Two-dimensional Quasiperiodic Structures Research Articles

Ionization and high harmonic generation of two-dimensional quasiperiodic structures in arbitrarily polarized strong laser fields

The ionization and high harmonic generation from two-dimensional quasiperiodic structures in strong laser fields are studied by numerically solving the time-dependent Schrodinger equation, in which a two-dimensional Kronig–Penney potential has been adopted to mimic the target. The two-dimensional photoelectron momentum distribution is significantly governed by the two-dimensional quasiperiodic potential, and thus is distinct from the single atom case. The low order harmonics initiated by a circularly polarized driving laser field are also circularly polarized and present alternative helicities; however, the high order harmonics deviate from circular polarization due to the unstable phase variation between different components. This calculation shows that elliptically polarized harmonics can be effectively generated by a circularly polarized driving laser field, and it sheds light on laser–surface interaction.

Ionization and high harmonic generation of two-dimensional quasiperiodic structures in arbitrarily polarized strong laser fields

Motivated by collapsing of Riemannian manifolds and inhomogeneous scaling of left invariant Riemannian metrics on a real Lie group $G$ with a sub-group $H$, we introduce a family of interpolation equations on $G$ with a parameter $\epsilon>0$, interpolating hypo-elliptic diffusions on $H$ and translates of exponential maps on $G$ and examine the dynamics as $\epsilon\to 0$. When $H$ is compact, we use the reductive homogeneous structure of Nomizu to extract a converging family of stochastic processes (converging on the time scale $\frac 1 \epsilon$), proving the convergence of the stochastic dynamics on the orbit spaces $G/H$ and their parallel translations, providing also an estimate on the rate of the convergence in the Wasserstein distance. Their limits are not necessarily Brownian motions and are classified algebraically by a Peter Weyl's theorem for real Lie groups and geometrically using a weak notion of the naturally reductive property; the classifications allow to conclude the Markov property of the limit process. This can be considered as `taking the adiabatic limit' of the differential operators ${\mathcal L}^\epsilon=\frac 1 {\epsilon} \sum_k (A_k)^2+ \frac 1{\epsilon} A_0+ Y_0$ where $Y_0, A_k$ are left invariant vector fields and $\{A_k\}$ generate the Lie-algebra of $H$.

About optical localization in photonic quasicrystals

We employ the two stage cut and project scheme to generate a dodecagonal two-dimensional quasiperiodic structure. The finite-differences-time-domain method is applied to simulate the propagation of electromagnetic modes in the system. We compute the transmission coefficients as well as the inverse participation ratio for a quasicrystal consisting of dielectric cylindrical rods. We find that for a small crystal the band gap forms due to destructive interference between extended states. The quasiperiodic geometry exhibits modes with enhanced transmission coefficients. The inverse participation ratio analysis indicates that these modes are localized and that the localization length is estimated to be 0.3207 in the inverse units of a lattice characteristic length scale.

An eightfold optical quasicrystal with cold atoms

We propose a means to realize two-dimensional quasiperiodic structures by trapping atoms in an optical potential. The structures have eightfold symmetry and are closely related to the well-known quasiperiodic octagonal (Ammann-Beenker) tiling. We describe the geometrical properties of the structures obtained by tuning parameters of the system. We discuss some features of the corresponding tight-binding models, and experiments to probe quantum properties of this optical quasicrystal.

Monte Carlo Simulation of the Potts Model on a Dodecagonal Quasiperiodic Structure

By means of a Monte Carlo simulation, we study the three-state Potts model on a two-dimensional quasiperiodic structure based on a dodecagonal cluster covering pattern. The critical temperature and exponents are obtained from finite-size scaling analysis. It is shown that the Potts model on the quasiperiodic lattice belongs to the same universal class as those on periodic ones.

Gap structures and wave functions of classical waves in large-sized two-dimensional quasiperiodic structures

We study thermal rectifying effect in two dimensional (2D) systems consisting of the Frenkel Kontorva (FK) lattice and the Fermi-Pasta-Ulam (FPU) lattice. It is found that the rectifying effect is related to the asymmetrical interface thermal resistance. The rectifying efficiency is typically about two orders of magnitude which is large enough to be observed in experiment. The dependence of rectifying efficiency on the temperature and temperature gradient is studied. The underlying mechanism is found to be the match and mismatch of the spectra of lattice vibration in two parts.

