Quasicrystal Model Research Articles

Local iron symmetry in quasicrystals

Iron sites in quasicrystalline Al(TFe)-alloys have been investigated with Mossbauer spectroscopy. The quadrupole splitting of these materials is about 0.4 mm/s, showing that the iron atom is not in an icosahedral\((m\bar 3\bar 5)\) site. A measurement in an external magnetic field ofHext≈5 T yields the asymmetry parameter η=(Vxx−Vyy)/Vzz. The value of η enables one to decide between the applicability of some mathematical structure models of quasicrystals.

Thermodynamic behavior of a Penrose-tiling quasicrystal.

The Penrose tiling provides a prototype for the quasiperiodic crystal model of quasicrystals. We report results of Monte Carlo simulations of a two-dimensional model in which a Penrose tiling is the ground state. A single energy is assigned to any violation of the Penrose matching rules. Our results support the existence of two separate phase transitions, corresponding to single- and double-arrow matching-rule disorder, respectively. Manifestations of these transitions in the behavior of ``perpendicular-space'' quantities are explored. A limited exploration of the effects of unequal double- and single-arrow matching-rule-violation energies is performed. Speculations that the Penrose pattern might be inherently prone to glassy behavior are shown to be incorrect.

Quasicrystal model based on a series of supercubic crystal structures consisting of triacontahedra or Mackay icosahedra

Abstract The Al-Mn or Al-Mn-Si icosahedral quasicrystal structure is shown to consist of Mackay icosahedra and double Mackay icosahedra packed closely together. The packing fraction f of the quasicrystal is found to be almost the same as that of a crystal structure with the same composition as the quasicrystal. An inflation rule with a scale of ϕ3 (ϕ = 1·618…) can be used to construct quasicrystal structures of any required order. The same packing method can be used to construct the (Al, Zn)49Mg32, quasicrystal structure if Mackay icosahedra and double Mackay icosahedra of the icosahedral phase are replaced with triacontahedra and double triacontahedra. If we arrange an nth-order structure at the vertices and body centre of a cubic unit cell, we obtain a series of supercubic crystal structures with lattice constants a = (ϕ3n a 0 and a space group Pm3, where a ois the lattice constant of the α-Al-Mn-Si crystal structure. The above structural model is in good agreement with previous higher resolution electr...

On the molecular-dynamics study of a phase transition in a quasicrystal model

A two-dimensional model of large (L) and small (S) atoms interacting via the Lennard-Jones potential has been studied by the molecular-dynamics technique. Slow cooling of the system L 0.435 S 0.565 from a disordered configuration leads to a defected quasicrystalline phase. Another run in which the L atoms were additionally interacting by the orientational three body potential, produced a phase with microcrystalline precipitates. This phase still preserves the pentagonal symmetry of the diffraction pattern. The crystalline precipitates have a unit cell of thin Penrose rhombuses Etude par la technique de dynamique moleculaire d'un modele bidimensionnel de gros atomes (L) et de petits atomes (S) en interaction via le potentiel de Lennard-Jones. Un refroidissement lent du systeme L 0,435 S 0,565 conduit a une phase quasicristalline

A quasi-crystalline sphere-packing with unexpected high density

Hendley and later Olamy and Kleman constructed circle packings in 2D-Penrose tilings of high packing densities. From these circle packings one obtains with an additional idea a sphere packing in a cylindrical 3D-Penrose tiling of packing density 0.67798.., which is higher than the best known packing densities of 0.62... for usual quasicrystals. This packing might be a model for the cylindrical quasicrystals discovered by Bendersky et al., Schaefer et al. and Fung et al.

