Rhombic Lattice Research Articles

Localized waves in the nonlinear rhombic waveguide array

Solitary electromagnetic waves propagating along the waveguides forming a rhombic one-dimensional lattice are considered. Two waveguides that are part of the unit cell are assumed to be made of an optical linear material, while the third waveguide from the same array is composed of material with the cubic nonlinearity. The equations of the coupled waves spreading in each waveguide are solved under some approximation. These solutions represent the breather like solitary waves, which are akin to three component soliton.

Numerical and theoretical analysis of the dynamic mechanical behaviour of a modified rhombic dodecahedron lattice structure

Metallic lattice structures have been widely utilized in different fields such as automobile, aerospace and so on, owing to their superior mechanical properties and energy absorption characteristics. In this paper, the enhanced dynamic mechanical properties of the modified rhombic dodecahedron (RD) lattice structures were investigated systematically through numerical simulation and theoretical analysis. Results indicated that compressive modulus, initial yield strength and normalized specific energy absorption (SEA) of the modified RD lattice structures all exhibited a quadratic growth trend with the increase of relative density beyond following the traditional law. Meanwhile, the plateau stress and normalized SEA exhibited certain sensitivity to the strain rate and almost linearly increased as the loading rate increased. Finally, a modified rigid-perfectly plastic-locking shock wave model was also proposed, and the predicted results of the proposed modified shock wave model were consistent with the simulation results.

An investigation into compressive responses of shape memory polymeric cellular lattice structures fabricated by vat polymerization additive manufacturing

A combination of additive manufacturing techniques with shape memory materials, so that the shape, property, or functionality of a 3D printed structure can change as a function of time, has recently created new progress in 4D printing. Low-density lattice structures, due to their unique mechanical properties and engineering characteristics, have been candidates for lightweight structures and energy absorbing applications. In the present work, Rhombic and Body-Centered Cubic (BCC) cellular lattice structures, as well as cylindrical bulk samples, were designed and fabricated with Digital Light Process (DLP) by using shape memory resin. The energy absorption of SMP samples was studied in terms of the capabilities of absorption and recovery. In addition, deformation mechanisms of the structures, the influence of strain rate, cyclic behavior and the strain recovery of the structures after each cycle were investigated. All the studies were done in three different cold, warm and hot programming schemes to evaluate the effects of temperature on shape memory effect of the products. Although both structures showed nearly the same strain recovery rates at all conditions, Rhombic structure was found to possess better functional and structural behaviors than BCC lattice in terms of strength, stiffness, and absorption as well as recovery of the induced energy.

Non-Abelian Aharonov-Bohm caging in photonic lattices

Aharonov-Bohm (AB) caging is the localization effect in translational-invariant lattices due to destructive interference induced by penetrated magnetic fields. While current research focuses mainly on the case of Abelian AB caging, here we go beyond and develop the non-Abelian AB caging concept by considering the particle localization in a 1D multi-component rhombic lattice with non-Abelian background gauge field. In contrast to its Abelian counterpart, the non-Abelian AB cage depends on both the form of the nilpotent interference matrix and the initial state of the lattice. This phenomena is the consequence of the non-Abelian nature of the gauge potential and thus has no Abelian analog. We further propose a circuit quantum electrodynamics realization of the proposed physics, in which the required non-Abelian gauge field can be synthesized by the parametric conversion method, and the non-Abelian AB caging can be unambiguously demonstrated through the pumping and the steady-state measurements of only a few sites on the lattice. Requiring only currently available technique, our proposal can be readily tested in experiment and may pave a new route towards the investigation of exotic photonic quantum fluids.

