One-dimensional Quasicrystals Research Articles

Bending Analysis of Functionally Graded One-Dimensional Hexagonal Piezoelectric Quasicrystal Multilayered Simply Supported Nanoplates Based on Nonlocal Strain Gradient Theory

In this study, the nonlocal strain gradient theory is adopted to investigate the static bending deformation of a functionally graded (FG) multilayered nanoplate made of one-dimensional hexagonal piezoelectric quasicrystal (PQC) materials subjected to mechanical and electrical surface loadings. The FG materials are assumed to be exponential distribution along the thickness direction. Exact closed-form solutions of an FG PQC nanoplate including nonlocality and strain gradient micro-size dependency are derived by utilizing the pseudo-Stroh formalism. The propagator matrix method is further used to solve the multilayered case by assuming that the layer interfaces are perfectly contacted. Numerical examples for two FG sandwich nanoplates made of piezoelectric crystals and PQC are provided to show the influences of nonlocal parameter, strain gradient parameter, exponential factor, length-to-width ratio, loading form, and stacking sequence on the static deformation of two FG sandwich nanoplates, which play an important role in designing new smart composite structures in engineering.

Antiplane Fracture Problem of Three Nanocracks Emanating from an Electrically Permeable Hexagonal Nanohole in One-Dimensional Hexagonal Piezoelectric Quasicrystals

Based on the Gurtin-Murdoch surface/interface model and complex potential theory, by constructing a new conformal mapping, the electrically permeable boundary condition with surface effect is established, and the antiplane fracture problem of three nanocracks emanating from a hexagonal nanohole in one-dimensional hexagonal piezoelectric quasicrystals with surface effect is studied. The exact solutions of the stress intensity factor of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate are obtained under the two electrically permeable and the electrically impermeable boundary conditions. The numerical examples show the influence of surface effect on the stress intensity factors of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate under the two boundary conditions. It can be seen that the surface effect leads to the coupling of the phonon field, phason field, and electric field, and with the decrease of cavity size, the influence of surface effect is more obvious.

Anti-plane problem of nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional hexagonal piezoelectric quasicrystals**Project supported by the National Key R&D Program of China (Grant No. 2017YFC1405605), the Innovation Youth Fund of the Ocean Telemetry Technology Innovation Center of the Ministry of Natural Resources, China (Grant No. 21k20190088), the Natural Science Foundation of Inner Mongolia, China (Grant No. 2018MS01005), and

By constructing a new conformal mapping function, we study the surface effects on six edge nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional (1D) hexagonal piezoelectric quasicrystals under anti-plane shear. Based on the Gurtin–Murdoch surface/interface model and complex potential theory, the exact solutions of phonon field, phason field and electric field are obtained. The analytical solutions of the stress intensity factor of the phonon field, the stress intensity factor of the phason field, the electric displacement intensity factor and the energy release rate are given. The interaction effects of the nano-cracks and nano-hole on the stress intensity factor of the phonon field, the stress intensity factor of the phason field and the electric displacement intensity factor are discussed in numerical examples. It can be seen that the surface effect leads to the coupling of phonon field, phason field and electric field. With the decrease of cavity size, the influence of surface effect is more obvious.

Electric field assisted alignment of monoatomic carbon chains

We stabilize monoatomic carbon chains in water by attaching them to gold nanoparticles (NPs) by means of the laser ablation process. Resulting nanoobjects represent pairs of NPs connected by multiple straight carbon chains of several nanometer lengths. If NPs at the opposite ends of a chain differ in size, the structure acquires a dipole moment due to the difference in work functions of the two NPs. We take advantage of the dipole polarisation of carbon chains for ordering them by the external electric field. We deposit them on a glass substrate by the sputtering method in the presence of static electric fields of magnitudes up to 105 V/m. The formation of one-dimensional carbyne quasi-crystals deposited on a substrate is evidenced by high-resolution TEM and X-ray diffraction measurements. The original kinetic model describing the dynamics of ballistically flowing nano-dipoles reproduces the experimental diagram of orientation of the deposited chains.

Tunable high Tc superconducting photonic band gap resonators based on hybrid quasi-periodic multilayered stacks

One-dimensional hybrid photonic quasicrystals made of superconductor (HgBa2Ca2Cu3O8+δ) and Dielectric (Si) materials are theoretically investigated by the Gorter Casimir two fluid model (GCTFM) and the transfer matrix method (TMM). The proposed quasicrystals super-lattices are built according the ternary configuration GF/GTM/GF. Where, GF and GTM are the generalized Fibonacci and Thue-Morse sequences, respectively. An interesting multi- photonic band gaps (m-PBGs) are achieved for suitable hybrid quasiperiodic system. The characteristic of these m-PBGs can be manipulated by the temperature of the superconductor and pressure of the system. The transmittance spectra exhibit tunable resonant peaks within the m-PBGs very sensitive to photonic system parameters. A superconducting resonator can be achieved by tuning the temperature, pressure, and thicknesses of the photonic system materials.

