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.     

Intermediate stages of a transformation between a quasicrystal and an approximant including nanodomain structures in the Al-Ni-Co system

Transition states between decagonal quasicrystal and periodic approximants are studied in the Al-Ni-Co system at a measured composition of Al_71.3Ni_11.3Co _17.4 by high-resolution transmission electron microscopy and electron diffraction. The nanodomain structures appearing after annealing at 1270 K show periodic fluctuations coherently embedded in domains with the coarse order of a one-dimensional quasicrystal. Further annealing at lower temperatures changes the features of nanodomain structures and results in an increase in more periodic structures. These can be strongly disordered and full of defects but tiling analysis and electron diffraction patterns show that they correspond to locked phason strain values of two closely related periodic approximants. We conclude that the periodic approximants do not result from a continuous increase in phason strain but from the growth of seeds with a locked phason strain.     

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.     

Photonic quasicrystals for nonlinear optical frequency conversion

We present a general method for the design of 2-dimensional nonlinear photonic quasicrystals that can be utilized for the simultaneous phase matching of arbitrary optical frequency-conversion processes. The proposed scheme--based on the generalized dual-grid method that is used for constructing tiling models of quasicrystals--gives complete design flexibility, removing any constraints imposed by previous approaches. As an example we demonstrate the design of a color fan--a nonlinear photonic quasicrystal whose input is a single wave at frequency omega and whose output consists of the second, third, and fourth harmonics of omega, each in a different spatial direction.     

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.     

一族一维准晶的局部电子性质

本文利用推广的实空间重整化群方法,研究按膨胀规则(A,B)→(AⁿB,A)构造的一族一维泛Fibonacci准晶系(A_n序列)的局部电子性质。所引入的2n²+1种基本变换可计算该族一维准晶中任一A_n序列在任意格点的局部格林函数和局部态密度。结果表明,该方法是有效的,A_n链的电子局部态密度象Fibonacci准晶一样,呈现临界性。 关键词：     

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.     

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.     

Jump kinetics on the Fibonacci quasilattice. Exactly solvable model of the layer growth and dislocation kinetics in quasicrystals

The jump kinetics on a quasiperiodic pinning potential is analyzed under small external force in a 1D Fibonacci quasilattice model. The model describes planar (layer) growth of stable quasicrystals from the melt and is also relevant to the movement of quasicrystal dislocations under small stress. An exact solution is found for the spectrum of jump length as function of the driving force. The solution describes the supercooling dependence of the spectrum of nucleus heights on the growing surface of a quasicrystal. The spectrum appears to be universal and its shape has a periodic dependence on the logarithm of the supercooling. The resulting quasicrystal growth kinetics agrees well with that found in computer simulations and in the analysis of continuous thermodynamic models. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 8, 531–536 (25 April 1999) Published in English in the original Russian journal. Edited by Steve Torstveit. Institute of Crystallography, Russian Academy of Sciences, 117333, Moscow, Russia     

Transition temperature to crystal phase of Al86Mn14 quasicrystal ultrafine particles determined by direct observation and characterization of surface oxide layer

Al–Mn quasicrystal ultrafine particles can be produced by the advanced gas evaporation method (AGEM), which is a method of preparing ultrafine alloy particles by coalescence growth among the particles near the evaporation sources. We investigated the phase transition temperature from a quasicrystal to a stable crystal, by examining successive electron diffraction patterns of an ultrafine particle in an in situ experiment using a transmission electron microscope. In spite of the report that the Al₈₆Mn₁₄ quasicrystal transforms into the crystal phase at around 400–670 °C on thin film specimens, the quasicrystal ultrafine particle transformed at 800 °C, i.e., the quasicrystal ultrafine particle is more stable. Since the cross-sectional view of the surface oxide layer of the quasicrystal ultrafine particles can be easily observed, the surface oxides of η-Al₂O₃ and MnO were characterized as a result of the oxidation of residual atoms on the surface of the produced alloy particles including the quasicrystals. The conditions required for Al–Mn quasicrystal ultrafine particle formation by the AGEM can be estimated under the cooling rate of 10⁵ K/s.     

Phase diagram of a random tiling quasicrystal

We study the phase diagram of a two-dimensional random tiling model for quasicrystals. At proper concentrations the model has 8-fold rotational symmetry. Landau theory correctly gives most of the qualitative features of the phase diagram, which is in turn studied in detail numerically using a transfer matrix approach. We find that the system can enter the quasicrystal phase from many other crystalline and incommensurate phases through first-order or continuous transitions. Exact solutions are given in all phases except for the quasicrystal phase, and for the phase boundaries between them. We calculate numerically the phason elastic constants and entropy density, and confirm that the entropy density reaches its maximum at the point where phason strains are zero and the system possesses 8-fold rotational symmetry. In addition to the obvious application to quasicrystals, this study generalizes certain surface roughening models to two-dimensional surfaces in four dimensions.     

