Phase-induced atomic permutations in icosahedral quasicrystals: a model for self-diffusion (original) (raw)

Diffusion in 2D Quasi-Crystals

Europhysics Letters (EPL), 1994

Self-diffusion induced by phasonic flips is studied in an octagonal model quasi-crystal. To determine the temperature dependence of the diffusion coefficient, we apply a Monte Carlo simulation with specific energy values of local configurations. We compare the results of the ideal quasi-periodic tiling and a related periodic approximant and comment on possible implications to real quasi-crystals. Recently, Kalugin and Katz [1] proposed a possible mechanism for bulk self-diffusion in quasi-crystals. Their model is based on specific geometric properties of the quasi-crystalline structure. In ordinary crystals, the diffusion process depends on the presence of vacancies in the lattice and is activated by the vacancy production rate and the hopping of atoms in the neighbourhood. Both lead to a Brownian-like motion, and the temperature dependence of the diffusion coefficient thus obeys an Arrhenius law [2]. Due to additional degrees of freedom of quasi-periodic systems-the so-called phasons-one can construct an additional process for bulk diffusion in quasi-crystals. Based on general arguments, Kalugin and Katz conclude that the phason degree of freedom leads to a deviation of the diffusion coefficient from the Arrhenius law. Unfortunately, because of the general nature of their considerations, they were neither able to determine at which temperature these deviations are to be expected nor to estimate their order of magnitude. In this article, we calculate the diffusion coefficient of the eightfold symmetric Ammann-Beenker tiling [3] quantitatively. We start from a mean-field model which takes the additional degrees of freedom into account and we apply a Monte Carlo method to estimate the phason contribution to the bulk diffusion in the quasi-periodic plane. We discuss general properties and similarities with the ansatz of Kalugin and Katz and compare the quasi-periodic tiling with a periodic approximant. The Arnmann-Beenker tiling is chosen because, on the one hand, it provides generic quasi-periodic properties while, on the other hand, it is as simple as possible. Nevertheless, all results stated below can easily be extended to other two-dimensional quasi-periodic tilings like the Penrose tiling [4] or the Ti.ibingen triangle tiling [5]. The Ammann-Beenker tiling consists of squares and 45-degree rhombi which form (modulo operations of the dihedral group d 8) 6 different vertex configurations. Combinatorially, another 10 (+ 3 mirror inverted) vertices are possible without gaps or overlaps, which were introduced and

Random tiling quasicrystals in three dimensions

Three-dimensional icosahedral random tilings with rhombohedral cells are studied in the semi-entropic model. We introduce a global energy measure defined by the variance of the quasilattice points in the orthogonal space. The internal energy, the specific heat, the configuration entropy and the sheet magnetization (as defined by Dotera and Steinhardt [Phys. Rev. Lett. 72 (1994) 1670]) have been calculated. The specific heat shows a Schottky anomaly which might indicate a phase transition from an ordered quasicrystal to a random tiling. But the divergence with the sample size as well as the divergence of the susceptibility are too small to pinpoint the phase transition conclusively. The self-diffusion coefficients closely follow an Arrhenius law, but show plateaus at intermediate temperature ranges which are explained by energy barriers between different tiling configurations due to the harmonic energy measure. There exists a correlation between the temperature behavior of the self-diffusion coefficient and the frequency of vertices which are able to flip (simpletons). Furthermore we demonstrate that the radial distribution function and the radial structure factor only depend slightly on the random tiling configuration. Hence, radially symmetric pair potentials lead to an energetical equidistribution of all configurations of a canonical random tiling ensemble and do not enforce matching rules.

