Deceptions in quasicrystal growth

We discuss a new general phenomenon pertaining to tiling models of quasicrystal growth. It is known that with Penrose tiles no (deterministic) local matching rules exist which guarantee defect-free tiling for regions of arbitrary large size. We prove that this property holds quite generally: namely, that the emergence of defects in quasicrystal growth is unavoidable for all aperiodic tiling models in the plane with local matching rules, and for many models inR ³ satisfying certain conditions.Research supported in part by NSF Grant No. DMS-9304269 and Texas ARP Grants 003658113 and 003658007.     

Self-Similar Transformation and Vertex Configurations of the Octagonal Ammann-Beenker Tiling

Based on the matching rules for squares and rhombuses,we study the self-similar transformation and the vertex configurations of the Ammann-Beenker tiling.The structural properties of the configurations and their relations during the self-similar transformation are obtained.Our results reveal the distribution correlations of the configurations,which provide an intuitive understanding of the octagonal quasi-periodic structure and also give implications for growing perfect quasi-periodic tiling according to the local rules.     

Substitutions of vertex configuration of Ammann-Beenker tiling in framework of Ammann lines

The Ammann-Beenker tiling is a typical model for two-dimensional octagonal quasicrystals. The geometric properties of local configurations are the key to understanding its formation mechanism. We study the configuration correlations in the framework of Ammann lines, giving an in-depth inspection of this eightfold symmetric structure. When both the vertex type and the orientation are taken into account, strict confinements of neighboring vertices are found. These correlations reveal the structural properties of the quasilattice and also provide substitution rules of vertex along an Ammann line.     

Entropy maximization in the force network ensemble for granular solids

A long-standing issue in the area of granular media is the tail of the force distribution, in particular, whether this is exponential, Gaussian, or even some other form. Here we resolve the issue for the case of the force network ensemble in two dimensions. We demonstrate that conservation of the total area of a reciprocal tiling, a direct consequence of local force balance, is crucial for predicting the local stress distribution. Maximizing entropy while conserving the tiling area and total pressure leads to a distribution of local pressures with a generically Gaussian tail that is in excellent agreement with numerics, both with and without friction and for two different contact networks.     

Weak matching rules for quasicrystals

Weak matching rules for a quasicrystalline tiling are local rules that ensure that fluctuations in perp-space are uniformly bounded. It is shown here that weak matching rules exist forN-fold symmetric tilings, whereN is any integer not divisible by four. The result suggests that, contrary to previous indications, quasicrystalline ground states are not confined to those symmetries for which the incommensurate ratios of wavevectors are quadratic irrationals. An explicit method of constructing weak matching rules forN-fold symmetric tilings in two dimensions is presented. It is shown that the generalization of the construction yields weak matching rules in the case of icosahedral symmetry as well.     

Quantum spins and quasiperiodicity: a real space renormalization group approach

We study the antiferromagnetic spin-1/2 Heisenberg model on a two-dimensional bipartite quasiperiodic structure, the octagonal tiling, the aperiodic equivalent of the square lattice for periodic systems. An approximate block spin renormalization scheme is described for this problem. The ground state energy and local staggered magnetizations for this system are calculated and compared with the results of a recent quantum Monte Carlo calculation for the tiling. It is conjectured that the ground state energy is exactly equal to that of the quantum antiferromagnet on the square lattice.     

Growing perfect decagonal quasicrystals by local rules

A local growth algorithm for a decagonal quasicrystal is presented. We show that a perfect Penrose tiling (PPT) layer can be grown on a decapod tiling layer by a three dimensional (3D) local rule growth. Once a PPT layer begins to form on the upper layer, successive 2D PPT layers can be added on top resulting in a perfect decagonal quasicrystalline structure in bulk with a point defect only on the bottom surface layer. Our growth rule shows that an ideal quasicrystal structure can be constructed by a local growth algorithm in 3D, contrary to the necessity of nonlocal information for a 2D PPT growth.     