Localization in non-Hermitian asymmetric rhombic lattice

This paper proposes a non-Hermitian asymmetric rhombic lattice which supports fully flat spectrum and Aharonov-Bohm caging formed by the oppositely oriented unidirectional couplings at the exceptional point. The localization area of excitation can be manipulated by the distribution of the gain and loss

Governing equations for stress distribution in rhombic disk packings

The governing equations of stress tensor fields are significant to understand the mechanical behaviors of static granular matter. A multiscale method based on the algebra of translation operators is proposed to derive the governing equations of stress transmission in rhombic lattice packings of disks with friction. Except for the two well-known equations, i.e. the symmetry of stress tensor and the stress equilibrium equation, we also obtain a stress-structure equation as the constitutive relation of the lattice packings. A key variable called friction-geometry tensor is introduced for the expression of the constitutive equation and the Janssen coefficient is given from the eigenvalues of this tensor. DEM (Discrete Element Method) simulations are performed to calculate the numerical values of Janssen coefficient and the results are in good agreement well with theoretical predictions.

Topological Corner Modes in Graphene-Coated Nanowire Waveguide

We study the plasmonic second-order topological modes in nanowire waveguides which are covered by monolayer graphene. The waveguide arrays are arranged in the kagome lattice. In a triangle-shaped lattice, the topological modes are localized at the corner of the triangle by tuning the spatial spacing between the different waveguides. The corner modes also depend on the corner shape, which only exist at one corner in a rhombic lattice. In addition to the corner modes, both structures also support the topological edge mode. We show that the corner modes experience a smaller modal wavelength, longer propagation distance, and smaller mode volume than the edge modes. The study may be utilized to explore the topological bound modes at the nanoscale.

Experimental and numerical study and multi-objective optimisation of quasi-static compressive test on three-dimensional printed lattice-core sandwich structures

This paper focuses on the energy absorption of additively-manufactured or 3D-printed lattice-core sandwich structures of different configurations. The rhombic dodecahedron lattice unit cell, which has been extensively investigated for energy absorption, is the starting point for this research. The compressive behaviour of 3D lattice structures with rhombic dodecahedron unit cells was examined at quasi-static loading along with the influence of the angle between struts, struts diameter, and the number of layers in the core. The angle between struts had the highest degree of influence on the specific energy absorption. The results showed that the diameter of struts has been the most influential parameter to increase mean and peak forces. The results of using the design expert software showed the optimal specific energy absorption and peak force to be 3.8779 J/gr and 3458.25 N, respectively.

Process-induced geometric defect sensitivity of Ti–6Al–4V lattice structures with different mesoscopic topologies fabricated by electron beam melting

Ti–6Al–4V lattice structures with different cell configurations are fabricated by electron beam melting (EBM) method. Two different cell topologies (octet truss and regular rhombic dodecahedron lattice) corresponding to stretching-dominated and bending-dominated respectively are taken into consideration. The mechanical properties of these specimens are examined by quasi-static compression experiments which are conducted by electronic universal machine, while the deformation evolutions of all the specimens are captured by a digital camera. The results indicate that the specimens with rhombic dodecahedron cells exhibit higher specific strength than the octet truss structures. This unusual phenomenon is proved to be caused by the geometric imperfections in the printed stretching-dominated structures. Then the X-ray computed tomography is adopted to capture the process-induced defects in the EBM-printed specimens, and virtual tests are performed by the X-ray tomography based 3D reconstruction model. The computational study explains that the collapse strength of the rhombic dodecahedron (RD) lattice is mainly affected by the surface roughness of the struts, while the weak connection at the joints of the struts leads to the significant reduction in the strength of the octet truss specimens. Meanwhile, the dynamic performance of the printed specimens is predicted by ideal shock wave model and validated by numerical simulation. The results indicate that the defects lead to the significant discrepancy between the theoretical prediction and numerical simulation of octet truss structure, but the failure mode transition of RD lattice structure can be well predicted by the analytical model.

