Quasiperiodic Plane Research Articles

Quasicrystalline surface order on decagonal Al 72.1Ni 11.5Co 16.4: An investigation with spot profile analysis LEED

The quasicrystalline order of a decagonal, Co-rich AlNiCo alloy was investigated using spot profile analysis low energy electron diffraction (SPA-LEED). The surface gives rise to a tenfold diffraction pattern with the peak positions being related by the golden mean, indicating quasicrystallinity in the surface region. From the half-widths of the specular spot as a function of the electron energy, the step height was determined to be (2.05±0.05) Å, a value which agrees with the bulk distance of the fivefold quasiperiodic planes of 2.04 Å. Kinematic model calculations revealed that the cluster structure of the bulk is intact also in the surface region. The existence of broad features in the diffraction pattern points to a certain degree of lateral disorder within the terraces.

Surface structure and structural transition of decagonal Al–Ni–Co quasicrystal

X-ray photoelectron diffraction (XPD) and reflection high-energy electron diffraction (RHEED) measurements have been performed to investigate the surface structure of the decagonal Al 72Ni 12Co 16 quasicrystal, which is known as a two dimensional quasicrystal. RHEED patterns for the surface parallel to the quasiperiodic planes are almost identical to the selected area electron diffraction (SAD) patterns taken along the twofold axes of the bulk sample. The XPD images show clear tenfold-symmetric patterns consistent with the long range orientational order of the decagonal quasicrystal. These facts suggest that the surface has the same quasiperiodic structure as in the bulk. The surface phase transition induced by Ar + ion bombardment is also investigated.

Inelastic neutron scattering study of the dynamics of the AlNiCo decagonal phase

The dynamics of the decagonal AlNiCo phase has been investigated on a single-grain quasicrystalline sample using inelastic neutron scattering. The decagonal structure can be viewed as a periodic stacking of quasiperiodic planes. The anisotropy between the modes propagating in the periodic and quasiperiodic directions is found to be much weaker than theoretically predicted. A strong resonance splitting is observed at an energy transfer of 15 meV for transverse modes polarized in the quasiperiodic plane.

Homoclinic jumping in the perturbed nonlinear Schr�dinger equation

We study the damped, periodically forced, focusing NLS equation with even, periodic boundary conditions. We prove the existence of complicated solutions that repeatedly leave and come back to the vicinity of a quasi-periodic plane wave with two time scales. For pure forcing, we prove the existence of a complicated, self-similar family of homoclinic bifurcations. For mode-independent damping, we construct “jumping” transients. For mode-dependent damping, we find generalized Silnikov-type solutions that connect a periodic plane wave to itself through repeated jumps. We also study the breakdown of the unstable manifold of plane waves through repeated jumping. Our results give a direct explanation for the numerical observations of Bishop et al. © 1999 John Wiley & Sons, Inc.

Low-temperature thermal and optical properties of single-grained decagonal Al-Ni-Co quasicrystals

We report measurements of the specific heat ${C}_{p}(T)$ and of the thermal conductivities ${\ensuremath{\lambda}}^{p}(T)$ along the periodic direction and ${\ensuremath{\lambda}}^{q}(T)$ along a direction in the quasiperiodic plane for decagonal Al-Ni-Co at low temperatures. The coefficient $\ensuremath{\gamma}=0.63$ mJ ${\mathrm{mol}}^{\mathrm{\ensuremath{-}}1}$ ${\mathrm{K}}^{\mathrm{\ensuremath{-}}2}$ of the linear term to ${C}_{p}(T)$ indicates a low density of electronic states at ${E}_{F}.$ The magnitude of the cubic-in-$T$ term to the low-temperature specific heat is in fair agreement with the acoustic contribution to ${C}_{p}(T)$ calculated from the results of low-temperature measurements of the elastic stiffness constants ${c}_{\mathrm{ij}}.$ The phonon contribution to ${\ensuremath{\lambda}}^{p}(T)$ shows a distinct maximum at 25 K, typical for periodic crystals. The dominant feature in the phonon contribution to ${\ensuremath{\lambda}}^{q}(T)$ is an extended plateau between 30 and 70 K, in agreement with the concept of generalized Umklapp processes in quasicrystals. This distinct difference between the phonon contributions to ${\ensuremath{\lambda}}^{p}(T)$ and ${\ensuremath{\lambda}}^{q}(T)$ is similar to that previously observed for ${\mathrm{Al}}_{65}{\mathrm{Cu}}_{20}{\mathrm{Co}}_{15}.$ The optical conductivity ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})$ is obtained from reflectivity data in the frequency range between 1 meV and 12 eV. At low frequencies the optical conductivity ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})$ is characterized by an anisotropic Drude contribution. At high frequencies an additional absorption for the periodic and the quasiperiodic direction is observed centered at approximately 1 and 2 eV, respectively, which can be associated with electronic excitations across a pseudogap.

