Continuum elastic sphere vibrations as a model for low-lying optical modes in icosahedral quasicrystals

The nearly dispersionless, so-called "optical" vibrational modes observed by inelastic neutron scattering from icosahedral Al-Pd-Mn and Zn-Mg-Y quasicrystals are found to correspond well to modes of a continuum elastic sphere that has the same diameter as the corresponding icosahedral basic units of the quasicrystal. When the sphere is considered as free, most of the experimentally found modes can be accounted for, in both systems. Taking into account the mechanical connection between the clusters and the remainder of the quasicrystal allows a complete assignment of all optical modes in the case of Al-Pd-Mn. This approach provides support to the relevance of clusters in the vibrational properties of quasicrystals.Comment: 9 pages without figure

Icosahedral multi-component model sets

A quasiperiodic packing Q of interpenetrating copies of C, most of them only partially occupied, can be defined in terms of the strip projection method for any icosahedral cluster C. We show that in the case when the coordinates of the vectors of C belong to the quadratic field Q[\sqrt{5}] the dimension of the superspace can be reduced, namely, Q can be re-defined as a multi-component model set by using a 6-dimensional superspace.Comment: 7 pages, LaTeX2e in IOP styl

Sound modes broadening for Fibonacci one dimensional quasicrystals

We investigate vibrational excitation broadening in one dimensional Fibonacci model of quasicrystals (QCs). The chain is constructed from particles with two masses following the Fibonacci inflation rule. The eigenmode spectrum depends crucially on the mass ratio. We calculate the eigenstates and eigenfunctions. All calculations performed self-consistently within the regular expansion over the three wave coupling constant. The approach can be extended to three dimensional systems. We find that in the intermediate range of mode coupling constants, three-wave broadening for the both types of systems (1D Fibonacci and 3D QCs) depends universally on frequency. Our general qualitative conclusion is that for a system with a non-simple elementary cell phonon spectrum broadening is always larger than for a system with a primitive cell (provided all other characteristics are the same).Comment: 2o pages, 15 figure

Surface structure of i-Al(68)Pd(23)Mn(9): An analysis based on the T*(2F) tiling decorated by Bergman polytopes

A Fibonacci-like terrace structure along a 5fold axis of i-Al(68)Pd(23)Mn(9) monograins has been observed by T.M. Schaub et al. with scanning tunnelling microscopy (STM). In the planes of the terraces they see patterns of dark pentagonal holes. These holes are well oriented both within and among terraces. In one of 11 planes Schaub et al. obtain the autocorrelation function of the hole pattern. We interpret these experimental findings in terms of the Katz-Gratias-de Boisseu-Elser model. Following the suggestion of Elser that the Bergman clusters are the dominant motive of this model, we decorate the tiling T*(2F) by the Bergman polytopes only. The tiling T*(2F) allows us to use the powerful tools of the projection techniques. The Bergman polytopes can be easily replaced by the Mackay polytopes as the decoration objects. We derive a picture of ``geared'' layers of Bergman polytopes from the projection techniques as well as from a huge patch. Under the assumption that no surface reconstruction takes place, this picture explains the Fibonacci-sequence of the step heights as well as the related structure in the terraces qualitatively and to certain extent even quantitatively. Furthermore, this layer-picture requires that the polytopes are cut in order to allow for the observed step heights. We conclude that Bergman or Mackay clusters have to be considered as geometric building blocks of the i-AlPdMn structure rather than as energetically stable entities

Random Tilings: Concepts and Examples

We introduce a concept for random tilings which, comprising the conventional one, is also applicable to tiling ensembles without height representation. In particular, we focus on the random tiling entropy as a function of the tile densities. In this context, and under rather mild assumptions, we prove a generalization of the first random tiling hypothesis which connects the maximum of the entropy with the symmetry of the ensemble. Explicit examples are obtained through the re-interpretation of several exactly solvable models. This also leads to a counterexample to the analogue of the second random tiling hypothesis about the form of the entropy function near its maximum.Comment: 32 pages, 42 eps-figures, Latex2e updated version, minor grammatical change

Structure of the icosahedral Ti-Zr-Ni quasicrystal

The atomic structure of the icosahedral Ti-Zr-Ni quasicrystal is determined by invoking similarities to periodic crystalline phases, diffraction data and the results from ab initio calculations. The structure is modeled by decorations of the canonical cell tiling geometry. The initial decoration model is based on the structure of the Frank-Kasper phase W-TiZrNi, the 1/1 approximant structure of the quasicrystal. The decoration model is optimized using a new method of structural analysis combining a least-squares refinement of diffraction data with results from ab initio calculations. The resulting structural model of icosahedral Ti-Zr-Ni is interpreted as a simple decoration rule and structural details are discussed.Comment: 12 pages, 8 figure