Structure and dielectric properties of solid solutions Bi7Ti4+xWxNb1−2x−0.1V0.1O21 (x=0.1−0.4)

The structural and electrophysical characteristics of a number of solid solutions of layered oxides of the perovskite-type Bi7[Formula: see text]Wx[Formula: see text][Formula: see text][Formula: see text] ([Formula: see text]) are studied. According to X-ray powder diffraction data, all the compounds studied are single-phase and have the structure of Aurivillius phases ([Formula: see text]) with a rhombic crystal lattice (space group I2[Formula: see text]cm, [Formula: see text]). Changes in tetragonal and rhombic distortions of perovskite-like layers in compounds were considered depending on their chemical composition. The temperature dependences of the relative permittivity [Formula: see text] (T) were measured. It was shown that the Curie temperature [Formula: see text] of the perovskite-type oxides Bi7[Formula: see text]Wx[Formula: see text][Formula: see text][Formula: see text] ([Formula: see text]) linearly decreases with increasing parameter [Formula: see text]. The activation energies of charge carriers were obtained in different temperature ranges. It was found that there are three temperature regions with very different activation energies due to the different nature of the charge carrier s in the studied compounds. The effect of substitution of [Formula: see text] ions by [Formula: see text] ions is investigated. It was found that for a number of compounds, the substitution of niobium ions by vanadium ions led to an increase in the dielectric constant and a decrease in the dielectric loss tangent.

Dynamic compressive behavior of a modified additively manufactured rhombic dodecahedron 316L stainless steel lattice structure

Abstract Dynamic compression properties of the modified rhombic dodecahedron (RD) lattice structures were investigated by using a Split Hopkinson Pressure Bar (SHPB) system. All the deformation processes were recorded and a digital image correlation (DIC) technique was employed to analyse the strain distribution during the compression process. Experimental results indicated that the modified lattice structure showed better compressive strength, plateau stress and energy absorption when compared with the original one under both quasi-static and dynamic loading conditions. The effect of the shape parameter on the deformation mechanisms of the RD lattice structure was discussed, where three different deformation modes were observed. Finite element analysis was also conducted to simulate the dynamic response of the modified lattice structure. The finite element results are in good agreement with the experimental results. Then, a mode classification map was plotted and discussed based on the numerical results.

Nonhexagonal Lattices From a Two Species Interacting System

A two species interacting system motivated by the density functional theory for triblock copolymers contains a long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. A minimal two species periodic assembly is one with the least energy per area. There is a parameter $b$ in [0,1] and the type of the lattice associated with a minimal assembly varies depending on $b$. There are several thresholds defined by a number $B=0.1867\ldots.$ If $b \in [0, B)$, a minimal assembly is associated with a rectangular lattice whose ratio of the longer side to the shorter side is in $[\sqrt{3}, 1)$; if $b \in [B, 1-B]$, a minimal assembly is associated with a square lattice; if $b \in (1-B, 1]$, a minimal assembly is associated with a rhombic lattice with an acute angle in $[\frac{\pi}{3}, \frac{\pi}{2})$. Only when $b=1$ is this rhombic lattice a hexagonal lattice. None of the other values of $b$ yield a hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where hexagonal lattices are ubiquitously observed.

Structure of Polycrystalline CeNi3-Based Intermetallic Hydrides at 293 and 5 K

The structure of polycrystalline CeNi3Н3.0 hydride samples at 5 and 293 K has been studied by neutron diffraction. The hydrides have a rhombic lattice (sp. gr. Pmcn, no. 62, Z = 12). The site parameters of metal and hydrogen atoms have been determined. It is established that the distribution of hydrogen atoms over the hydride structure sites at 5 K is similar to the hydrogen distribution at room temperature. A comparison of the neutron diffraction patterns obtained at different temperatures indicates a change in the sample microstructure upon cooling.

Observation of quincunx-shaped and dipole-like flatband states in photonic rhombic lattices without band-touching

We demonstrate experimentally the existence of compact localized states (CLSs) in a quasi-one-dimensional photonic rhombic lattice in presence of two distinct refractive-index gradients (i.e., a driven lattice ribbon) acting as external direct current (dc) electric fields. Such a lattice is composed of an array of periodically arranged evanescently coupled waveguides, which hosts a perfect flatband that touches both remaining dispersive bands when it is not driven. The external driving is realized by modulating the relative writing beam intensity of adjacent waveguides. We find that a y-gradient set perpendicularly to the ribbon preserves the flatband while removing the band-touching. The undriven CLS - which occupies two lattices sites over one unit cell - turns into a quincunx-shaped CLS spanned over two unit cells. Instead, an x-gradient acting parallel to the ribbon yields a Stark ladder of CLS whose spatial profile is unchanged with respect to the undriven case. We notably find that their superposition leads to Bloch-like oscillations in momentum space.