The structure of a Frank-Kasper decagonal quasicrystal in the Zn-Mg-Dy system: Comparison with the Al-Ni-Co system

The structure of a new decagonal quasicrystal in the Zn-Mg-Dy system has been studied by transmission electron microscopy and compared with that of a decagonal Al-Ni-Co (d-(Al-Ni-Co)) phase with primitive structure. Electron diffraction patterns of these two d phases reveal the same appearance of the Bragg reflections but with different unit lengths. The characteristic length, in and normal to the quasiperiodic plane, are a = 0.227 nm and c = 0.255nm for d-(Zn-Mg-Dy), and a = 0.20 nm and c = 0.20 nm for d-(Al-Ni-Co); the c / a ratios are therefore also different. The local atomic arrangement of the d-(Zn-Mg-Dy) is proposed to be a tetrahedrally close-packed structure.

Electronic structure and transport properties of decagonal Al-Cu-Co alloys

The electronic structure of the decagonal phase Al-Cu-Co and of related crystalline phases has been calculated using the self-consistent tight-binding linear-muffin-tin-orbital technique. The description of the atomic structure is based on different variants of the model proposed by Burkov [Phys. Rev. Lett. 67, 614 (1991); Phys. Rev. B 47, 12 325 (1993)]. The chemical decoration of these models is optimized on the basis of the electronic structure calculations. This is shown to be important for describing correctly the positions of the $d$ bands of Co and Cu. The observed shift of the Co $d$ band to relatively high binding energies is related to a particular type of order. We also show that the Cu and Co $d$-band shifts play a more important role for the stabilization of the decagonal phase than the formation of a pseudogap in the Al $\mathrm{sp}$ band. Satisfactory agreement with photoemission and x-ray emission and adsorbtion spectra is obtained for a special chemical order with preferred Co-Co coordination. The low-temperature conductivity is investigated within the Boltzmann formalism. A strong anisotropy $({\ensuremath{\sigma}}_{p}/{\ensuremath{\sigma}}_{q}\ensuremath{\approx}50)$ of the conductivity in the model with idealized coordinates results from a pronounced anisotropy of the band structure. It is shown that thermal disorder breaks the coherence of the electronic eigenstates in the quasiperiodic plane and leads to an enhancement of the conductivity and a reduction of the anisotropy ratio.

Calculation and illustration of section-projection diagrams of quasicrystals

On the basis of cut and projection description of quasicrystals, a method for calculating section-projection diagrams of quasicrystals is developed and applied to decagonal quasicrystals. Firstly, a section-projection diagram parallel to the quasiperiodic plane of perfect decagonal quasicrystal is calculated. By using the existing elastic displacement field formulae, section-projection diagrams parallel to the quasiperiodic plane of a decagonal quasicrystal containing a straight dislocation along periodic direction have been calculated when only phason strain and both phason and phonon strains are introduced respectively.

Electronic distributions and pseudo-gap in quasicrystalline decagonal and alloys

Electronic state distributions of decagonal and quasicrystals have been investigated by experimental means using the soft-x-ray spectroscopy technique complemented by photoemission spectroscopy measurements. In both quasicrystals, interaction is shown within an energy range of 2 eV below the Fermi level between the Al states and either the Co 3d states or both the Ni 3d and the Co 3d states. In , the Cu 3d states are observed in the middle of the occupied band. The formation of a significant pseudo-gap at the Fermi level is shown in the Al 3p and 3s - d distributions of these alloys. The comparison with available photoemission data an and samples shows good agreement in the energy positions of Cu, Co and Ni occupied states. The results for are compared with calculations of densities of states for two models of the Al - Co - Cu decagonal quasicrystals. Very good agreement between experiment and one of these calculations is found for the Al distributions. For , fair agreement between the present results and data for crystalline is shown. The Al p and Ni s - d unoccupied states are considered as well in . It is suggested that a slight charge transfer may occur from Al to Ni in this sample and that most of the Al p unoccupied states presumably occur from the contribution of the quasiperiodic planes in this specimen.