十二次对称准周期结构的自相似变换及准晶胞构造

研究十二次对称准晶体模型的几何结构性质.修正了Socolar在正方形-菱形-六边形拼图模型中自相似变换的错误.在Stampfli-Ghler正方形-菱形-三角形拼图模型的基础上,成功地构造出了准晶胞,使得十二次对称准周期结构可以用覆盖理论描述. 关键词： 准晶体 拼图模型 自相似变换 覆盖理论     

Crystallization of a dusty plasma in the positive column of a glow discharge

The formation of macroscopic ordered structures in the standing striations of a stationary glow discharge in Ne is observed. A Coulomb quasicrystal is formed by spherical glass particles with diameters of 50–63 μm and charge Z _p~7·10⁵ e. The interparticle distance is approximately 300 μm. This corresponds to a nonideality parameter Γ~5·10⁴, which leads to crystallization in the Yukawa model. The factors leading to the formation of a quasicrystal in the striations are discussed. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 2, 86–91 (25 July 1996)     

The structure of an Al–Rh–Cu decagonal quasicrystal studied by spherical aberration (Cs)-corrected scanning transmission electron microscopy

The structure of an Al–Rh–Cu decagonal quasicrystal formed with two quasiperiodic planes along the periodic axis in an Al₆₃Rh_18.5Cu_18.5 alloy has been studied by spherical aberration (Cs)-corrected high-angle annular detector dark-field (HAADF)- and annular bright-field (ABF)-scanning transmission electron microscopy (STEM). Heavy atoms of Rh and mixed sites (MSs) of Al and Cu atoms projected along the periodic axis can be clearly represented as separate bright dots in observed HAADF-STEM images, and consequently arrangements of Rh atoms and MSs on the two quasiperiodic planes can be directly determined from those of bright dots in the observed HAADF-STEM image. The Rh atoms are arranged in pentagonal tiling formed with pentagonal and star-shaped pentagonal tiles with an edge-length of 0.76 nm, and also MSs with a pentagonal arrangement are located in the pentagonal tiles with definite orientations. The star-shaped pentagonal tiles in the pentagonal tiling are arranged in τ²(τ: golden ratio)-inflated pentagonal tiling with a bond-length of 2 nm. From arrangements of Rh atoms placed in pentagonal tilings with a bond-length of 2 nm, which are generated by the projection of a five-dimensional hyper-cubic lattice, occupation domains in the perpendicular space are derived. Al atoms as well as Rh atoms and MSs are represented as dark dots in an observed ABF-STEM image, and arrangements of Al atoms in well-symmetric regions are discussed.     

Structure of the AlRhCu decagonal quasicrystal: I. A unit-cell approach

Structure of the Al---Rh---Cu decagonal quasicrystal has been studied by high-resolution electron microscopy. The high-resolution structure image shows an aperiodic tiling composed of three kinds of subunits, namely flattened hexagon, crown and five star. Therefore, a structural model of the Al---Rh---Cu decagonal quasicrystal has been constructed in a unit-cell approach, in which the atom arrangements in the subunits have been proposed. It is known that the phase has two layers in a period of 0.4 nm along the unique tenfold axis according to the previous study by electron diffraction method. The ideal model of the Al---Rh---Cu decagonal quasicrystal is proposed as periodic stacking of the layers with quasiperiodic tessellation of the three kinds of subunits, in each layer the two-colour Penrose tiling is obtained if different atom decorations for the same shape subunits are distinguished by white and black colours. Calculated images reproduces well the contrast features of the observed images, which means that the present model is basically correct. Structural relationship between the Al---Rh---Cu decagonal quasicrystal and the previously reported Al---Ni---Co decagonal quasicrystal, which has also a period of 0.4 nm, has also been discussed.     

Application of symmetry operations for construction of quasiperiodic structures in a plane

A method is proposed for constructing a quasiperiodic structure of symmetry elements — regular pentagons and five-pointed stars — in a plane. The growth of the structure is determined by the action of the symmetry operations, whose effect is not completely identical to that of similar operations in classical crystallography. The tiling (“flower of pentagons”), consisting of a central pentagon and five side pentagons joined along the edges, is studied. The growth of this tiling is accompanied by the appearance of a “flower of stars” and by the formation of isolated pores in the form of rhombi. The relation between the obtained structure and Penrose tiling is examined, and it is noted that some vertices of the Penrose tiling coincide with all vertices of the polygons of the packing obtained. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 8, 635–640 (25 April 1997)     

Absence of weak local rules for the planar quasicrystalline tiling with the 8-fold rotational symmetry

Levitov's theory of local rules (the preceding paper of this issue) gives no prediction for the planar quasicrystalline tiling having 8-fold rotational symmetry. Absence of weak local rules for this particular tiling is proven.     

Degenerate quasicrystal of hard triangular bipyramids

We report a degenerate quasicrystal in Monte Carlo simulations of hard triangular bipyramids each composed of two regular tetrahedra sharing a single face. The dodecagonal quasicrystal is similar to that recently reported for hard tetrahedra [Haji-Akbari et al., Nature (London) 462, 773 (2009)] but degenerate in the pairing of tetrahedra, and self-assembles at packing fractions above 54%. This notion of degeneracy differs from the degeneracy of a quasiperiodic random tiling arising through phason flips. Free energy calculations show that a triclinic crystal is preferred at high packing fractions.     

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     

Growing mechanisms in the QKPZ equation aaand the DPD models

The roughening of interfaces moving in inhomogeneous media is investigated by numerical integration of the phenomenological stochastic differential equation proposed by Kardar, Parisi, and Zhang [Phys. Rev. Lett. 56, 889 (1986)] with quenched noise (QKPZ) [Phys. Rev. Lett. 74, 920 (1995)]. We express the evolution equations for the mean height and the roughness into two contributions: the local and the lateral one in order to compare them with the local and the lateral contributions obtained for the directed percolation depinning models (DPD) introduced independently by Tang and Leschhorn [Phys. Rev A 45, R8309 (1992)] and Buldyrev et al. [Phys. Rev A 45, R8313 (1992)]. These models are classified in the same universality class of the QKPZ although the mechanisms of growth are quite different. In the DPD models the lateral contribution is a coupled effect of the competition between the local growth and the lateral one. In these models the lateral contribution leads to an increasing of the roughness near the criticality while in the QKPZ equation this contribution always flattens the roughness. Received 7 April 2000 and Received in final form 7 March 2001