Real Quasicrystals Research Articles

Spectral Structures and Topological Methods in Mathematical Quasicrystals

The mathematical theory of aperiodic order grew out of various predecessors in discrete geometry, harmonic analysis and mathematical physics, and developed rapidly after the discovery of real world quasicrystals in 1982 by Shechtman. Many mathematical disciplines have contributed to the development of this field. In this meeting, the goal was to bring leading researchers from several of them together to exchange the state of affairs, with special focus on spectral aspects, dynamics and topology.

Local cluster networks and the number of valence states in aluminium–transition metal face-centred icosahedral quasicrystals

Abstract Coordination environments of a special class of clusters which are related to the valence states in aluminium–transition metal face-centred icosahedral quasicrystals were analysed on the basis of the six-dimensional model of Katz and Gratias for the atomic sites and that of Yamamoto for the centres of the clusters. The number of the clusters per atom in the model (≈0.263) is comparable with that of Al–Cu–Ir cubic approximant crystal (≈0.258), and it indicates that the model reasonably represents the distribution of the clusters. We identified all the possible local cluster networks of short distances, which are related to the inter-cluster bonds, and evaluated the number of the valence states by assuming two possible configurations of inter-cluster bonds. The number of the valence states per atom was derived as a function of the fraction of transition-metal atoms, and the value is in good agreement with that expected from the compositions of real quasicrystals. Notably, the number of the part of the valence states which do not depend on the fraction of transition-metal atoms is numerically similar to the electron to atom ratio known as e/a, which is phenomenologically related to the stability of quasicrystals.

Delaunay sets and condensed matter: The dialogue continues

Since the discovery of quasicrystals began to percolate through the scientific community, thirty years ago, Delaunay sets have been the tool of choice for describing their structures geometrically. These descriptions have gradually evolved from tiling vertex models to random cluster models, as the structures of real and simulated quasicrystals have been clarified experimentally. In this paper I outline these developments and explain why this productive dialogue between mathematicians and materials scientists will continue.

In-situ HRTEM observations of growth process of decagonal quasicrystals

Quasicrystals possess quasiperiodicity, where the structure cannot be described simply by the repetition of unit cell like conventional crystals. This fact raises the question of how quasicrystals grow, i.e., what physical mechanism makes the growth of quasicrystals possible. While crystals can grow by copying a unit cell via local atomic interactions, nonlocal structural information seems to be required in the growth of quasicrystals. This problem has attracted much attention ever since the first discovery of a quasicrystal in 1984, and several theoretical growth models [1] have been proposed. However, no experimental studies have so far been reported, and it is still unclear whether these theoretical growth models apply to real quasicrystals. In the present study, we have conducted in-situ high-temperature electron microscopic (HRTEM: High-Resolution Transmission Electron Microscopy) observations of the growth process of decagonal quasicrystals to elucidate the growth mechanism. The growth processes of a decagonal quasicrystal of Al70.8Ni19.7Co9.5were observed by HRTEM in the temperature range 1073-1173K. Tiling patterns with edge length of about 2nm were constructed from a series of HRTEM images. They were analysed in the framework of the projection method. Here, we followed the procedures in our previous work [2]. We have already reported the results of some observations and analyses elsewhere [3]. However, the growth processes of them were on a small scale, and the results were indefinite. Recently, we have succeeded in observing a growth process on a massive scale. In this paper, we present the results of this observation and subsequent analyses, and discuss the growth mechanism of the quasicrystal.

Atomic clusters and atomic surfaces in icosahedral quasicrystals.

This paper presents the basic tools commonly used to describe the atomic structures of quasicrystals with a specific focus on the icosahedral phases. After a brief recall of the main properties of quasiperiodic objects, two simple physical rules are discussed that lead one to eventually obtain a surprisingly small number of atomic structures as ideal quasiperiodic models for real quasicrystals. This is due to the fact that the atomic surfaces (ASs) used to describe all known icosahedral phases are located on high-symmetry special points in six-dimensional space. The first rule is maximizing the density using simple polyhedral ASs that leads to two possible sets of ASs according to the value of the six-dimensional lattice parameter A between 0.63 and 0.79 nm. The second rule is maximizing the number of complete orbits of high symmetry to construct as large as possible atomic clusters similar to those observed in complex intermetallic structures and approximant phases. The practical use of these two rules together is demonstrated on two typical examples of icosahedral phases, i-AlMnSi and i-CdRE (RE = Gd, Ho, Tm).

