Decagonal quasicrystals – What has been achieved?

Where are we now, 25 years after the discovery of the first stable decagonal quasicrystal (DQC)? In this critical review, the status of research into these axial quasicrystals, which are quasiperiodic in two dimensions and periodic along the third, is discussed, and some of the open questions are addressed. We conclude that the structures of DQC are essentially known now, a few of them even as a function of temperature. Some hypotheses concerning DQC formation, growth and stability have still to be confirmed.     

Diffuse scattering. X-rays vs. neutrons

Fundamental and experimental differences between X-ray and neutron scattering methods are outlined and the advantages of either radiation for the study of disorder in crystals are described. Special emphasis is laid on their complementarity, e.g. to identify the atomic species involved in the disorder or to decide between the elastic or inelastic nature of the diffuse scattering (static or dynamic disorder). This is illustrated by three examples: LiNbO₃ (dynamic nature of chain-like disorder), doped Zr0₂ (identification of cationic and anionic disorder components) and decagonal quasicrystals (distinction between quasi-isoelectronic elements).     

十次对称准晶的光电导率

讨论了十次对称准晶的光电导率，推导了十次对称准晶在周期方向和准周期平面不同的光电导率的计算公式，并以此计算了十次对称准晶Al₆₅Co₁₇Cu₁₈的光电导率，得到了与实验值相当符合的结果. 关键词： 准晶 光电导率     

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.     

Chemical ordering in Al(72)Ni(20)Co(8) decagonal quasicrystals

First-principles total energy calculations of the 2-nm clusters seen in high-perfection Al (72)Ni(20)Co(8) decagonal quasicrystals demonstrate that chemical ordering between Al and transition metals in the central ring is energetically highly favorable. The chemical ordering introduces extensive structure relaxation and results in broken decagonal symmetry. Such broken symmetry is sufficient to enforce the perfect quasiperiodic tiling.     

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.     

Aluminium diffusion in decagonal quasicrystals

Aluminium is the majority element in many quasicrystals and expected to be the most mobile element, but its diffusion properties are hardly accessible to experiment. Here we investigate aluminum diffusion in decagonal Al-Ni-Co and Al-Cu-Co quasicrystals by molecular dynamics simulations, using classical effective pair potentials. Above two-thirds of the melting temperature, strong aluminum diffusion is observed. The diffusion constant is measured as a function of temperature and pressure, from which the activation enthalpies and activation volumes are determined. As there are no vacancies in the samples, the diffusion, which is anisotropic, must use a direct mechanism. The high mobility of aluminium is also relevant for structure determination, and will contribute to diffuse scattering. The qualitative behavior of the dynamics is confirmed by ab initio simulations.     

Elastohydrodynamic problems in quasicrystal elasticity theory and wave propagation

Elastohydrodynamic problems of decagonal quasicrystals are analysed where the phonon field obeys wave equation and the phason field obeys diffusive wave equation. Basic equations are solved in the quasiperiodic plane and periodic plane, respectively. Final governing equations of dynamic behaviours of decagonal quasicrystals are obtained. A general solution is derived in terms of introduced three auxiliary functions, where two individually satisfy a fourth-order partial differential equation and one satisfies a second-order hyperbolic diffusion equation. Using the derived governing equations, elastic waves propagating in the quasiperiodic plane and a plane containing the period axis are analysed. Secular equations are obtained. It is found that differing from conventional crystals, at least four branches of elastic waves exist when the phonon–phason coupling is present. Moreover, acoustic waves have attenuation during wave propagation. Phason fluctuations exhibit exponential decaying behaviour due to kinematic viscosity and damping. The phase speeds are isotropic in the quasiperiodic plane and anisotropic in a plane with the periodic axis. The section of the slowness surfaces is plotted.     

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. Received: 18 November 1998 / Accepted: 27 November 1998     

准晶理论的新进展

准晶的高分辨电子显微像常显示为直径为2nm、呈十重对称的圆盘,它们还经常重叠。据此,Gummelt在1995年设计出一种内部涂有两种颜色的正十边形,并且证明只要在覆盖时两个十边形中同一颜色的部分相重,就会给出准周期的Penrose图。这种覆盖理论只需要一种正十边形的“准单胞”,覆盖部分是相邻“准单胞”生长的核。覆盖越多,结构越稳定。Gummelt在1999年将正十边形中的两种颜色去掉,既可给出准周期结构,也可给出周期结构,以及介于它们两者之间的各种中间结构。     

Structure and composition of cleaved and heat-treated tenfold surfaces of decagonal Al-Ni-Co quasicrystals

We investigated cleavage surfaces perpendicular to the tenfold direction of as-grown decagonal Al-Ni-Co quasicrystals by scanning tunneling microscopy, Auger electron spectroscopy, and scanning electron microscopy. The cleavage surface is determined by a cluster-subcluster structure. The image contrast of the smallest features, 1-2 nm in diameter, is related to the columnar atom arrangements extending perpendicular to the cleavage plane, which are predicted by current models of the decagonal quasicrystal structure. No voltage dependence of the STM images is observed. The presence of surface states and an enhanced density of states are discussed. Heat-treatments of the cleaved Al-Ni-Co quasicrystal surfaces show nearly no changes in chemical composition and structure up to about 750 °C. This is correlated with a much lower concentration of vacancies in as-grown decagonal Al-Ni-Co quasicrystals as compared to that in as-grown icosahedral Al-Pd-Mn quasicrystals.     

