Quasiperiodic Octagonal Tiling Research Articles

Vibrational properties of quasi-periodic beam structures

Quasi-periodic structures have been widely studied, notably in the atomic vibration domain. In this paper a beam structure based on Octagonal quasi-periodic tiling is studied. We provide a complete description of its vibrational response, including the density of its vibrational states, a detailed description of its vibration modes, and the computation of the dynamical structure factor (spectral density of energy) for transverse and for longitudinal waves. It is shown that quasi-periodic structures exhibit localized low frequency vibration modes that are due to resonant vibrations of isolated patterns in the quasi-periodic structure, but in opposite, high-frequency modes are (non-trivially) extended. Moreover, the paper shows the possible existence of band gaps in the vibrational response of periodic and quasi-periodic beam lattices as a function of the ratio between the bending and the tensile stiffness of the beams.

Nonperiodic Octagonal Patterns from a Jali Screen in the Mausoleum of Muhammad Ghaus in Gwalior and Their Periodic Relatives

One of the large jali screens adorning the mausoleum of Muhammad Ghaus in Gwalior (N India), built in 1565, contains panels composed of disordered composite octagons and Salomon stars. These elements show a rotational disorder with some interdependence. Analysis of these partially disordered patterns with rotatable configurations of the above elements suggested that they may be approximants of a quasiperiodic octagonal tiling based on a new type of composite tiles. Comparisons with the Amman’s quasiperiodic tiling were made. Instances of similar or related periodic ornamental patterns at other northern Indian localities are analyzed as well.

Temperature-driven transition of the eight-state Potts model on the quasiperiodic octagonal tiling

Attention is focussed on a certain class of crystalline structures wherein the atomic arrangements can be described in terms of periodic arrangements of tiles of two different types—squares and rhombi. Such structures can be regarded as rather simple (unmodulated) approximants of various two-dimensional quasiperiodic structures. Some of the metallic phases which belong to this class are the beta-Mn and the sigma phases which are so related to the octagonal and the dodecagonal quasicrystals respectively, as also certain AlTi intermetallic phases which are derivatives of a quasiperiodic superlattice structure. It is well established that some of these phases can transform into their quasiperiodic counterparts in a continuous manner. Application of the method of projection from higher dimensions to such continuous transformations shows that the transformation to quasiperiodicity initiates through the formation of faults (antiphase boundaries in the case of superlattice structures) in the unmodulated structures.

A real-space renormalization group approach to the ferromagnetic Potts model on the two-dimensional octagonal quasi-periodic tiling

A one-step real-space renormalization group (RSRG) transformation is used to study the ferromagnetic (FM) q-state Potts model on the two-dimensional (2D) octagonal quasi-periodic tiling (OQT). The critical exponents of the correlation length for different values of q and the critical temperature of the Ising model are obtained. The results are shown to be not sensitive to the choice of parameters. The comparison of the results with previous results for the OQT and the square lattice (SQL) seems to show that the universal classes of the q-state Potts models on the OQT and the SQL are the same for the range from q = 1 to q = 3, in accordance with previous research.

Real-space renormalization group approach of the Potts model on the octagonal quasi-periodic tiling

A one-step real-space renormalization group (RSRG) transformation is used to study the ferromagnetic (FM) Potts model on the two-dimensional (2D) octagonal quasi-periodic tiling (OQT). The critical exponents of the correlation length in theq=1,2,3,4 cases and the crtitical surface of the Ising model are obtained. The results are discussed by comparing with previous results on the OQT and the square lattice (SQL).

Monte Carlo investigation of the eight-state Potts model on quasiperiodic tilings

A Monte Carlo investigation of the eight-state Potts model on the two-dimensional (2D) quasiperiodic octagonal tiling with free boundary conditions is performed in order to determine the nature of a temperature-driven transition. It is shown that numerical data suffer from drastic free boundary effects that strongly disturb the probability distributions of the internal energy and, consequently, the scaling behavior of the specific heat. An alternative way consisting in analysing the core of the tilings is applied to pass over free boundary effects. This analysis combined with the Lee–Kosterlitz method allows one to evidence that the system undergoes a first-order transition as in 2D periodic lattices. The first-order type of scaling is observed for the maximum in the susceptibility of the core of the tilings but not for the maximum in the specific heat.