Exact Solutions for Interaction of Parallel Screw Dislocations with a Wedge Crack in One-Dimensional Hexagonal Quasicrystal with Piezoelectric Effects

The purpose of this paper is to consider the interaction between many parallel dislocations and a wedge-shaped crack and their collective response to the external applied generalized stress in one-dimensional hexagonal piezoelectric quasicrystal, employing the complex variable function theory and the conformal transformation method; the problem for the crack is reduced to the solution of singular integral equations, which can be further reduced to solving Riemann–Hilbert boundary value problems. The analytical solutions of the generalized stress field are obtained. The dislocations are subjected to the phonon field line force, phason field line force, and line charge at the core. The positions of the dislocations are arbitrary, but the dislocation distribution is additive. The dislocation is not only subjected to the external stress and the internal stress generated by the crack, but also to the force exerted on it by other dislocations. The closed-form solutions are obtained for field intensity factors and the image force on a screw dislocation in the presence of a wedge-shaped crack and a collection of other dislocations. Numerical examples are provided to show the effects of wedge angle, dislocation position, dislocation distribution containing symmetric configurations and dislocation quantities on the field intensity factors, energy release rate, and image force acting on the dislocation. The principal new physical results obtained here are (1) the phonon stress, phason stress, and electric displacement singularity occur at the crack tip and dislocations cores, (2) the increasing number of dislocations always accelerates the crack propagation, (3) the effect of wedge angle on crack propagation is related to the distribution of dislocations, and (4) the results of the image force on the dislocation indicate that the dislocations can either be attracted or rejected and reach stable positions eventually.

Non-Hermitian mobility edges in one-dimensional quasicrystals with parity-time symmetry

We investigate localization-delocalization transition in one-dimensional non-Hermitian quasiperiodic lattices with exponential short-range hopping, which possess parity-time ($\mathcal{PT}$) symmetry. The localization transition induced by the non-Hermitian quasiperiodic potential is found to occur at the $\mathcal{PT}$-symmetry-breaking point. Our results also demonstrate the existence of energy dependent mobility edges, which separate the extended states from localized states and are only associated with the real part of eigen-energies. The level statistics and Loschmidt echo dynamics are also studied.

A Hamiltonian model of the Fibonacci quasicrystal using non-local interactions: simulations and spectral analysis

This article presents a Hamiltonian architecture based on vertex types and empires for demonstrating the emergence of aperiodic order in one dimension by a suitable prescription for breaking translation symmetry. At the outset, the paper presents different algorithmic, geometrical, and algebraic methods of constructing empires of vertex configurations of a given lattice. These empires have non-local scope and form the building blocks of the proposed lattice model. This model is tested via Monte Carlo simulations beginning with randomly arranged N tiles. The simulations clearly establish the Fibonacci configuration, which is a one-dimensional quasicrystal of length N, as the final relaxed state of the system. The Hamiltonian is promoted to a matrix operator form by performing dyadic tensor products of pairs of interacting empire vectors followed by a summation over all permissible configurations. A spectral analysis of the Hamiltonian matrix is performed and a theoretical method is presented to find the exact solution of the attractor configuration that is given by the Fibonacci chain as predicted by the simulations. Finally, a precise theoretical explanation is provided which shows that the Fibonacci chain is the most probable ground state. The proposed Hamiltonian is a mathematical model of the one dimensional Fibonacci quasicrystal.

Conformal Quasicrystals and Holography

Recent studies of holographic tensor network models defined on regular tessellations of hyperbolic space have not yet addressed the underlying discrete geometry of the boundary. We show that the boundary degrees of freedom naturally live on a novel structure, a conformal quasicrystal, that provides a discrete model of conformal geometry. We introduce and construct a class of one-dimensional conformal quasicrystals, and discuss a higher-dimensional example (related to the Penrose tiling). Our construction permits discretizations of conformal field theories that preserve an infinite discrete subgroup of the global conformal group at the cost of lattice periodicity.

