Green's functions for infinite bi-material planes of cubic quasicrystals with imperfect interface

The problem of an infinite plane which is composed of two half-planes with different cubic quasicrystal materials subjected to line phonon and phason forces is investigated. By virtue of the general solution of cubic quasicrystals, a series of displacement functions is adopted to obtain Green's functions in the closed form. For the bonding along the bi-material interface three different models account for different coupling conditions.     

Complex variable function method for the plane elasticity and the dislocation problem of quasicrystals with point group 10 mm

General complex variable function method for solving plane elasticity and the dislocation problems of quasicrystals with point group 10 mm has been proposed. All the fields variables are expressed by four arbitrary analytic functions. Analytical displacement expressions for the dislocation problem of the quasicrystal is obtained. The interaction between two parallel dislocations is also discussed in detail.     

The electronic spectrum of a three-dimensional quasicrystal

The electronic spectrum of an icosahedral quasicrystal with a central-atom decoration of the Amman-Mackay network is investigated in the tight-binding approximation. The quasicrystal is described as a structural limit of the optimal cubic approximants with increasing period. The electronic spectra for the first four optimal cubic approximants do not contain the hierarchical gap structure which is typical for the Cantor set of the spectrum of a one-dimensional quasicrystal. At the same time, as the order of the approximant increases, the spectrum becomes singular throughout the entire energy scale. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 8, 557–562 (25 April 1998) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Three-dimensional photonic quasicrystal with a complete band gap

The band structure of three-dimensional cubic approximants of a photonic quasicrystal has been determined by numerical calculation. The approximants of different orders appear to have large, almost isotropic, band gaps in a wide range of relative permittivity values. The existence of the complete band gap in the photonic quasicrystal with the six-dimensional bcc lattice is shown.     

Electron spectrum and wave functions of icosahedral quasicrystals

Electron spectra and wave functions of icosahedral quasicrystals have been investigated in the tight-binding approximation using the two-fragment structural model (the Amman-MacKay network) with “central” decoration. A quasicrystal has been considered as a limiting structure in a set of optimal cubic approximants with increasing lattice constants. The method of level statistics indicates that the energy spectrum of an icosahedral quasicrystal contains a singular (nonsmooth) component. The density of electron states has been calculated for the first four optimal cubic approximants of the icosahedral quasicrystal, and the respective Lebesgue measures of energy spectra of these approximants have been obtained. Unlike the case of a one-dimensional quasiperiodic structure, the energy spectrum of an icosahedral quasicrystal does not contain a hierarchical gap structure typical of the Cantor set of measure zero in a one-dimensional quasicrystal. Localization of wave functions in an icosahedral quasicrystal has been studied, and their “critical” behavior has been detected. The effect of disorder due to substitutional impurities on electron properties of icosahedral quasicrystals has been investigated. This disorder makes the electron spectrum “smoother” and leads to a tendency to localization of wave functions. Zh. éksp. Teor. Fiz. 113, 1009–1025 (March 1998)     

Ni deposition on the pentagonal surface of an icosahedral Al-Pd-Mn quasicrystal

We have evaporated Ni on the pentagonal surface of an icosahedral Al-Pd-Mn quasicrystal kept at room temperature. At the initial stage of growth, Ni intermixes with the substrate surface. Subsequently, Al from the quasicrystal matrix migrates to growing layers. The modified chemical composition in an initially icosahedral region near the surface induces a structural transformation. An Al-Pd-Mn alloy is formed which consists of five cubic domains with dimensions in the nm-range exposing their (1 1 0) faces parallel to the surface. These domains are azimuthally rotated by 2π/5 with respect to each other and aligned with symmetry directions of the icosahedral substrate. Al-Mn-Ni, Al-Ni, and Ni overlayers adopt both structure and orientation of these domains which stabilises Ni in a novel body-centred cubic phase. Ni-rich overlayers exhibit out-of-plane magnetic ordering.     

