Random-tiling quasicrystal in three dimensions

Physical Review B, 1990

A two-dimensional rhombus tiling model with a matching-rule-based energy is analyzed using real-space renormalization-group methods and Monte Carlo simulations. The model spans a range from T=O quasiperiodic crystal (Penrose tiling) to a random-tiling quasicrystal at high temperatures. A heuristic picture for the disordering of the ground-state quasiperiodicity at low temperatures is proposed and corroborated with exact and renormalization-group calculations of the phason elastic energy, which shows a linear dependence on the strain at T=O but changes to a quadratic behavior at T&0 and sufficiently small strain. This is further supported by the Monte Carlo result that phason fluctuations diverge logarithmically with system size for all T &0, which indicates the presence of quasi-long-range translational order in the system, meaning algebraically decaying correlations. A close connection between the rhombus tiling model and the general surface-roughening phenomena is established. Extension of the results to three dimensions and their possible implication to experimental systems is also addressed.

Journal of Non-crystalline Solids, 2004

Numerical experiments have been performed on the propagation of mode-I cracks in both ordered and randomized two-dimensional decagonal model quasicrystals. In particular, the dependence on temperature, applied load and underlying structure has been investigated. The samples are endowed with an atomically sharp crack and subsequently loaded by linear scaling of the displacement field. The response of the system is followed by molecular dynamics simulations. For temperatures below 30 % of the melting temperature T m the crack velocity grows monotonically with the applied load and the model quasicrystal shows brittle fracture. For large overloads the crack becomes unstable and branches. In this low temperature regime crack tip velocities are in the range of 20 %-50 % of the shear wave velocity v s. For temperatures above 30 % of T m the crack does not remain atomically sharp but blunts spontaneously. To circumvent this and to nevertheless study a sharp crack a linear

Journal of Alloys and Compounds, 2002

Using the generalized dual method, close analytical expressions for the coordinates of the quasiperiodic lattice are given. This allows to define the lattice as an average plus a fluctuation part. The average is a superposition of crystalline lattices, and the dynamical structure factor or the diffraction pattern of the quasiperiodic structure can be expressed in terms of the average plus the fluctuation part. The average lattice dominates the response for long-wave modes of a probe particle or field, which is relevant in some recent application of quasiperiodic structures. The method can be extended for quasiperiodic grids, and is a very efficient algorithm to perform calculations in quasicrystals. Finally, the present approach can be used to define a Brillouin zone without ambiguities in the reciprocal space. 

Journal of Non-Crystalline Solids, 1990

The electronic wave functions of an icosahedral quasicrystal are investigated with the numerical method based on the tight-binding model. It is found that the spectral density of states exhibits a dispersion-relation-like pattern. The critical points (stationary points) of the 'dispersionrelation' appear at the special points which correspond to high-symmetry points in the Brillouin zone of a periodic lattice. On the other hand, the Fourier-intensity vs. energy profile of the wave functions exhibits a size dependence obeying a power law with a fractal exponent. This shows that the wave functions are critical with respect to localization. Moreover, the wave functions obey weakly the 'generalized Bloch's theorem'

Journal of physics, 1994

The process of phase-induced self-diffusion has been investigated for the threedimensional Ammann-Kramer-Penrose tiling. This happens along the lines of Kalugin and Katz for the octagonal planar tiling. lt is found that for any permutntion within two subsets of the ten vertices in a triacontahedral cage there is a corresponding loop in phase space. There are other loops by which vertices are exchanged between interlocked triacontahedral cages. Along chains of such triacontahedral cages, percolative diffusion is possible. This self-diffusion, however, occurs along two separated sublattices. Thus it is a two-component diffusion.

Journal of Alloys and Compounds, 2002

For a two-dimensional binary tiling model quasicrystal, the full set of (zero temperature) elastic constants is determined. It is found that the elastic energy is a perfect quadratic form in the phonon and phason strains. One of the phason elastic constants turns out to be negative, implying that the quasicrystal is only metastable at zero temperature.

The European Physical Journal B, 2014

In a recent paper we proposed a means to realize a quasicrystal with eight-fold symmetry by trapping particles in an optical potential created by four lasers. The quasicrystals obtained in this way, which are closely related to the well-known octagonal tiling, offer unique possibilities to study the effects of quasiperiodicity on physical properties. This method allows, furthermore, to transform the structures, to inflate or deflate them, include interactions or disorder and thus realize a large variety of theoretical models, both classical and quantum. In this paper we present the model, derive a number of interesting geometrical properties of the optical quasicrystals, as well as some results obtained by numerical calculations.

Physical Review B, 2010

The dynamics of quasicrystals is more complicated than the dynamics of periodic solids and difficult to study in experiments. Here, we investigate a decagonal and a dodecagonal quasicrystal using molecular dynamics simulations of the Lennard-Jones-Gauss interaction system. We observe that the short time dynamics is dominated by stochastic particle motion, so-called phason flips, which can be either single-particle jumps or correlated ring-like multi-particle moves. Over long times, the flip mechanism is efficient in reordering the structure of the quasicrystals and can generate diffusion. The temperature dependence of diffusion is well described by an inverse Arrhenius law. We also study the spatial distribution and correlation of mobile particles by analyzing the dynamic propensity.

Philosophical Magazine A, 2001

A quasiperiodic covering of a plane by regular decagons is described, and an analogous structure in three dimensions is deduced. This consists of a pattern of interpenetrating congruent triacontahedral clusters, related to the ½ 3 in¯ation rule for quasiperiodic Ammann tiling patterns. The overlap regions are triacontrahedron faces, oblate hexahedra, rhombic dodecahedra and rhombic icosahedra. The structure leads to a plausible model for T2 icosahedral quasicrystalline phases.

Physical review. B, Condensed matter, 1991

Current model networks for amorphous Ge contain five-membered rings and pentagonal dodecahedra to explain why in the radial distribution function the third peak of the diamond structure is missing. By presenting an algorithm based on a decoration of the three-dimensional Penrose quasilattice, we prove that this local pentagonal symmetry can be extended globally to an icosahedral quasicrystalline tetracoordinated network. Its structural elements and topological properties coincide with previous hand-built models of random networks. Thus it is suitable for simulating bulk properties of amorphous semiconductors.