From a single slit to periodic, modulated, and quasiperiodic crystals – a physical space diffraction analysis of aperiodic systems

Using a statistical approach to diffraction analysis, it was shown that the scale of difficulty in the analysis of the diffraction image of a one-dimensional quasicrystal is not significantly different from the analysis of the diffraction image for a single slit. In both cases we get a rectangular probability distribution, for which the Fourier transform leads to analytical expressions on the envelopes of diffraction peaks expressed by means of simple trigonometric functions (sinc). The generalization to 2d (decagonal quasicrystals) and 3d (icosahedral quasicrystals) requires more complex calculations, but conceptually, it does not significantly differ from the calculations for model one-dimensional quasicrystals.

一维六方准晶中带双裂纹的椭圆孔口问题的塑性模拟

一维六方准晶中带双裂纹的椭圆孔口问题的塑性模拟

Design of output-graded narrow polychromatic filter by using photonic quasicrystals

The properties of one-dimensional photonic quasicrystals built according to one-dimensional-generalized Fibonacci (GF)/Generalized Thue-Morse (GTM) were investigated in order to design an output polychromatic filter. This main aperiodic multilayered structure was made up of alternate two dielectrics materials Silica and Bi4Ge3O12(BGO)with higher and lower refractive indices respectively. The Transfer Matrix Method (TMM) was adopted to calculate the photometric response. The transmittance spectrum exhibited a stacking of similar Bragg gaps (BGs) for specific arrangement (m = n) of GF and GTM. We noted that the positions and the number and size of BGs were sensitive to the constituent of the Photonic quasicrystal (PQC) system and lattice parameter of quasiperiodic sequence. Therefore, these configurations of multilayered stacks can be useful as a graded polychromatic filter.

Analysis of 3D planar crack problems in one-dimensional hexagonal piezoelectric quasicrystals with thermal effect. part I: Theoretical formulations

Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are performed systematically. Part I of this work derives a series of theoretical formulations that are then used to study the 3D planar crack problems in the QCs. The simple layer potential functions with the extended displacement discontinuities (EDDs) as the unknown variables and the general solution based on quasi-harmonic functions for the QCs under consideration are used to deduce the boundary equations that govern 3D planar crack problems. The hypersingular integral equation method is used to analyze the asymptotic singularities of the coupled thermal-electrical-phonon-phason fields near the crack edge. Expressions are then presented for the extended stress intensity factors (ESIFs) of a mixed model crack in terms of EDDs for arbitrarily-shaped cracks in the QCs, and the basic relationships between the energy release rate and the ESIFs are established. Closed-form solutions for some typical cracks, including an elliptical crack that is subjected to coupled electrical-phonon-phason loadings and a penny-shaped crack that is subjected to antisymmetric thermal loading, are determined via Fabrikant's analysis method. Additionally, both the physical quantities on the crack plane and the corresponding variables in the coupled thermal-electrical-phonon-phason field in the full space are given. The theoretical formulations derived in this paper provide a fundamental basis for development of the numerical approach proposed in Part II of our work, and can also serve as benchmarks for numerical solutions.

Analysis of 3D planar crack problems of one-dimensional hexagonal piezoelectric quasicrystals with thermal effect. part II: Numerical approach

Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are carried out systematically. Part II of the work aims to develop a general numerical approach to study 3D planar crack problems in 1D hexagonal piezoelectric QC media. Based on the theoretical formulations presented in Part I, a direct and effective method is proposed to derive the Green's functions for point extended displacement discontinuities (EDDs). These Green's functions are presented explicitly by a series of potential functions in a compact form. Using the superposition principle, the Green's functions for uniformly distributed EDDs over the crack elements are obtained. Related element solutions are used to construct the numerical approach, known as EDD boundary element method, for 1D hexagonal piezoelectric QCs. The proposed numerical method can be applicative to many complicated planar crack problems, such as multiple cracks, and cracks with non-uniform loadings, for 3D media composed of 1D hexagonal piezoelectric QCs with thermal effect. A comparison of the results obtained from the theoretical solutions given in Part I of the work with those obtained from the numerical method proposed here validates of the present investigation.

Three-dimensional static analysis of multilayered one-dimensional orthorhombic quasicrystal spherical shells with the piezoelectric effect

The static analysis of multilayered one-dimensional orthorhombic piezoelectric quasicrystals spherical shells is presented by using the state vector method. Based on two separated state equations, the solution of a homogenous spherical shell is derived, and the solution for the corresponding multilayered case is obtained via the propagator matrix method. Mechanical behaviors of spherical shells subjected to two kinds of symmetric loads are analyzed, respectively. Under uniformed loads, smaller phonon displacements are achieved in a quasicrystal-skin structure than those in crystal-skin one. Under the compression of rigid half-plane, the radius of the circular contact area increases with decreasing thickness of quasicrystal layers.