Conservation laws of a decagonal quasicrystal in elastodynamics

A special version of Noether's theorem for the sake of absolute invariance on invariant variational principles is used to obtain conservation laws of a three-dimensional solid periodically stacked in a two-dimensional quasiperiodic structure with decagonal symmetry. By applying Lie's infinitesimal criterion on the invariance to the Lagrangian density function under a continuous transformation group, it is found that there exist fourteen conservation laws (nine in physical space and five in material space). Conservation laws in physical space correspond to the known physical facts. Influence of the phason displacement field on the conservation laws in material space is presented. The infinitesimal symmetry-transformations from scale change and translation of coordinates as well as one coordinate rotation are verified to be true. Especially, although the angular momentum conservation laws do not contain the physical quantities in phason field, and the conservation of angular momentum of the phason displacement field itself does not hold, the physical quantities in phason field play an important role in the conservation law in material space derived from an infinitesimal coordinate rotation.

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.

Structural Phase Transitions Of Decagonal Quasicrystals

AbstractA two step model for the transformation of decagonal quasicrystals (DQC) to their crystalline approximants is presented. First, a partially disordered approximant domain structure results rapidly from small atomic displacements. Subsequently, atomic diffusion leads very slowly to an ordered approximant crystal (AC) structure. The model is discussed on atomic scale. Its basic idea is to describe the structure of a quasicrystal (QC) as incommensurately modulated structure (IMS). In that description, the average structure (AS) of a QC is at the same time the AS of all its irrational and rational approximants. In this approach, the positional quasicrystal- to-approximant-crystal (QC-AC) transformation can be performed by atomic displacements smaller than any interatomic distance. This mechanism, however, leads inherently to to a certain amount of chemical disorder. In most cases also some positional disorder will result. The tools developed for the description of phase transitions of IMS can analogously be used for QC. Examples for one- and two-dimensional quasiperiodic structures are discussed.

Third sound in one and two dimensional modulated structures

An experimental technique is developed to study acoustic transmission in one and two-dimensional modulated structures by employing third sound of a superfluid helium film. In particular, the Penrose lattice, which is a two-dimensional quasiperiodic structure, is studied. In two dimensions, the scattering of third sound is weaker than in one dimensions. Nevertheless, we find that the transmission spectrum in the Penrose lattice, which is a two-dimensional prototype of the quasicrystal, is observable if the helium film thickness is chosen around, 5 atomic layers. The transmission spectra in the Penrose lattice are explained in terms of dynamical theory of diffraction.

On the evolution of quasiperiodicity through faulting—A projection formalism approach

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.

Fcc superlattice structure: A novel quasiperiodic structure without forbidden symmetries.

A two-dimensional quasiperiodic structure, based on tiles of three different shapes, has been generated using a modified strip-projection method. This structure belongs to the 4mm point symmetry group and can be viewed as an ordered derivative of fcc. The constituent tiles have been identified to be unimolecular subunit cells of three well known ordered structures of the fcc 1(1/20 family. The concept of a ``rational modification of a truly quasiperiodic structure'' is introduced.

Stable tenfold faceted single-grain decagonal quasicrystals of Al65Cu15Co20.

We report stable ${\mathrm{Al}}_{65}$${\mathrm{Cu}}_{15}$${\mathrm{Co}}_{20}$ decagonal quasicrystals which exhibit faceted columnar morphology. The column axis is along the tenfold direction, and it is faceted by the ten twofold planes. Faceted single grains, several millimeters in size were grown and characterized. Peritectic growth is fastest along the periodic tenfold axis and slowest along the nucleation controlled, quasiperiodic twofold directions. Single-crystal x-ray scattering verifies the two-dimensional quasiperiodic structure. The two-dimensional quasicrystalline structure is defect free on a length scale of 2000 A\r{} and is layered with a thickness of 8.26 A\r{}.