Quasiperiodic spectra and orthogonality for iterated function system measures

We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a “small perturbation” of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs X admit spectral duality. When X is given, we identify geometric conditions on X for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices. Research supported in part by a grant from the National Science Foundation DMS-0704191.     

Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices,II

Recent results of several authors have led to constructions of parallelotopes which are bounded remainder sets for totally irrational toral rotations. In this brief note we explain, in retrospect, how some of these results can easily be obtained from a geometric argument which was previously employed by Duneau and Oguey in the study of deformation properties of mathematical models for quasicrystals.     

Crack and indentation problems for one-dimensional hexagonal quasicrystals

In this paper we develop a general method to solve elastic three-dimensional problems for one-dimensional hexagonal quasicrystals with point groups 6mm, 62_h2_h, m2_h and 6/mmm, including crack and indentation problems. Exact solutions are obtained by using Fourier series and Hankel transform methods. These results automatically reduce to those in the classical elasticity theory when the phason field is absent. Received 6 October 2000     

Condensate oscillations in a Penrose tiling lattice

We study the dynamics of a Bose-Einstein condensate subject to a particular Penrose tiling lattice. In such a lattice, the potential energy at each site depends on the neighbour sites, accordingly to the model introduced by Sutherland [16]. The Bose-Einstein wavepacket, initially at rest at the lattice symmetry center, is released. We observe a very complex time-evolution that strongly depends on the symmetry center (two choices are possible), on the potential energy landscape dispersion, and on the interaction strength. The condensate-width oscillates at different frequencies and we can identify large-frequency reshaping oscillations and low-frequency rescaling oscillations. We discuss in which conditions these oscillations are spatially bounded, denoting a self-trapping dynamics.     

Formation and crystallographic orientation study of quasicrystal, 2/1 and 1/1 approximants in Cd–Mg–Y system using electron backscatter diffraction (EBSD)

A partial isothermal ternary phase diagram of the Cd–Mg–Y system including icosahedral quasicrystal (i-QC) and 1/1 approximant (APP) was obtained at 673?K. The stability of the i-QC and APPs in terms of composition and temperature was studied, as a result of which the 2/1 APP phase was discovered at 773?K. Moreover, the i-QC phase was found to be stable at lower Y concentrations of ～12–13?at.%, as compared to the 1/1 APP, which showed ～15–16?at.% Y. Single i-QC, 2/1 and 1/1 APP grains were also synthesized applying centrifuging system. Electron backscatter diffraction (EBSD) characterization of the obtained i-QC and APPs revealed almost the same Kikuchi patterns for the i-QC and 2/1 APP, indicating high resemblance of their structures. However, the Kikuchi pattern acquired from the 1/1 APP was distinguishable from the one obtained from the i-QC as it showed split bands. The occurrence of the split bands was associated with significant deviation from a perfect icosahedral symmetry. This was also confirmed by analysing calculated electron diffraction and Kikuchi patterns along pseudo-five-fold axes of the 2/1 and 1/1 APPs. Finally, orientation relationships between the i-QC, 1/1 APP and α-Mg were investigated by analysing acquired EBSD Kikuchi patterns from the respective domains. Accordingly, three perpendicular two-fold axes of the i-QC and {100} axes of the 1/1 APP occur along three mutually orthogonal {0001}, {10-10} and {11-20} axes of the α-Mg.     

Ni deposition on the pentagonal surface of an icosahedral Al-Pd-Mn quasicrystal

We have evaporated Ni on the pentagonal surface of an icosahedral Al-Pd-Mn quasicrystal kept at room temperature. At the initial stage of growth, Ni intermixes with the substrate surface. Subsequently, Al from the quasicrystal matrix migrates to growing layers. The modified chemical composition in an initially icosahedral region near the surface induces a structural transformation. An Al-Pd-Mn alloy is formed which consists of five cubic domains with dimensions in the nm-range exposing their (1 1 0) faces parallel to the surface. These domains are azimuthally rotated by 2π/5 with respect to each other and aligned with symmetry directions of the icosahedral substrate. Al-Mn-Ni, Al-Ni, and Ni overlayers adopt both structure and orientation of these domains which stabilises Ni in a novel body-centred cubic phase. Ni-rich overlayers exhibit out-of-plane magnetic ordering.     

A classification of (some) Pisot-Cyclotomic numbers

A complete classification of degree 2, 3 and degree 4 Pisot-Cyclotomic numbers is given. Some examples of higher degrees are also given. Pisot-Cyclotomic numbers have applications to quasicrystals and quasilattices.     

The standard description of quasicrystal linear elasticity may produce non-physical results

In static setting, the standard formulation of quasicrystal linear elasticity, obtained neglecting phason diffusion in addition to inertia, implies - it is shown here - non-physical results for common choices of the constitutive tensors. Two examples of this type are presented: they involve a 2D-quasicrystal filling half-plane, subjected to in-plane concentrated forces.     

Electrical transport, magnetic and thermal properties of icosahedral Al–Pd–Mn quasicrystals

We report measurements of electrical resistivity (ρ), Hall coefficient (R_H), magnetization (M) and specific heat (C_p(T)) of high-quality icosahedral Al_70.4Pd_20.8Mn_8.8 phases with different thermal treatment. An improvement in the quasi-crystallinity upon the annealing treatment caused a drastic increase in ρ up to 7000 μΩ cm accompanied by a very small electronic specific heat coefficient γ. The low temperature ρ(T) data has been analyzed in terms of weak localization and electron–electron interaction effects. The Hall resistivity (ρ_H) is found to be strongly temperature-dependent and varies linearly with the magnetization (M) for the same field and temperature. Magnetization measurement reveals that more conductive samples are more magnetic and vice versa. Magnetic susceptibility (χ) data of all the annealed samples agrees with the Curie–Weiss-like behavior implying the existence of localized moments. The negative Curie–Weiss temperature (θ) indicates strong antiferromagnetic coupling between individual Mn atoms. The magnetic Mn concentration is found to be small, ranging from 1.73×10^-4 for the less magnetic sample studied up to 3×10^-3 for the more magnetic one. The small electronic specific heat coefficient obtained for all the samples suggests a significant reduction in the electronic density of states (DOS) at the Fermi level (E_F) upon thermal annealing treatment.