Output multichannel optical filter based on hybrid photonic quasicrystals containing a high-Tc superconductor

In this study, we determined the properties of one-dimensional photonic quasicrystals (PQC), including superconductors and dielectric materials, based on two representative quasiperiodic sequences comprising the generalized Thue–Morse (GTM) and generalized Fibonacci sequences in order to design a multichannel optical filter. By using the transfer matrix method and two-fluid model, we determined that the transmittance spectra exhibited a series of photonic band gaps (PBGs). In order to improve the characteristics of the PBGs, we considered the GTM configuration and applied a deformation comprising y=xh+1 along the PQC device, where h represents the deformation degree, and x and y are the thicknesses of the layers before and after deformation, respectively.

Bending Deformation of Multilayered One-Dimensional Quasicrystal Nanoplates Based on the Modified Couple Stress Theory

The modified couple stress theory has been applied in many nanomaterials except for nano-quasicrystals. In this paper, the modified couple stress theory is firstly adopted to analyze the static bending deformation of multilayered one-dimensional (1D) hexagonal quasicrystal (QC) nanoplates under surface loadings. The general solutions for the extended displacement and traction vectors in a simply supported and homogeneous QC nanoplate are derived by solving an eigenvalue problem reduced from the governing equations. Utilizing the propagator matrix method, the analytical solutions of multilayered 1D QC nanoplates are then obtained by assuming that the layer interfaces are continuous. Numerical examples for some kinds of nanoplates made up of QC and crystal ($$\hbox {BaTiO}_{3}$$) are provided to illustrate the effects of the material length parameter, the number of layers and the stacking sequence of nanoplates on the phonon and phason fields, which is helpful for the application of QCs to surface coating and solar energy selective absorber.

Accurate fracture analysis of electrically permeable/impermeable cracks in one-dimensional hexagonal piezoelectric quasicrystal junction

An accurate fracture analysis of a multi-material junction of one-dimensional hexagonal quasicrystals with piezoelectric effect is performed by using Hamiltonian mechanics incorporated in the finite element method. Two idealized electrical assumptions, including electrically permeable and impermeable crack-face conditions, are considered. In the Hamiltonian system, the analytical solutions to the multi-material piezoelectric quasicrystal around the crack tip (singular domain) are obtained and expressed in terms of symplectic eigensolutions. Therefore, the large number of nodal unknowns in the singular domain is reduced into a small set of undetermined coefficients of the symplectic series. The unknowns in the non-singular domain remain unchanged. Explicit expressions of phonon stresses, phason stresses, and electric displacement in the singular domain and newly defined fracture parameters are achieved simultaneously. Comparisons are presented to verify the proposed approach and very good agreement is reported. The key influencing parameters of the crack are discussed in detail. The effects of electrical assumptions and positions of the crack on the fracture parameters are discussed in detail.

One-Dimensional Quasicrystals with Power-Law Hopping.

One-dimensional quasiperiodic systems with power-law hopping, 1/r^{a}, differ from both the standard Aubry-André (AA) model and from power-law systems with uncorrelated disorder. Whereas in the AA model all single-particle states undergo a transition from ergodic to localized at a critical quasidisorder strength, short-range power-law hops with a>1 can result in mobility edges. We find that there is no localization for long-range hops with a≤1, in contrast to the case of uncorrelated disorder. Systems with long-range hops rather present ergodic-to-multifractal edges and a phase transition from ergodic to multifractal (extended but nonergodic) states. Both mobility and ergodic-to-multifractal edges may be clearly revealed in experiments on expansion dynamics.

Magnons in a Quasicrystal: Propagation, Extinction, and Localization of Spin Waves in Fibonacci Structures

Magnonic quasicrystals exceed the possibilities of spin wave (SW) manipulation offered by regular magnonic crystals, because of their more complex SW spectra with fractal characteristics. Here, we report the direct x-ray microscopic observation of propagating SWs in a magnonic quasicrystal, consisting of dipolarly coupled permalloy nanowires arranged in a one-dimensional Fibonacci sequence. SWs from the first and second band as well as evanescent waves from the band gap between them are imaged. Moreover, additional mini-band gaps in the spectrum are demonstrated, directly indicating an influence of the quasiperiodicity of the system. The experimental results are interpreted using numerical calculations and we deduce a simple model to estimate the frequency position of the magnonic gaps in quasiperiodic structures. The demonstrated features of SW spectra in one-dimensional magnonic quasicrystals allows utilizing this class of metamaterials for magnonics and makes them an ideal basis for future applications.