The quasicrystal-crystal interface between icosahedral Al-Pd-Mn and deposited Co: Evidence for the affinity of the quasicrystal structure to the CsCl structure

We have evaporated Co on the pentagonal surface of an icosahedral Al-Pd-Mn quasicrystal kept at room temperature. For submonolayer Co deposits, five AlCo domains are formed exposing their (1 1 0) faces parallel to the surface and rotated by 72° with respect to each other. The orientational relationship between these domains and the substrate is determined by the optimum matching of the average quasicrystal structure and the CsCl structure with a lattice constant of about 2.88 Å. For further deposition, body-centred cubic Co domains grow epitaxially on the AlCo domains.     

The standard description of quasicrystal linear elasticity may produce non-physical results

In static setting, the standard formulation of quasicrystal linear elasticity, obtained neglecting phason diffusion in addition to inertia, implies - it is shown here - non-physical results for common choices of the constitutive tensors. Two examples of this type are presented: they involve a 2D-quasicrystal filling half-plane, subjected to in-plane concentrated forces.     

Elastic fields around a nanosized elliptic hole in decagonal quasicrystals

Based on the variational principle, a continuum theory of surface elasticity and new boundary conditions for qua- sicrystals is proposed. The effect of the residual surface stress on a decagonal quasicrystal that is weakened by a nanoscale elliptical hole is considered. The explicit expressions for the hoop stress along the edge of the hole are obtained using the Stroh formalism. The results show that the residual surface stress and the shape of the hole have a significant effect on the elastic state around the hole.     

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.     

Microstructure of a composite with a quasicrystalline phase fraction obtained by directional solidification of Al61Cu27Fe12 alloy

The microstructure of a composite containing a quasicrystal phase, i.e. so-called crystal–quasicrystal (CQ) composite, was studied. The CQ composite was obtained by the Bridgman method via solidification of Al₆₁Cu₂₇Fe₁₂ alloy (numbers indicate at%). The process was conducted at a pull out rate of v = 0.07 mm/min. The average temperature gradient in the heating zone was 43 K/cm. The composite matrix consisted of cubic β phase Al(Fe, Cu), with reinforcement of λ-phase rod-shaped fibres surrounded by a quasicrystal icosahedral ψ phase, which also existed in the fibre core. The fibres were rhomboidal in cross-section. The composite was studied using X-ray and electron diffraction, light-optical and scanning electron microscopy (SEM), X-ray topography and Laue diffraction.     

Generalized 2D problem of icosahedral quasicrystals containing an elliptic hole

The generalized 2D problem of icosahedral quasicrystals containing an elliptic hole is considered by using the extended Stroh formalism.The closed-form solutions for the displacements and stresses are obtained under general loading conditions.The solution of the Griffith crack problem as a special case of the results is also observed.The stress intensity factor and strain energy release rate are given.The effect of the phonon–phason coupling elastic constant on the mechanical behavior is also discussed.     

Elasticity and dislocations in quasicrystals with 18-fold symmetry

Elasticity problems of quasicrystals with 18-fold rotational symmetry are studied. Constitutive equations and governing equations are obtained. For static elasticity problems, the displacement vectors in two phason fields are expressed in terms of two pairs of associated harmonic functions or two analytic functions. For dynamic problems, the displacement vectors can be represented in terms of an auxiliary function satisfying a fourth-order partial differential equation. A general solution of phasons is given by the solution of two diffusion equations. Phason elastic fields induced by a dislocation in a quasicrystal with 18-fold symmetry are determined and exhibit an inverse singularity.     

Isotropic and anisotropic physical properties of quasicrystals

Since quasicrystals have positional and orientational long-range order, they are essentially anisotropic. However, the researches show that some physical properties of quasicrystals are isotropic. On the other hand, quasicrystals have additional phason degrees of freedom which can influence on their physical behaviours. To reveal the quasicrystal anisotropy, we investigate the quasicrystal elasticity and other physical properties, such as thermal expansion, piezoelectric and piezoresistance, for which one must consider the contributions of the phason field. The results indicate that: for the elastic properties, within linear phonon domain all quasicrystals are isotropic, and within nonlinear phonon domain the planar quasicrystals are still isotropic but the icosahedral quasicrystals are anisotropic. Moreover, the nonlinear elastic properties due to the coupling between phonons and phasons may reveal the anisotropic structure of QCs. For the other physical properties all quasicrystals behave like isotropic media except for piezoresistance properties of icosahedral quasicrystals due to the phason field.     