特集・準結晶研究の発展 準結晶の構造モデル

特集・準結晶研究の発展 準結晶の構造モデル

Ionic mobility in sulphofluorinated ion-exchange membranes

The mobility, x × 10 7 (cm 2/V s), of ions in sulphofluorinated ion-exchange membranes has been measured in the presence of an electrical field. For Na +, it equals 0.25–2.8, for H 3O +, 0.5–6.0 and for OH −, 2–16 as the equivalent mass is varied from 1350 to 1820 g/mol. The nature of the change in the ionic mobility may be described either by a free-volume model or by the model of an open quasi-crystal. Both models correctly described the nature of the change in mobility but the open quasi-crystal model takes into account the special features of the polymer's structure and the nature of the ion.

Structure factors for icosahedral quasicrystals

The restriction of a hypercubic lattice in six dimensions to a subspace of three dimensions yields a well known quasiperiodic description of quasicrystals with non-crystallographic icosahedral point symmetry. A quasicrystal model is considered where this description is further reduced to a non-periodic quasilattice formed from two types of rhombohedra. Given the density on two representative rhombohedral cells, the full Fourier transform is expressed in closed form through structure factors, quasilattice factors and kinematical factors. The diffraction from point scatterers in the quasilattice is computed as an example.

Transfer-matrix analysis of a two-dimensional quasicrystal.

We investigate the quasicrystalline state of a two-dimensional binary alloy in a discrete tiling approximation. Through transfer-matrix calculations we determine the configurational entropy over a range of concentrations. We find that the entropy density is maximized by a state with tenfold symmetry at the quasicrystal concentration. Derivatives of the entropy density at its maximum yield values for the phason elastic constants. Our results confirm the existence of quasi-long-range translational order in equilibrium quasicrystalline alloys and lend support to the random-tiling model of quasicrystals.

From quasicrystals to ordinary crystals

Abstract A series of electron diffraction patterns of T2-AI6CuLi3 with various degrees of distortion have been obtained and interpreted. It is shown that a certain quenched linear phason strain can introduce imperfections into icosahedral quasicrystals such that the structure deviates from its idealized form and changes towards a corresponding b.c.c. lattice. A possible approach to constructing an icosahedral quasicrystal structural model from the structure of the b.c.c. phase is pointed out.

Hierarchical clustering in the spectra of incommensurate systems

Using degenerate perturbation theory to all orders the authors derive the Hofstadter cluster rules (1976) for the spectrum of Harper's equation (1955). They also extend their method to calculate the fractal dimension, Df, of the spectrum and speculate that Df=1/2 for typical values of the incommensurability parameter phi . The clustering rules are also shown to be valid for a one-dimensional model of quasicrystals.

DEDUCTION OF QUASILATTICE WITH FIVE-FOLD SYMMETRY AND PARTICLE FRACTAL STRUCTURE MODEL

In the structure of quasicrystal, the coordination icosahedron has long ordering but no translation ordering. The author dealt with the building principle ofquasicrystal and thought that two principles played a certain role in the quasicrystal structure, i.e. the icosahedron principle and the golden mean principle. We obtained the most simple.structure model of quasicrystals, and could explain all details of the high-resolution electron microscopic image of the A1-Mn quasicrystal based on the two principles. The author's model has the characteristic of fractal structure, therefore, we call it the particle fractal structure madeh The author has made a systematic deduction of quasicrystal point group, forms, possible type of quasicrystal lattice.

Neutron-diffraction approach to the atomic decoration of the Al-Mn icosahedral quasicrystal.

Neutron-diffraction data were obtained from single-phase icosahedral powder of the system ${\mathrm{Al}}_{74}$${\mathrm{Si}}_{5}$${\mathrm{Mn}}_{21}$ and its modification by isomorphous substitution on the Mn sites. Amplitudes and phase differences of the partial structure factors (${F}_{\mathrm{Al}}, {F}_{\mathrm{Mn}}$) were determined. From their ${Q}_{\ensuremath{\perp}}$ dependences, and within a strip-projection approach, phases were reconstructed. Atomic densities calculated in the physical 3D space show that Mackay icosahedra, the structural units usually invoked for quasicrystal models, do not emerge as the basic ingredients of the atomic arrangements.

Equilibrium faceting shapes for quasicrystals.