The Equivalence Between Unit-Cell Twinning and Tiling in Icosahedral Quasicrystals

Scientific Reports

It is shown that tiling in icosahedral quasicrystals can also be properly described by cyclic twinning at the unit cell level. The twinning operation is applied on the primitive prolate golden rhombohedra, which can be considered a result of a distorted face-centered cubic parent structure. The shape of the rhombohedra is determined by an exact space filling, resembling the forbidden five-fold rotational symmetry. Stacking of clusters, formed around multiply twinned rhombic hexecontahedra, keeps the rhombohedra of adjacent clusters in discrete relationships. Thus periodicities, interrelated as members of a Fibonacci series, are formed. The intergrown twins form no obvious twin boundaries and fill the space in combination with the oblate golden rhombohedra, formed between clusters in contact. Simulated diffraction patterns of the multiply twinned rhombohedra and the Fourier transform of an extended model structure are in full accord with the experimental diffraction patterns and can be indexed by means of three-dimensional crystallography. The alternative approach is fully compatible to the rather complicated descriptions in a hyper-space. Ever since quasicrystals (QCs) were first reported 1 they attracted great interest, because they apparently contradicted some basic concepts of crystallography 2-4. Contrary to fully disordered solids and perfectly grown single crystals, characterized by their rotational and translational symmetries, QCs with their forbidden five-fold rotational symmetry and with the apparently lost translational order represented something in between the two categories. However, some of their properties contradict this distinction. Their shapes can be well developed and the corresponding diffraction patterns (DPs) show exceptionally sharp reflections, without any diffuse scattering, characteristic of short-range order, modulation, or any other deviation from an ideal crystalline structure. Pauling was convinced that none of the existing crystallographic rules was violated in the newly discovered materials 5-10. He believed these crystals were composed of twinned cubic domains with huge unit cells, whose basic building elements were composed of one Mn atom linked to twelve Al atoms. Although the existing experiments seemingly supported his model, he after all run into problems. Another major problem with Pauling's approach was, that no twin boundaries were ever detected in QCs 11. Contrary to Pauling, a number of researchers 12-18 considered QCs an exception to the known solid state structures, which required a novel approach. Their explanation was based on the so-called Amman tiling 19 , the three-dimensional equivalent of the two-dimensional Penrose tiling. Likewise to two Penrose rhombic tiles filling a plane, their three-dimensional equivalents, the prolate and the oblate golden rhombohedra, will fill the space and form the QC structure. It is shown in the present work that tiling in the icosahedral QC structure can also be properly explained by cyclic unit cell twinning 20,21 , applied on primitive golden rhombohedra, forming thus intergrown twins without explicit twin-boundaries. Results Procedure. Instead of describing a QC structure in a hyper-space, the present description is based on twinning of the basic building units. Dependent on the sizes and the composition of the constituent atoms a hypothetical parent face-centered cubic structure of an alloy may collapse into a primitive rhombohedral one along four equivalent directions. If the resulting rhombohedral angle is close to 63.43°, i.e. the angle of a prolate golden rhombohedron, it will lock

Quasicrystals: tiling versus clustering

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.

Tilings, coverings, clusters and quasicrystals

2000

A quasiperiodic covering of the plane by regular decagons and an analogous structure in three dimensions are described. The 3D pattern consists of interpenetrating triacontahedral clusters, related to the ttmathrn3tt^{\mathrn{3}}ttmathrn3 inflation rule for the 3D Penrose tiling patterns. The overlap regions are triacontahedron faces, rhombic dodecahedra and rhombic icosahedra, The structure leads to a plausible model for the T2 icosahedral quasicrystalline phases.

Tetracoordinated quasicrystals

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.

Sub-diffusive electronic states in octagonal tiling

Journal of Physics: Conference Series, 2017

We study the quantum diffusion of charge carriers in octagonal tilings. Our numerical results show a power law decay of the wave-packet spreading, L(t) ∝ t β , characteristic of critical states in quasicrystals at large time t. For many energies states are sub-diffusive, i.e. β < 0.5, and thus conductivity increases when the amount of defects (static defects and/or temperature) increases.

Random-tiling quasicrystal in three dimensions

Physical Review Letters, 1990

A three-dimensional random-tiling icosahedral quasicrystal is studied by a Monte Carlo simulation. The hypothesis of long-range positional order in the system is confirmed through analysis of the finitesize scaling behavior of phason fluctuations and Fourier peak intensities. By investigating the diffuse scattering we determine the phason stiffness constants. A finite-size scaling form for the Fourier intensity near an icosahedral reciprocal wave vector is proposed.