Short-range-order diffuse scattering in decagonal Ni-rich Al-Ni-Co quasicrystals

An analytical expression of the diffuse scattering (DS) intensity due to short-range order derived in a recent paper [A. Yamamoto, Acta Crystallogr. Sect. A 66, 372 (2010)] is applied to analyze DS intensities of the S1 phase at 1120 K and the quenched basic Ni rich (b-Ni) phase in decagonal Al${}_{70}$Ni${}_{22}$Co${}_{8}$ (d-Al${}_{70}$Ni${}_{22}$Co${}_{8}$) quasicrystals. The analysis is based on a microdomain model consisting of 20 \AA{} clusters. The S1 and b-Ni phases are low- and high-temperature phases of d-Al${}_{70}$Ni${}_{22}$Co${}_{8}$ so that microdomains with the S1 phase structure are assumed to be formed in the b-Ni phase and vice versa. The intercluster correlation of the 20 \AA{} clusters within the microdomains is assumed to be the same as that in a corresponding completely ordered structure up to the third intercluster distance $\ensuremath{\simeq}32$ \AA{}. It is demonstrated that simulated DS intensity distributions reproduce characteristic features in the observed DS intensity in these phases. This suggests that the analysis of DS intensity is efficient for understanding phase transition mechanisms of quasicrystals. It also shows the effectiveness of the theory and applicability of cluster-based DS intensity calculations to real quasicrystals, which reduce the number of short-range-order parameters drastically.

準結晶研究の最近の展開 ４．準結晶のはたらき（準結晶の物性と応用）近似結晶の電子構造の第一原理計算

First-principles calculation of electronic structures of approximant crystals is a reliable tool for studying cohesion mechanism and electronic properties of real quasicrystals. This article aims at explaining the first-principles calculation as methodology of quasicrystal reaseach. After brief introduction of approximant and methods of the first-principles calculations, we present several approaches, including the tight-binding method and the plane-wave-basis method, for Cd-M (M = Ca, Mg) binary systems.

Temperature-dependent phason elasticity in a random tiling quasicrystal

Both phason elastic constants have been measured from Monte Carlo simulations of a random-tiling icosahedral quasicrystal model with a Hamiltonian. The low-temperature limit approximates the ``canonical-cell'' tiling used to describe several real quasicrystals. The elastic constant ${K}_{2}$ changes sign from positive to negative with decreasing temperature; in the ``canonical-cell'' limit, ${K}_{2}∕{K}_{1}$ appears to approach $\ensuremath{-}0.7$, about the critical value for a phason-mode modulation instability. We compare to the experiments on $i$-AlPdMn and $i$-AlCuFe.

Structure defects of icosahedral quasicrystals as electrically active centres

By dilatometric experiments on rapidly quenched quasicrystalline Al–Cu–Fe, in the course of their annealing it is proposed that icosahedral (I-) phases with free carriers of different signs (either electrons or holes) have different senses of the volume changes upon ordering. The effect discovered serves as a direct experimental support that the structural defects in I-quasicrystals are electrically active centers. This circumstance allows the identification of intrinsic and extrinsic properties of I-phases and opens novel opportunities for understanding the physical properties of real quasicrystals.

Polyhedral and chemical orders in icosahedral Al–Pd–Mn quasicrystals

Diffraction data and quasicrystallography provide average quasiperiodic structures for real quasicrystals. In the present paper it is reported a detailed analysis of one of these quasiperiodic structures, with the emphasis on complementarity of Mackay- and Bergman-like polyhedra as well as on chemical order aspects.

Dodecagonal tilings almost covered by a single cluster

A single covering cluster cannot produce a rigorously quasiperiodic dodecagonal structure but may almost do so and thus be relevant to real quasicrystals. Such ‘almost covering” patches are shown for the ship, shield, square-triangle and Socolar tiling. The ‘almost covering7rdquo; of the Socolar tiling has a fractal feature.

Frequency dependent response of a Thue-Morse aperiodic lattice

We present an exact calculation for the dynamic structure factor of a Thue-Morse aperiodic lattice using a real space renormalization-group (RSRG) method proposed by S. N. Karmakar, A. Chakrabarti, and R. K. Moitra [Phys. Rev. B 46, 3660 (1992)]. However, the ordering inherent in a Thue-Morse sequence prohibits a direct application of the RSRG method. We overcome this difficulty by observing that the Thue-Morse lattice can be obtained from a more general deterministic structure that is constructed using a four-letter substitution rule. This general deterministic sequence has the advantage that, first, both the aperiodic Thue-Morse sequence as well as the quasiperiodic period-doubling sequence can be extracted from it at two different limits, and second, within a decimation renormalization technique, it can be split into two equivalent self-similar sublattices. This facilitates the application of the renormalization method. Exact recursion relations for the Hamiltonian parameters for these sublattices are provided. Results are given both for this general nonperiodic structure and the Thue-Morse lattice as its special case. In addition, within the same framework we calculate the average density of states of the Thue-Morse lattice. Some features in the response of the aperiodic chains are found to be compatible with experiments on real quasicrystals.