Correlation functions and the dynamical structure factor of quasicrystals

The icosahedral or decagonal symmetry of quasicrystals is well described by a periodic structure in higher dimensions. One consequence is the existence of dynamic phason modes in addition to the phonon modes. In an atomistic model phasons show up as correlated atomic jumps. We detect the phasons by the calculation of correlation functions and the dynamical structure factor in molecular dynamics simulations similar to the procedure used for phonons. In the simulations it is also possible to observe atomic jump processes directly. The models studied here represent icosahedral AlCuLi and decagonal AlCuCo quasicrystals. Ring processes are observed in the icosahedral case, and flips in the decagonal model. Received 17 March 2000 and Received in final form 8 June 2000     

Decagonal quasicrystals

Following the discovery of two dimensional quasicrystals in rapidly solidified Al-Mn alloys by us and L. Bendersky in 1985, a number of fascinating studies has been conducted to unravel the atomic configuration of quasicrystals with decagonal symmetry. A comprehensive mapping of the reciprocal space of decagonal quasicrystals is now available. The interpretation of the diffraction patterns brings out the comparative advantages of various indexing schemes. In addition, the nature of the variable periodicity can be addressed as a form of polytypism. The relation between decagonal quasicrystals and their crystalline homologues will be explored with emphasis on Al₆₀Mn₁₁Ni₄ and ‘Al₃Mn’. It will also be shown that decagonal quasicrystals are closely related to icosahedral quasicrystals, icosahedral twins and vacancy ordered phases.     

Continuous transformation from quasicrystals to approximants in Al–Cu–Fe alloys

Continuous transformation of icosahedral quasicrystals as observed in Al-Cu-Fe alloys proceeds through intermediate modulated structures towards rational approximants with a rhombohedral structure. Corresponding to the diffuse scattering in the electron diffraction during the transformation, a tweed contrast emerges throughout the icosahedral phase matrix. High-resolution electron microscopy reveals a complex modulated structure which tends to evolve into rhombohedral microdomains. The observed distortion of the reciprocal quasilattice due to the structural modulation has been simulated on a computer by introducing linear phason strains into the quasicrystals.     

Decagonal and Dodecagonal Quasicrystals Obtained by Molecular Dynamics Simulations

Double-well potentials are used for molecular dynamics simulation in monatomic systems. The potentials change as their parameters are adjusted, resulting in different structures. Of particular interest, we obtain decagonal and dodecagonal quasicrystals by simulations. We also verify the results and explain the formation of quasicrystals from the perspective of potential energy.     

Covering Rules and Nearest Neighbour Configurations of the Dodecagonal Quasiperiodic Structure

A dodecagonal quasiperiodic structure can be generated by duster covering and the quasilattice is described in terms of the quasi-unit cell. We study the structural properties of the two-dimensional dodecagonal quasilattice in the covering scheme. The pair covering rules and the nearest neighbour configurations are determined. It is shown that the dodecagonal structure is more complicated than those of decagonal and octagonal quasilattices.     

Quantum antiferromagnetism in quasicrystals

The antiferromagnetic Heisenberg model is studied on a two-dimensional bipartite quasiperiodic lattice. Using the stochastic series expansion quantum Monte Carlo method, the distribution of local staggered magnetic moments is determined on finite square approximants with up to 1393 sites, and a nontrivial inhomogeneous ground state is found. A hierarchical structure in the values of the moments is observed which arises from the self-similarity of the quasiperiodic lattice. The computed spin structure factor shows antiferromagnetic modulations that can be measured in neutron scattering and nuclear magnetic resonance experiments. This generic model is a first step towards understanding magnetic quasicrystals such as the recently discovered Zn-Mg-Ho icosahedral structure.     

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.     

Low-energy lattice excitations in the decagonal Al-Ni-Fe and icosahedral Al-Cu-Fe quasicrystals and the (Al,Si)-Cu-Fe cubic phase

The specific heat of decagonal Al_71.3Ni_24.0Fe_4.7 and icosahedral Al₆₂Cu_25.5Fe_12.5 quasicrystals and the Al_55.0Si_7.0Cu_25.5Fe_12.5 cubic phase approximating the structure of the icosahedral alloy has been studied in the temperature range 4.2–40.0 K. All the three compounds exhibit low coefficients of the electronic heat capacity and pronounced deviations of the low-temperature lattice heat capacity from a cubic temperature law in the range 5–10 K. The results obtained by the thermodynamic method and inelastic neutron scattering have been compared and analyzed. It has been established that, at energies ɛ < 14 meV, the spectral density of thermal vibrations in the icosahedral quasicrystal is substantially higher than those in the cubic approximant and in decagonal quasicrystal.