Photoelasticity in quasicrystals

The maximum number of non-vanishing and independent second order photoelastic coefficients required by the seven pentagonal and the two icosahedral point groups 5(C₅), (S₁₀), (C_5h ), m2(D_5h ), 52(D₅), 5m(C_5v ), 2m(D_5d ); 235(I), 2/m (I _h )—that describe the quasicrystals symmetry groups in two and three dimensions—is obtained. The schemes of non-vanishing and independent coefficients have been calculated and listed. Finally the results of this group-theoretical study are briefly discussed.     

Quasipatterns versus superlattices resulting from the superposition of two hexagonal patterns

We present a short review of the experimental observations and mechanisms related to the generation of quasipatterns and superlattices by the Faraday instability with two-frequency forcing. We show how two-frequency forcing makes possible triad interactions that generate hexagonal patterns, twelvefold quasipatterns or superlattices that consist of two hexagonal patterns rotated by an angle α relative to each other. We then consider which patterns could be observed when α does not belong to the set of prescribed values that give rise to periodic superlattices. Using the Swift–Hohenberg equation as a model, we find that quasipattern solutions exist for nearly all values of α. However, these quasipatterns have not been observed in experiments with the Faraday instability for $\alpha \ne \mathrm{\pi }/6$. We discuss possible reasons and mention a simpler framework that could give some hint about this problem.     

Anomalous electronic transport in quasicrystals and related complex metallic alloys

We analyse the transport properties in approximants of quasicrystals α-AlMnSi, 1/1-AlCuFe and for the complex metallic phase λ-AlMn. These phases present strong analogies in their local atomic structures and are related to existing quasicrystalline phases. Experimentally, they present unusual transport properties with low conductivities and a mix of metallic-like and insulating-like characteristics. We compute the band structure and the quantum diffusion in the perfect structure without disorder and introduce simple approximations that allow us to treat the effect of disorder. Our results demonstrate that the standard Bloch–Boltzmann theory is not applicable to these intermetallic phases. Indeed their dispersion relations are flat, indicating small band velocities, and corrections to quantum diffusion, which are not taken into account in the semi-classical Bloch–Boltzmann scheme, become dominant. We call this regime the small velocity regime. A simple relaxation time approximation to treat the effect of disorder allows us to reproduce the main experimental facts on conductivity qualitatively and even quantitatively.     

Surface layer self diffusion in icosahedral Al-Pd-Mn quasicrystals

The self diffusion of Mn and Pd in a single grain icosahedral Al_69.9Pd_20.5Mn_9.6 quasicrystal has been determined by low energy ion scattering (LEIS). The diffusion was determined by depositing different elements (Pd, Mn) on the surface and measuring the rate of change in surface composition as a function of temperature by LEIS. The surface composition was monitored over the temperature range of 355-575 K for Mn and 440-745 K for Pd and compared to model calculations to allow the activation energy for diffusion to be determined. Activation energies of 0.20 ± 0.01 eV for Mn and 0.64 ± 0.03 eV for Pd have then been measured for self diffusion in i-Al-Pd-Mn, respectively. No deviation from Arrhenius behavior was detected in the temperature range covered by the present experiments. From the low values of activation energy we propose that this range of diffusion is phason related, reflecting the specific nature of the icosahedral structure.     

Phonon spectra in one-dimensional quasicrystals

The propagation of phonons in one-dimensional quasicrystals is investigated. We use the projection method which has been recently proposed to generate almost periodic tilings of the line. We define a natural Laplace operator on these structures, which models phonon (and also tight-binding electron) propagation. The selfsimilarity properties of the spectrum are discussed, as well as some characteristic features of the eigenstates, which are neither extended nor localized. The long-wavelength limit is examined in more detail; it is argued that one is the lower critical dimension for this type of models.     

Grain growth and phason-strain field in quasicrystalline Al-Li-Cu

We report on grain growth and related structure change in single phased Al-Li-Cu quasicrystals. The icosahedral phase grains have been investigated using scanning ion microscopy and transmission electron microscopy. Regular boundaries between large grains have been observed both before and after high temperature annealing. The electron diffraction study shows that the grain growth is accompanied by a reduction of the phason-strains. The orientation relation between grains sets the 2-fold icosahedral axes parallel, and the coincidence of the planes depends on the phason strain-field. The effect of phason-strain field on these boundaries is discussed. It is proposed that the phason strain elimination can play a role in the grain growth. Received 1 February 1999 and Received in final form 12 May 1999     

Analysis of one-dimensional Fibonacci-class quasicrystals using perturbation theory

In this work, we present a new and semi-analytical method to investigate one-dimensional Fibonacci-class photonic quasicrystals from waveguiding properties point of view. Usually, the exact treatment for obtaining band diagram in these situations is very hard and the numerical methods are used. In this paper, we investigate the waveguiding properties of one-dimensional quasicrystals by using perturbation method. In this direction, the band diagram and field distribution of this structure are investigated.     

Multi-tiling and Riesz bases

Let S   be a bounded, Riemann measurable set in R^d${\mathbb{R}}^d$, and Λ be a lattice. By a theorem of Fuglede, if S   tiles R^d${\mathbb{R}}^d$ with translation set Λ, then S has an orthogonal basis of exponentials. We show that, under the more general condition that S multi-tiles  R^d${\mathbb{R}}^d$ with translation set Λ, S has a Riesz basis of exponentials. The proof is based on Meyer?s quasicrystals.     

On the perfect penrose tiling and its basic building blocks

We show that the basic building blocks of a perfect Penrose pattern (PPT) in two dimensions can be established by adding another condition to the Penrose’s original edge rules. The implications of this result are discussed in the context of recent papers by Onada, Jaric, Ronchetti and others concerning growth algorithm for PPT’s.     

Walk of a Line Defect in Quasicrystals

Quasicrystalline alloys are quasiperiodic aggregates in which local atomic changes (phason activity) influence prominently the gross deformative behavior. The effects of these rearrangements on a line defect are analyzed here. Balances of standard and phason interactions are derived along the line together with the balance of actions driving eventually it the former are found to be covariant while the latter follows from a requirement of invariance with respect to the action of the group of isochoric diffeomorphisms. The results can be applied to the analysis of dislocations in quasicrystals.