G. Kasner

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.

Atomic decoration of Katz–Gratias–de Boissieu–Elser model applied to the surface structure of i-Al–Pd–Mn

Abstract The Katz–Gratias–de Boissieu–Elser (KGdeBE) model for icosahedral quasicrystals is studied in order to provide an atomic bulk model for i-Al–Pd–Mn and i-Al–Cu–Fe. We interpret the KGdeBE model as the tiling T ∗(2 F ) decorated by Bergman and Mackay polytopes. Using information stemming from the inflation properties of the tiling T ∗(2 F ) (arrows on the edges of the tiles), we propose rules to make a unique choice between split positions and derive the corresponding vertex-windows. The resulting bulk model is used to provide atomic models of the five-, three-, and twofold surfaces of i-Al–Pd–Mn, as well as the terrace structure found in these materials. In order to stimulate new LEED calculations for these surfaces, we make the data sets (ranging up to some 10 6 atoms) public.

The Efficiency of Delone Coverings of the Canonical Tilings Τ*(a4) and Τ*(d6)

This chapter is devoted to the coverings of the two quasiperiodic canonical tilings [1] Τ *(a4) [2] and Τ *(d6) ≡ Τ *(2f) [3, 4], obtained by projection [3] from the root lattices A4 and D6, respectively [5]. The projection from the “high-dimensional lattice” L onto the space of a quasiperiodic tiling, called “parallel space” and denoted by \( \mathbb{E}_\parallel \), is defined by the representations of noncrystallographic groups [1]. We consider a canonical quasiperiodic tiling Τ *(l) [1] projected from a lattice L onto parallel space \( \mathbb{E}_ \bot \) , whose coding window in perpendicular space \( \mathbb{E}_\parallel \) is a projected Voronoi cell [5, 1] V ⊥, and whose tiles in parallel space are projected boundaries of Delone cells [1, 5] X *‖. These tiles are, in the case of Τ*(a4), two golden triangles with edge lengths ➁, a standard length parallel to a 2-fold axis of an icosahedron, and τ➁, where τ = (1 + √5)/2. The “2-fold direction” is defined only in 3-dimensional space, and indeed the decagonal tiling Τ *(a4) can be seen as a subtiling of the icosahedral tiling Τ *(d6) [3, 4]. The tiles of Τ *(d6) are six golden tetrahedra [3, 6] of edge lengths ➁ and τ ➁, as above. The tiles are coded in perpendicular space \( \mathbb{E}_ \bot \) by corresponding dual Voronoi boundaries projected onto \( \mathbb{E}_ \bot \). We denote these projected boundaries by X⊥. The codings X⊥ are, in the case of Τ*(a4) two rhombuses of the same shape as the prototiles of the Penrose tiling P2, with angles of 36° and 72° [2, 7]. In the case of Τ *(d6) the codings X⊥ are acute and obtuse rhombohedra and four pyramids [3] (see also Fig. 5.18).

LATTICE DYNAMICS OF AN F-TYPE ICOSAHEDRAL QUASICRYSTAL

By using a matrix-continued fraction approach we calculate the density of vibrational states (DOS) of a tiling model based on the 6D face centered lattice. Parameters of the Lennard-Jones pair interaction are obtained from relaxation calculations. With a ternary decoration the tiling was found to be stable. The DOS was approximated by weighting the local DOS (LDOS) of the allowed vertex configurations by their relative frequencies in the infinite tiling. These frequencies were obtained by using extended deflation rules. Results of this approach are compared to exact finite cluster calculations and an embedded cluster approach. We find the DOS to consist of a single band (in our resolution) and a rich structure at higher frequencies.

Electronical properties of quasicrystalline model systems: influence of phason flips

Abstract The influence of single phason flips on the electronic properties of quasicrystalline model systems is investigated instead of using random tiling models. In a tight binding model of the Penrose lattice there are seven different types of possible phason flips according to the involved vertexstars. Tolerating a random number of flips of only one type simultaneously and avoiding multiple flips the spectrum, the eigenstates and the dc-conductance of this model system as a function of the flip type are calculated. Every flip in the lattice leads to a defect state localized at the flipped atom sites with an eigenenergy outside the spectrum of the unflipped lattice. All types of flips result in smoothing out the density of states and the very strong fluctuations of the conductance in most of the energy regions. But the strength of this effect at a given energy depends strongly on the type of the flips in the lattice. A certain quasicrystalline pattern showing resonance at this energy can be more or less influenced by the flips resulting in a strong or weak effect to the conductance. So, the investigation of the conductance under the influence of special flips offers the possibility to study the relevance of several local configurations to the electronic properties at a given energy.

Vibrational density of states, dynamical structure factor and eigenstates of icosahedral models

Abstract Extending our previous studies on icosahedral models, we numerically calculate the integrated density of states (IDOS), the dynamical structure factor, and special eigenstates of the ABCK tiling decorated by Danzer and Talis. We find a broadening of the structure factor at frequencies where the IDOS exhibits kinks. These kinks arise from a strong increase of the imaginary part of the Greens function (strong damping of plane-wave modes) and indicate localization. The centers of the localized vibrations mainly form icosahedra. The occurrence of distinct regimes in the IDOS may be related to the hierarchical structure of the tiling. The flatness of the optical branches in the dynamical structure factor implies strong localization of optical modes.