of E . . . . . . . . . . 9 1.3.4 Fundamental Domain Compatible with a Tiling . . . . . . . . . . . . . . . . 10 1.3.5 Linked Delone Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.6 Delone Covering of the Fibonacci Tiling . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Decagonal Voronoi Clusters and Covering of the Penrose Tiling . . . . . 11 1.5 Coverings of Aperiodic and Quasiperiodic Sets . . . . . . . . . . . . . . . . . . . . . . 14 1.5.1 Covering and Cluster Density in 2D Systems . . . . . . . . . . . . . . . . . . .14 1.5.2 Shelling of Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.3 Covering of Atomic Positions in Icosahedral Quasicrystals . . . . . .15 1.5.4 Fundamental Domains and Unit Cells for Quasiperiodic Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.5 Clusters in Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.6 Surfaces of Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.6 Perspectives on the Theory of Covering for Discrete Quasiperiodic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Symmetries of Icosahedral Quasicrystals

The icosahedral group A(5) as the point symmetry group of quasicrystals is by now well established for a variety of materials. Quasicrystals have been grown up to macroscopic scale and display polyhedral shapes with this symmetry, compare the review by Guyot, Kramer and de Boissieu 1990 [1]. The non-crystallographic icosahedral point group, considered in crystallography for local sites only, is defined more precisely through its three-dimensional (3D) faithful representation, the symmetry group of the regular icosahedron and dodecahedron. This representation requires, for the embedding into a periodic lattice, at least the dimension 6D. The embedding into a lattice and the study of 3D sections through the embedding periodic structure is one of the main theoretical tools for the analysis of ideal icosahedral quasicrystals. In this survey we shall develop some aspects of this embedding for centered hypercubic 6D lattices. The primitive hypercubic lattice has been studied before, but will be considered for comparison. A detailed analysis is given for the root lattice D 6. We hope to show that the 3D sections of this lattice display a rich geometric structure which we expect to encounter in the geometry and physics of the corresponding quasicrystals.

On the theory of collective motion in nuclei. III. From Hamiltonian dynamics to vortex-free fluidity

Abstract It is assumed that the Hamiltonian for collective motion in nuclei is invariant under the orthogonal group O(n, R ). For degenerate orbits in phase space it is shown that the classical Hamiltonian equations reduce to the equations of a vortex-free fluid with a velocity field determined by independent equations of motion.

Tiling theory applied to the surface structure of icosahedral AlPdMn quasicrystals

Surfaces of as observed in scanning tunnelling microscopy (STM) and low-energy electron diffraction (LEED) experiments show atomic terraces in a Fibonacci spacing. We analyse them in a bulk tiling model due to Elser which incorporates many experimental data. The model has dodecahedral Bergman clusters within an icosahedral tiling and is projected from the six-dimensional face-centred hypercubic lattice. We derive the occurrence and Fibonacci spacing of atomic planes perpendicular to any fivefold axis, compute the variation of planar atomic densities, and determine the (auto-) correlation functions. Upon interpreting the planes as terraces at the surface, we find quantitative agreement with the STM experiments.

Projection of the Danzer tiling

We derive the icosahedrally-symmetric octahedral tiling of Danzer(1991), denoted by T(D), locally from the tiling T(2F) of Kramer et al(1989), obtained by icosahedral projection from the root lattice D6. Moreover, we determine all windows such that the tiling T(D) can be obtained by projection from the 6D root lattice D6. We reconstruct all vertex configurations of the tiling T(D) using the tools of the projection method.

Local configurations on surfaces of icosahedral quasicrystals

The surfaces of icosahedral quasicrystals of an F-phase, orthogonal to a fivefold symmetry axis have a terrace-like appearance. On highly resolved images of these terraces one observes different local configurations. We search for possible corresponding configurations in the bulk model of atomic positions. based on the icosahedral tiling *(2 F ).

Delone covering of canonical tilings T*(D6)

Abstract We consider canonical quasiperiodic tiling projected from a lattice τ*(D6) whose window in perpendicular space is projected Voronoi cell, V┴, and whose tiles in parallel space are projected boundaries of Delone cells, X*∥ six golden tetrahedra. The tiles are coded by corresponding dual Voronoi boundaries projected to IE┴, denoted by X┴. We try to obtain the Delone covering of the tiling by projecting to the parallel space some of the Delone cells (Dh, h = a, c, b) of the D 6 lattice. We determine the coding in orthogonal space for these Delone clusters and their fillings by the tiles. The covering of the tiling by the corresponding Delone clusters is considered.

A geometric classical model of collective motion in nuclei

Abstract A collective phase space of dimension 12 is introduced to study a classical model of nuclear collective motion. The model employs the 6 components of the coordinate quadrupole and 6 corresponding generalized momenta and can be related to properties of closed-shell nuclei. Vibrational and rotational coordinates are introduced, and purely rotational solutions are studied. The model demonstrates hamiltonian non-rigid motion with a fixed shape of the nucleus. The relation between the coordinate quadrupole tensor and the ellipsoids related to the angular momentum and angular velocity is analyzed for simple forms of the collective potential.

Hilbert spaces of analytic functions and representations of the positive discrete series of Sp(6, R)

Hilbert spaces of analytic functions are constructed which carry irreducible representations of the positive discrete series of Sp(6, R).

A quantum model of collective motion in nuclei and its dequantization

Abstract Collective coherent states of Perelomov type are denned by acting with unitary operators from a representation of the symplectic group on the ground state of closed-shell nuclei. A dequantization scheme associates with quantum observables classical ones, and with the state space a phase space and a generalized classical dynamics. Applications to the nuclei 4 He, 16 O and 40 Ca are derived from microscopic interactions.