Quantum diffusion in two-dimensional octagonal quasicrystals

We have studied the quantum diffusion in an octagonal quasiperiodic tiling. Paticular attention is paid to the analysis of long-time behavior of the autocorrelation function C(t) and the relationship between diffusion and fractal dimensions of energy spectrum. For the pure hopping model, numerical calculations show that C(t) ~ t-1, which corresponds to the case of periodic systems. For the combined model, in which the contribution of site potentials is considered, C(t) ~ t-δ with 0<δ ≤ 1. The scaling factor δ is related to Hamiltonian parameters and the type of site where electron is initially located. Moreover, we find a crossover from δ = 1 to 0<δ<1 with increasing site potentials.

Finite-size behavior of the three-state Potts model on the quasiperiodic octagonal tiling

The static critical behavior of the three-state Potts model on the two-dimensional (2D) quasiperiodic octagonal tiling with free boundary conditions is investigated by means of the importance-sampling Monte Carlo method and the single histogram technique. The results strongly suggest that the static critical exponents \ensuremath{\nu} and \ensuremath{\gamma} are the same as in 2D periodic lattices whereas the different estimates of \ensuremath{\alpha} are not really consistent with the 2D periodic value. The infinite tiling critical temperature, ${\mathrm{kT}}_{c}/J\ensuremath{\approx}1.557,$ is slightly higher than the critical temperature of the three-state Potts model on the square lattice, in agreement with previous studies on the Ising model on quasiperiodic tilings.

Static critical behavior of a weakly frustrated Ising model on the octagonal tiling.

The static critical behavior of a weakly frustrated ferromagnetic Ising model on the two-dimensional (2D) quasiperiodic octagonal tiling is studied by means of Monte Carlo simulations and finite-size scaling analysis. Our results strongly suggest that this frustrated Ising model on the octagonal tiling belongs to the same universality class as the ferromagnetic Ising model on 2D periodic lattices. The infinite tiling critical temperature, ${\mathit{kT}}_{\mathit{c}}$/J=1.49\ifmmode\pm\else\textpm\fi{}0.02, agrees with previous studies indicating that tendency to ferromagnetic ordering is higher in quasiperiodic tilings than in periodic lattices. \textcopyright{} 1996 The American Physical Society.

Critical temperature of the non-frustrated ferromagnetic ising model on the quasiperiodic octagonal tiling

The critical temperature of some non-frustrated ferromagnetic Ising spin systems on the two-dimensional octagonal tiling is calculated by Monte Carlo simulations using the simulated annealing method. The ferromagnetic interactions are limited to the third neighbours ( J 1, J 2, J 3) where only J 2 is a percolating interaction. The infinite-tiling critical temperature is estimated at kT c/ J = 2.39 ± 0.02 ( J 2 = J and J 1 = J 3 = 0), kT c/ J = 3.05 ± 0.03 ( J 2 = J 1 = J and J 3 = 0), kT c/ J = 3.60 ± 0.04 ( J 2 = J 3 = J and J 1 = 0) and kT c/〈 z〉 J is generally slightly higher in the octagonal tiling than in the square and triangular lattices indicating that the tendency to ferromagnetic ordering is higher in quasiperiodic tilings. It is found that the critical temperature, T c, varies linearly with the different exchange integrals.

Static critical behavior of the ferromagnetic Ising model on the quasiperiodic octagonal tiling

The static critical behavior of the nonfrustrated ferromagnetic Ising model on the two-dimensional (2D) quasiperiodic octagonal tiling with free boundary conditions is studied by means of Monte Carlo simulations and finite-size scaling analysis. Several estimates of the critical temperature are clearly consistent and provide the final value ${\mathit{kT}}_{\mathit{c}}$/J=2.39\ifmmode\pm\else\textpm\fi{}0.01. This result shows that tendency to ferromagnetic ordering is higher in the octagonal quasilattice than in the square lattice (the mean number of interacting neighbors is equal to 4 in the two lattices). The estimates of the static critical exponents \ensuremath{\nu}, \ensuremath{\beta}, and \ensuremath{\gamma} are in agreement with previous studies on the Penrose tiling and evidence that the nonfrustrated ferromagnetic Ising model on 2D quasilattices belongs to the same universality class as the ferromagnetic Ising model on 2D periodic lattices.

Ising model on the quasiperiodic octagonal tiling

Ferromagnetic transition of the Ising model on the 2D octagonal tiling is studied by Monte-Carlo simulations using the simulated annealing method. The ferromagnetic interactions are limited to the third neighbours (J1, J2, J3). The Curie temperature for the infinite tiling is estimated at kTC/J = 2.39 ± 0.02 (J2 = J), kTC/J = 3.05 ± 0.03 (J2 = J1 = J) kTC/J = 3.60 ± 0.04 (J2 = J3 = J) and kTC/J = 4.39 ± 0.04 (J2 = J1 = J3 = J).