The Local Structure of Tilings and Their Integer Group of Coinvariants

The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group furnishes part of the K ₀-group of the groupoid C ^*-algebra for tilings which reduce to decorations of . The group itself as well as the image of its state is computed for substitution tilings in case the substitution is locally invertible and v-primitive. This yields in particular the set of possible gap labels predicted by K-theory for Schr?dinger operators describing the particle motion in such a tiling. Received: 22 September 1995 / Accepted: 2 December 1996     

Theory of matching rules for the 3-dimensional Penrose tilings

We consider packings of the two Ammann rhombohedra used for tiling the three dimensional space. We define decorations for the facets of the rhombohedra. Using elementary algebraic topology, we prove that any tiling by these rhombohedra with matching decorations is a quasiperiodic Penrose tiling. The proof does not involve any reference to self similarity.     

Magnetic phase structure on the Penrose lattice

The Ising model on a two-dimensional Penrose tiling is studied by means of the Migdal-Kadanoff scheme. This approximate renormalization method closely follows the inflation rules of the tiling, which are easily described in terms of Robinson triangles, and lead to the consideration of four types of nearest neighbor couplings. The ferromagnetic phase transition is similar to the usual one encountered on periodic lattices. When the couplings have both signs, the presence of frustration without randomness yields a fairly intricate phase diagram, essentially made up of two regions with a very complicated border. Region I consists of quasiferromagnetic models, which exhibit long-range order below some finite critical temperature. The models of region II are paramagnetic at nonzero (low) temperature, but may become ordered (reen-trant phases) in a higher temperature range.     

Vertex configuration rules to grow an octagonal Ammann–Beenker tiling

Based on vertex configurations in the Ammann–Beenker tiling, we propose an algorithm for aggregation of square and rhombus tiles to generate an octagonal quasilattice, which mimics the growth process of a two-dimensional quasicrystal. Local matching rules with configuration selection are used to guide the way that tiles are joined to a cluster and form Ammann lines according to a generalized Fibonacci sequence. Our results reveal that vertex configuration selection can improve the performance of the algorithm, which provides an approach for growing a perfect octagonal quasiperiodic structure.     

Penrose tiling fractality in coordination Cayley’s tree graphs representation

We offer the mathematical apparatus for mapping lattice and cellular systems into the generalized coordination Cayley’s tree graphs. These Cayley’s trees have a random branchiness property and an intralayer interbush local intersection. Classical Bethe-Cayley tree graphs don’t have these properties. Bush type simplicial decomposition on Cayley’s tree graphs is introduced, on which the enumerating polynomials or enumerating distributions are built. Within the entropy methodology three types of fractal characteristics are introduced, which characterize quasi-crystalline pentagonal Penrose tiling. The quantitative estimate for the frontal-radial fractal percolation on a Cayley’s tree graph of a Penrose tiling leading to the overdimensioned effect is calculated.     

The exact solution of an octagonal rectangle-triangle random tiling

We present a detailed calculation of the recently published exact solution of a random tiling model possessing an eightfold-symmetric phase. The solution is obtained using the Bethe Ansatz and provides closed expressions for the entropy and phason elastic constants. Qualitatively, this model has the same features as the square-triangle random tiling model. We use the method of P. Kalugin, who solved the Bethe Ansatz equations for the square-triangle tiling which were found by M. Widom.     

Transition state theory without time-reversal symmetry: chaotic ionization of the hydrogen atom

We present the first application of transition state theory to a system that evolves from an initial to a final state without time-reversal symmetry. The problem studied is the chaotic ionization of a hydrogen atom in crossed electric and magnetic fields. The stable manifolds of the transition state reveal a fractal tiling which connects the geometrical properties of the tiling to the ionization rate, leading to a theoretical explanation for the computational and experimental observation of "prompt" and "delayed" electrons in this problem.     

Dispersion-based short-time Fourier transform applied to dispersive wave analysis

Although time-frequency analysis is effective for characterizing dispersive wave signals, the time-frequency tilings of most conventional analysis methods do not take into account dispersion phenomena. An adaptive time-frequency analysis method is introduced whose time-frequency tiling is determined with respect to the wave dispersion characteristics. In the dispersion-based time-frequency tiling, each time-frequency atom is adaptively rotated in the time-frequency plane, depending on the local wave dispersion. Although this idea can be useful in various problems, its application to the analysis of dispersive wave signals has not been made. In this work, the adaptive time-frequency method was applied to the analysis of dispersive elastic waves measured in waveguide experiments and a theoretical investigation on its time-frequency resolution was presented. The time-frequency resolution of the proposed transform was then compared with that of the standard short-time Fourier transform to show its effectiveness in dealing with dispersive wave signals. In addition, to facilitate the adaptive time-frequency analysis of experimentally measured signals whose dispersion relations are not known, an iterative scheme for determining the relationships was developed. The validity of the present approach in dealing with dispersive waves was verified experimentally.     