Low-Temperature Thermodynamic and Thermal-Transport Properties of Decagonal Al65Cu20Co15.

We report measurements of the specific heat ${C}_{p}(T)$ and of the thermal conductivities ${\ensuremath{\lambda}}^{p}(T)$ along the periodic direction and ${\ensuremath{\lambda}}^{q}(T)$ along a direction in the quasiperiodic plane for decagonal ${\mathrm{Al}}_{65}{\mathrm{Cu}}_{20}{\mathrm{Co}}_{15}$ at low temperatures. The phonon contribution to ${\ensuremath{\lambda}}^{p}(T)$ shows a maximum at 25 K, typical for periodic crystals. The dominant feature in the phonon contribution to ${\ensuremath{\lambda}}^{q}(T)$ is an extended plateau between 30 and 70 K, in agreement with the concept of generalized umklapp processes in quasicrystals. The coefficient $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.512\mathrm{mJ}\mathrm{g}{\mathrm{atom}}^{\ensuremath{-}1}{\mathrm{K}}^{\ensuremath{-}2}$ of the linear term to ${C}_{p}(T)$ indicates a low density of electronic states at ${E}_{F}$.

Low-temperature thermal properties of decagonal Al65Cu20Co15

We report measurements of the specific heatC p (T) and of the thermal conductivity λ(T) along both periodic and quasiperiodic directions on single-grained samples of decagonal Al65Cu20Co15 quasicrystals at low temperatures. The electronic contribution to λ(T) in the periodic direction is compatible with the prediction of the Wiedemann-Franz law in the limits of low and high temperatures. The phonon thermal conductivity λph in the periodic direction shows a maximum at 25 K. The dominant feature in λph (T) measured along a direction in the quasiperiodic plane is an extended plateau between 30 and 60 K, in agreement with the concept of generalized Umklapp processes in quasicrystals. The coefficient γ=0.512 mJg-atom−1K−2 of the linear term toC p (T) indicates a low density of electronic states atE F .

Propagating and localized elementary excitations in decagonal quasicrystals.

We have studied the properties of elementary excitations in decagonal $(d)$ quasicrystals as an example of vibrational eigenstates in a model $d$-AlMn alloy. We show that the presence of a periodic direction leads to the following marked differences compared to icosahedral ( $i$) quasicrystals: distinctly higher eigenvalues, a much more structured density of states, and a stronger tendency towards the localization of certain eigenstates. These effects are related to the lower dimensionality of the quasiperiodicity, and to the coupling between longitudinal modes propagating along the periodic direction and transverse modes in the quasiperiodic plane, but polarized parallel to the periodic axis.

Ray theory for scattering by two-dimensional quasiperiodic plane finite arrays

Many scattering configurations of interest include finite portions with periodic or quasiperiodic features. Several recent investigations have dealt with this problem for the planar two-dimensional case and have developed high-frequency asymptotic solutions that include multibeam reflections obeying the Bragg condition and Bragg-modulated edge diffractions. These constituents have been interpreted as wave objects in a generalized geometrical theory of diffraction (GTD). The present investigation adds to these previous results and formalizes them into a ray theory. This allows the scattered fields due to a finite quasiperiodic array of obstacles, excited by an arbitrary incident field, to be constructed entirely by ray tracing. Scattered ray plots and caustics for various shapings of incident fields and array parameters illustrate the variety of phenomena associated with this class of scattering environments.

Generalization of Eshelby's method to the anisotropic elasticity theory of dislocations in quasicrystals

The general expressions of the elastic fields induced by straight dislocations in quasicrystals have been given according to Eshelby's method which was used to treat the anisotropic elasticity of dislocations in crystals. As an example, the elastic displacement vector, the stress tensor and the elastic energy density of a screw dislocation line lying on the quasiperiodic plane of decagonal quasicrystals are calculated.

Anisotropic optical conductivity of decagonal quasicrystals.