Numerical solution of 3D Green's function for the dynamic system of anisotropic elasticity

In this Letter, we used homotopy perturbation method to obtain numerical solution of the 3D Green's function for the dynamic system of anisotropic elasticity. Application of homotopy perturbation method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. The numerical results obtained from convolution of Green's function and data of the Cauchy problem are compared with the exact solution for cubic media. The results reveal that the proposed method is very effective and simple.     

Green’s tensor for crystals of the hexagonal system

Expressions for components of the Green’s tensor of the basic equation of the elasticity theory for hexagonal system crystals have been obtained using the Lifshitz-Rozentsveig method. A problem is in principle reduced to finding the roots of a sixth-order algebraic equation. They are either complex or purely imaginary for all known hcp metals. In both cases, the desired components of the Green’s tensor are calculated exactly in contrast to metals of the cubic system. A limiting transition to the isotropic approximation is shown.     

An electronmicroscopic study of hexagonal phases related to quasicrystals in Zn-Mg-RE alloys

A number of hexagonal phases with inflated/deflated a lattice parameters but equal c parameters are found to exist with compositions close to that of the icosahedral phase. Surprisingly, no cubic or orthorhombic approximants are found. The closeness of the hexagonal phases to the icosahedral phase is shown through the electron diffraction features. Through stereographic analysis of the hexagonal phases it is shown that they consist of three icosahedral units rotated by 120 degrees about a common twofold axis, forming the sixfold axis. It is suggested that this phase falls into a class of crystals closely related to both the icosahedral and decagonal quasicrystals, but are not approximants to them. These phases can instead be considered to be approximant to the hexagonal quasicrystal.     

Crystalline structures as an approximant of quasicrystal and distortion of B12 icosahedron in boron-rich solids: Search for semiconducting quasicrystal

Analyzing the crystalline structure of α- and β-rhombohedral boron as an approximant structure of a quasicrystal, atomic structures of two unit cells (prolate and oblate rhombohedra) of icosahedral quasicrystal are constructed. According to molecular orbital calculation, a neutral B ₁₂ H ₁₂ icosahedral cluster is distorted to a cubic or rhombohedral type by the Jahn–Teller effect. The distortion of B ₁₂ in the β-rhombohedral boron and K₂B₁₂H₁₂ has the same sign as the lowest energy distortion in the calculation. Among the α-rhombohedral type structures, the distortion of B₁₂ is correlated with the rhombohedral axis angle, and the angle is the closest to the icosahedral angle for B₆O and the distortion is the smallest for B–C. A meta-stable phase has been found on the way towards crystallization of amorphous B–C films.     

Analytic solutions to a finite width strip with a single edge crack of two-dimensional quasicrystals

In this paper, we investigate the well-known problem of a finite width strip with a single edge crack, which is useful in basic engineering and material science. By extending the configuration to a two-dimensional decagonal quasicrystal, we obtain the analytic solutions of modes I and II using the transcendental function conformal mapping technique. Our calculation results provide an accurate estimate of the stress intensity factors K_I and K_II, which can be expressed in a quite simple form and are essential in the fracture theory of quasicrystals. Meanwhile, we suggest a generalized cohesive force model for the configuration to a two-dimensional decagonal quasicrystal. The results may provide theoretical guidance for the fracture theory of two-dimensional decagonal quasicrystals.     

Stress fluctuations and elasticity in one- and two-component cubic crystals

The elastic constants of the Lennard-Jones system and a binary repulsive particle system (in 1:13 composition) obeying the Weeks-Chandler-Andersen (WCA) potential (WCA2-AB13 system) have been computed using the stress fluctuation formalism by equilibrium molecular dynamics simulations. The systems are under finite fixed pressures. The evolution of the elastic moduli as a function of the temperature is characterised. In both systems, the melting transition is signalled by a jump of the shear modulus, in consistence with the behaviour of the specific volume. At zero temperature, while the elasticity is totally affine for the Lennard-Jones face-centered cubic (fcc) crystal, it clearly contains a non-affine part for the AB13 cubic superlattice. The evolution of the non-affine elasticity as a function of the temperature is investigated, and its behaviour is compared to that of a typical glass-former.