The equilibrium shapes (ES's) of icosahedral quasicrystals are analyzed for a wide class of lattice models which incorporate finite-range two-body interactions. Completely faceted shapes have been predicted for such models at temperature T=0. We prove that a number of simple shapes cannot be ES's for any model in this class for which the atomic interactions are constrained to be pure-attractive. This extends the result of Ho et al. [Phys. Rev. Lett. 59, 1116 (1987)], who showed that the pentagonal dodecahedron, a shape observed in grains of icosahedral Al-Cu-Fe and Ga-Mg-Zn, is a forbidden shape for pure-attractive models. We then introduce a lattice model for quasicrystals which incorporates mixed attractive and repulsive interactions and show that the possible ES's include the dodecahedron and other previously forbidden shapes.

Crystallography of Quasicrystals; Application to Icosahedral Symmetry

Crystallographic concepts are extended to quasicrystalline structures. The fact is used that any d-dimensional non-crystallographic point group has D-dimensional representations (D > d) which are compatible with periodicity, i.e. can be viewed as point groups of D-dimensional periodic structures. These periodic structures can be classified by conventional crystallographic methods: Bravais lattices, point groups and space groups can be calculated. This program is carried out for icosahedral symmetry, for which the minimal dimension D is 6. There are surprisingly few types of 6-dimensional crystals with icosahedral symmetry: 3 Bravais lattice types, 2 point groups and totally 11 space groups can occur. The non-symmorphic space groups lead to systematic extinctions of Bragg peaks, similar to the case of ordinary crystals. These extinctions, if present, can help to distinguish between quasiperiodic and competing glass models for icosahedral quasicrystals.

準周期模様の数理 １２回対称準結晶の構造モデルを中心にして

準周期模様の数理 １２回対称準結晶の構造モデルを中心にして

Structural relationships between the quasicrystalline and the amorphous state in alloys

A brief review is given on essential features of structural quasi-crystal models. In these an important part is played by large chemically and structurally ordered atom clusters based on the icosahedron. Subsequently results of experiments carried out in order to investigate the structural relationships of amorphous and quasi-crystalline phases are compiled. In the discussion the amorphous structure is described as that of a dense random atom packing with little or no chemical short-range order. The transitions to and from the quasi-crystalline state are characterized by the build-up or the destruction of the large chemically and structurally ordered atom clusters.

Local environment of iron sites in icosahedral Al65Cu

A novel icosahedral alloy, ${\mathrm{Al}}_{65}$${\mathrm{Cu}}_{20}$${\mathrm{Fe}}_{15}$, produced in both a thermodynamically metastable and stable phase, has been studied with $^{57}\mathrm{Fe}$ M\"ossbauer spectroscopy, x-ray diffraction, and differential thermal analysis. It was found that the M\"ossbauer spectrum of either phase can be successfully analyzed in terms of a distribution of quadrupole-splitting magnitudes. This is interpreted as evidence for a continuous distribution of iron sites. The similarities of M\"ossbauer parameters of both phases indicate that a high degree of atomic disorder is an intrinsic property of icosahedral materials. Critical examination of previous M\"ossbauer studies of icosahedral materials together with other published experimental data show that the assertion of two distinct transition-metal sites in these materials is unfounded. Our results can be interpreted in terms of the distorted-icosahedral quasicrystal model.

Growth model of dodecagonal quasicrystal based on correlated tiling of squares and equilateral triangles.

We present a growth model of the dodecagonal quasicrystal consisting of hexagonal antiprisms on a square and equilateral-triangle tesselation derived from the close structural relationship with the well-known $\ensuremath{\sigma}$ phase.

Electron energy spectrum in 2D-quasicrystals

The electron energy spectrum of a tight-binding two-dimensional quasicrystal (2DQS) model is numerically investigated. It is shown that there exist strongly localized states throughout the energy band, their density forming a devil's staircase. There is also a region around the band center, in which roughly 70% of all the states lie. These states are not well localized, and the properties of this sub-band resemble those of a disordered system.