The ‘random’ square–triangle tiling: simulation of growth

Some alloy systems, such as Ni–Cr, V–Ni–Si and Ta–Te, have quasicrystalline phases with 12-fold symmetry. These structures may be described in terms of dodecagonal tilings by equilateral triangles and squares. The formation of quasicrystals still poses a problem, since local information is insufficient for the construction of a perfect quasiperiodic structure. The growth of real quasicrystals may be due to several mechanisms. We have simulated the growth of a quasicrystal from a melt, consisting of squares and equilateral triangles of equal edge length. We are interested in the abundancies of the vertex configurations formed, both regular and defective. Unrestricted random growth tends to result in segregation of triangles from squares. Favoring triangles to attract squares and vice versa brings about nearly perfect patterns with nearly perfect vertex abundancies, as well as realistic defect concentrations. We have also calculated the exact vertex frequencies of the ideal square–triangle tiling by relying on inflation symmetry.

Reply to the comment by G. Coddens on 'Hierarchical porosity in real quasicrystals'

Reply to the comment by G. Coddens on 'Hierarchical porosity in real quasicrystals'

Comment on 'Hierarchical porosity in real quasicrystals'

Comment on 'Hierarchical porosity in real quasicrystals'

Coherency of phason dynamics in Fibonacci chains

The effects of phason disorder on the dynamical structure factor of Fibonacci chains are studied, and the existence of a coherent phason field in real quasicrystals is addressed. The neutron-scattering response is modeled for coherent and random phasons. The results show that coherent and random phasons can be distinguished for high values of the momentum transfer. However, for both sorts of phasons the response in the acoustic-mode region is quite similar, since the only important quantity is the average length between atoms. In particular, it is shown that a random phason produced in the quasicrystal's hyperspace leads to a coherent phason field in real space.

Electronic structure of quasicrystals

The state-of-the-art of the theory of electronic properties of quasicrystals is reviewed. Although the existing calculations are generally in very good agreement with experimental spectroscopic data, open questions concerning the character (extended, critical, localized) of the eigenstates in real three-dimensional quasicrystals remain. As long as this fundamental problem subsists, a deeper understanding of certain aspects of electronic transport, magnetism, etc. will remain incomplete.

The interest of x-ray imaging for the study of defects in realquasicrystals

The understanding of defects in quasicrystals has improved through x-ray imaging at ESRF, one of the first operating third-generation synchrotron radiation sources. These sources, while enhancing the possibilities of x-ray topography, opened the way to new techniques like phase contrast radiography and tomography. The combination of these techniques makes it possible to visualize and characterize both structural defects and inhomogeneities in the bulk of real quasicrystals. Defects exhibiting a loop-shaped contrast, previously observed by x-ray topography in both AlCuFe and AlPdMn quasicrystals, were related to inhomogeneities (holes and precipitates) revealed by phase radiography and tomography. The evolution of the defects after annealing provides clues on their nature and formation. The experimental results are discussed with reference to the existing structural models for quasicrystals. Those concerning holes are in fair agreement with the predictions of a theoretical model introduced by Janot et al to describe the structure of icosahedral AlPdMn quasicrystals in terms of hierarchical self-similar packing of overlapping atomic clusters.

Self-Similar Porosity In Quasicrystals

AbstractSingle grains of real quasicrystals have been investigated using X-ray imaging techniques at a third-generation synchrotron radiation source (ESRF). Facetted microholes have been observed. The experimental results are discussed with reference to a description of the quasicrystal structure in terms of overlapping atomic clusters (self-similar packing or random covering modifications). Relations between holes and quasicrystal formation are also discussed.

Wulff Shape for Nonperiodic Arrangements

We define an analogue of the Gibbs–Curie energy for quasi-crystals. We show that there exists a Wulff-type shape optimizing this energy, which is always a convex polytope. This way we can model the triacontahedron and the dodecahedron as shapes of quasi-crystals, according to reality. We also provide a shorter proof for the well-known formula for the number of points of a non-periodic set in a large convex domain, which also yields a non-trivial error term in the cases connected to real quasi-crystals.