The optical conductivity of high quality decagonal quasicrystals of ${\mathrm{Al}}_{65}$${\mathrm{Co}}_{17}$${\mathrm{Cu}}_{18}$ and ${\mathrm{Al}}_{62}$${\mathrm{Co}}_{15}$${\mathrm{Cu}}_{20}$${\mathrm{Si}}_{3}$ shows a strong anisotropy between the periodic and the quasiperiodic directions. We suggest that this is due to the dramatic enhancement of scattering in the quasiperiodic planes. The analysis of the conductivity reveals a drastic difference between the electronic structure of the decagonal quasicrystals and the icosahedral quasicrystals.

A comparative study of zone axis pattern maps from decagonal phases with varying periodicity

Tile discovery ill 1985 of the decagonal phase with periodicity in one direction and quasiperiodicity in the other two directions has excited considerable interest (1,2). This was followed by the observation that the periodicity perpendicular to tim quasiperiodic plane can assume nmltiple values of n=[l,2,3,4]x0.4 nm (3,4).

Synchrotron X-ray studies of diffuse scattering in an Al–Cu–Co two-dimensional decagonal quasicrystal

Abstract Diffuse scattering along the (10000) direction in the quasiperiodic plane in a single-phase decagonal Al–Cu–Co quasicrystal has been studied by the synchrotron X-ray diffraction technique at room temperature. The line shapes along both longitudinal (L) and transverse (T) directions were measured. A peculiar anisotropic line shape along the L direction was found. Two-dimensional contour maps reveal that some peaks have strong diffuse scattering components and that they are elongated along the T direction. The observed diffuse scattering profile varies from peak to peak, although all the peak positions are aligned in the same direction in reciprocal space. The diffuse scattering component, which is elongated along the T direction, can be fitted by a single Lorenzian with a linewidth about 20 times broader than that of the Bragg peak. No simple G‖ or G⊥ dependences for the diffuse scattering intensity were observed, in disagreement with the current status of the theories. The origin of discrepancies between experiment and theory is suggested.

High-order periodic approximants of a decagonal quasicrystal in Al70Ni15Co15

Abstract Two kinds of high-order periodic approximant phase of a decagonal quasi-crystal have been found to form in a slowly cooled Al70Ni15Co15 alloy. For both of them, the two-dimensional decagonal symmetry, 10 mm, breaks down to one of its subgroups, 2 mm, in the quasi-periodic plane of the decagonal phase. One of them has periodicity along two mutually perpendicular twofold axes in the quasi-periodic plane of the decagonal phase with periods, 3·8 nm and 5·2 nm. The other is periodic along one of the twofold axes with a period of 6·2 nm, and it apparently maintains quasi-periodicity along the other twofold axis within the experimental resolution. Structural relations of the two phases to the decagonal phase are discussed in the framework of the projection scheme from a five-dimensional hyperspace.

A novel algorithm for a quasiperiodic plane lattice with fivefold symmetry

Conventionally, Penrose tilings with fivefold symmetry are constructed with the aid of two characteristic rhombic tiles and sets of rules based on either matching of markings on the tiles or their subdivision. Both these procedures involve decision making when tiling is to be done extensively. In the present communication, a fool-proof method of producing Penrose tilings using a set of operations that can be repeated ad infinitum is described. The steps in the present procedure are akin to conventional crystallographic operations and can be expressed in simple mathematical terms which bring out some interesting aspects of Penrose tilings.

Giant Al-M (M=transitional metal) crystals as Penrose-tiling approximants of the decagonal quasicrystal.

The two-dimensional decagonal quasicrystal has been found in many Al-M (M=transitional metal) alloys, very often coexisting with a large-unit-cell phase of similar composition and local structure. By introducing phasons in two orthogonal directions in the quasiperiodic plane perpendicular to the periodic tenfold axis, i.e., approximating the irrational golden mean \ensuremath{\tau} with rational ratios of two consecutive Fibonacci numbers ${\mathit{F}}_{\mathit{n}+1}$/${\mathit{F}}_{\mathit{n}}$, the Penrose pattern in this plane gradually becomes periodic with fairly large unit-cell parameters (1--3 nm). Some giant Al-M crystals with cubic, orthorhombic, and monoclinic symmetries have thus been derived as Penrose-tiling approximants of the decagonal quasicrystal, and their simulated electron diffraction patterns agreed fairly well with experiments.