Ballistic transport in aperiodic Labyrinth tiling proven through a new convolution theorem

In this article, we report a distinct convolution theorem developed for the Kubo-Greenwood formula in Labyrinth tiling by transforming the two-dimensional lattice into a set of independent chains with rescaled Hamiltonians. Such transformation leads to an analytical solution of the direct-current conductance spectra, where quantized steps with height of 2g₀ are found in Labyrinth tiling with periodic order along the applied electric field direction, in contrast to the step height of g₀ observed in the corresponding square lattices, being g₀ the conductance quantum. When this convolution theorem is combined with the real-space renormalization method, we are able to address in non-perturbative way the electronic transport in macroscopic aperiodic Labyrinth tiling based on generalized Fibonacci chains. Furthermore, we analytically demonstrate the existence of ballistic transport states in aperiodic Labyrinth tiling. This finding suggests that the periodicity should not be a necessary condition for the single-electron ballistic transport even in multidimensional fully non-periodic lattices.     

Partition function zeros for aperiodic systems

The study of zeros of partition functions, initiated by Yang and Lee, provides an important qualitative and quantitative tool in the study of critical phenomena. This has frequently been used for periodic as well as hierarchical lattices. Here, we consider magnetic field and temperature zeros of Ising model partition functions on several aperiodic structures. In 1D, we analyze aperiodic chains obtained from substitution rules, the most prominent example being the Fibonacci chain. In 2D, we focus on the tenfold symmetric triangular tiling which allows efficient numerical treatment by means of corner transfer matrices.     

Matched wavelength and incident angle for the diagnostic beam to achieve coherent grating tiling

Design and operation of a practical, accurate alignment diagnostic system is important for the grating tiling technology, which is supposed to be applied in a chirped-pulse amplification system to increase the output power. A diagnostic method is proposed and demonstrated for grating tiling. Provided that the wavelength and incident angle of the diagnostic beam are properly set, the far-field of the main laser beam and that of the diagnostic beam can vary in the same way with the tiling errors between the sub-aperture gratings. Therefore, rotational and translational errors can be controlled and compensated according to the far-field of the diagnostic beam. The real-time monitoring and alignment can be achieved without disturbing the main beam.     

Self-similarity and self-inversion of quasicrystals

The discovery of quasicrystals played a revolutionary role in the condensed matter science and forced to renounce the dogma of the classical crystallography that the regular filling of the space by identical blocks is reduced solely to the Fedorov space groups. It is shown that aperiodic crystals, apart from the similarity, exhibit the self-inversion property. In a broadened sense, the self-inversion implies the possible composition of the inversion with translations, rotations, and homothety, whereas pure reflection by itself in a circle can be absent as an independent symmetry element. It is demonstrated that the symmetry of aperiodic tilings is described by Schottky groups (which belong to a particular type of Kleinian groups generated by the linear fractional Möbius transformations); in the theory of aperiodic crystals, the Schottky groups play the same role that the Fedorov groups play in the theory of crystal lattices. The local matching rules for the Penrose fractal tiling are derived, the problem of choice of the fundamental region of the group of motions of a quasicrystal is discussed, and the relation between the symmetry of aperiodic tilings and the symmetry of constructive fractals is analyzed.     

High resolution transmission electron microscopy observation of thermally fluctuating phasons in decagonal Al-Cu-Co

In situ high-temperature, high resolution transmission electron microscopy (HRTEM) was performed on an Al-Cu-Co decagonal quasicrystal, to investigate thermal fluctuation of phasons. A tiling pattern constructed from the HRTEM image was analyzed in the framework of the strip-projection method. Transitions between two local tile arrangements were observed at high temperature for the first time, and were shown to correspond to a thermal phason fluctuation.