A note on quasielastic neutron scattering

We present a theorem that in coherent quasielastic neutron scattering on a confined and limited system of identical particles only the configuration-configuration correlations have to be examined without any hindsight on the distribution of the particles inside the configuration. This constitutes a strong simplification with respect to the problem where all particle-particle correlations would have to be accounted for. The method is illustrated on an example of correlated hopping on a piece of an octagonal quasiperiodic tiling. With some modifications the method can also be used for incoherent scattering. A further simplification of the calculations both for coherent and incoherent scattering functions is obtained by introducing a new matrix notation that allows fully for the symmetry properties of the problem

Electronic properties of 2D quasicrystals: level spacing distribution and diffusion

We present numerical calculations of the band spectrum of periodic approximants of the octagonal quasiperiodic tiling. We have studied a one-parameter family of Hamiltonians including the pure hopping case, the Laplacian and a regime where atomic potentials prevail. Working on approximants allows us to avoid boundary effects and to identify properly the nature of the spectrum. We have found a range of the parameter for which the spectrum consists in only one band. Moreover, the Lebesgue measure is finite except maybe in a small domain of the parameter. We have also studied the statistics of levels. We have obtained a transition between a regime of level repulsion (quantum chaos) and another of level clustering (Cantor spectrum). As a result, in the strong coupling regime the integrated density of spacings behaves like P( s) ∼ s − α , s → 0. We have related α to the quantum diffusion exponent of a wave packet.

Anomalous diffusion and conductivity in octagonal tiling models.

We present numerical calculations of the quantum diffusion over an octagonal quasiperiodic tiling. We have studied a one-parameter family of Hamiltonians including the pure hopping case, the Laplacian, and a regime where atomic potentials prevail. We have found that unlimited diffusion occurs with anomalous exponents both in the hopping regime, where the spectrum has a band structure, and in the strong-coupling regime, where the spectrum has a Cantor structure. Upon introducing disorder in the lattice through phasonic fluctuations, the diffusion exponent increases in the pure hopping regime, while localization appears in the strong-coupling regime. The consequences on the conductivity of real quasicrystals are considered.

FRUSTRATED ISING MODELS ON QUASIPERIODIC LATTICES IN TWO DIMENSIONS

Competing short-range interactions are considered between Ising spins placed on vertices of the two-dimensional quasiperiodic octagonal tiling. For physically reasonable competing interactions, various types of magnetic ground states appear for different choices of the local environments of active — magnetic — sites, consequences jointly of the frustration and quasiperiodicity.

Band spectrum of the octagonal quasicrystal: Finite measure, gaps, and chaos.

In this paper, we present numerical calculations of the band spectrum of periodic approximants of the octagonal quasiperiodic tiling. We have studied a one-parameter family of Hamiltonians, including the pure hopping case, the Laplacian, and a regime in which atomic potentials prevail. We have found a parameter range over which the spectrum consists of only one band. Moreover, the Lebesgue measure is finite except maybe in a small domain of the parameter range. We have also studied the statistics of the levels. We have obtained a transition between a regime of level repulsion (quantum chaos) and another of level clustering (Cantor spectrum).

Geometric study of a 2D tiling related to the octagonal quasiperiodic tiling

A quasicrystal built with three types of tiles is related to the well-known octagonal tiling. The relationships between both tilings are investigated. More precisely, we show that the coordinates of the vertices can be obtained in two different but equivalent ways. The structure factor is calculated exactly. We emphazise the difficulty one can have to define the order of the symmetry of a quasicrystal, from a practical point of view, exhibiting a quasiperiodic tiling whose spectrum has a «quasi» eight-fold symmetry. Finally, we show how to recover easily a class of octagonal-like quasicrystals Au moyen de trois tuiles, nous construisons un pavage quasiperiodique du plan, que nous relions au quasicristal octogonal. Ainsi, nous montrons que les coordonnees des nœuds peuvent etre obtenues de deux manieres differentes. Le facteur de structure est calcule exactement. Ce pavage qui possede «presque» une symetrie d'ordre huit, souleve la difficulte de la determination pratique de la symetrie d'un quasicristal. Finalement, nous montrons comment construire une large classe de pavage du type de l'octogonal, a partir de ce nouveau pavage