New group V graphyne: two-dimensional direct semiconductors with remarkable carrier mobilities, thermoelectric performance, and thermal stability

The past decades have witnessed the great progress and successes in the research and applications of two-dimensional (2D) carbon materials such as graphene, graphdiyne, and so on. Similar to pure 2D carbon materials, 2D carbon nitride–like h-BN also possesses excellent electronic, mechanical, and optical properties. In this work, stimulated by the chemical tuition of atomic substitution, a new family of monolayer group V graphyne (C16N4, C16P4, and C16As4) with rhombic lattice is designed by replacing some C atoms with group V elements of N, P, or As in 2D graphyne. By using first-principles approach, we investigated their thermal stability, electronic/thermal transport properties, and thermoelectric performance and found that N(P,As)-graphyne monolayers are semiconductors with considerable direct bandgap values of 0.87 eV (0.59 eV, 0.71 eV), respectively. The ab initio molecular dynamics results demonstrate that N(P,As)-graphyne monolayers remain stable up to 1500 K. They all possess high carrier mobilities with the order of 105cm2V−1s−1 for electrons along the zigzag direction. Under the uniaxial tensile strains in the range of 0% to 10%, N(P,As)-graphyne monolayers keep direct-bandgap properties, and the effective mass of carriers can be efficiently tuned. Moreover, the calculated thermoelectric figure of merits at room temperature for the new monolayer group V graphyne are 0.62∼0.69 owing to the low lattice thermal conductivity, which are comparable with some conventional thermoelectric materials. Their excellent electronic transport and thermoelectric performance make N(P,As)-graphyne monolayers promising in high-speed (opto)electronic and thermoelectric devices, and the strain-engineering properties may lead to applications in flexible nanoelectronics.

Tensor network simulation of the Kitaev-Heisenberg model at finite temperature

We investigate the Kitaev-Heisenberg (KH) model at finite temperature using the exact environment full update (eeFU), introduced in Phys. Rev. B 99, 035115 (2019), which represents the purification of a thermal density matrix on an infinite hexagonal lattice by an infinite projected entangled pair state (iPEPS). We show that thanks to a dynamical mapping from a hexagonal to a rhombic lattice, the eeFU on the hexagonal lattice is as efficient as the simple full update (FU) algorithm. Critical temperatures for coupling constants in the stripy and the antiferromagnetic phase are estimated. They are an order of magnitude less than the couplings in the Hamiltonian. By a duality transformation, these results can be mapped to, respectively, the ferromagnetic and zigzag phases. For the special case of the pure Kitaev model, which is tractable by quantum Monte-Carlo but the most challenging for tensor networks, the algorithm is benchmarked against the Monte-Carlo results. It recovers accurately the crossover to spin ordering and qualitatively the one to flux ordering.

Post-buckling and dynamic response of angled struts in elastic lattices

This paper presents analytical and reduced-order numerical solutions describing the non-linear response of slender, elastic struts (or plates) inclined relative to the loading direction. The solutions provide a highly efficient framework to predict post-buckling behaviors in cellular structures, including: stability regimes, peak strains during and after buckling, the work dissipated via cyclic loading, the impact of biaxial loading, and the role of geometric imperfections in the struts. Regime maps are presented that illustrate configurations that lead to snap-through, permanent deformation after unloading, strut failure, and enhanced hysteresis during cyclic loading. The maps illustrate that reversible snap-through events only occur within a very specific range of relative density (e.g.∼0.25−0.4 for rhombic lattices). A highly efficient non-linear single degree-of-freedom dynamics model is derived from the statics solution, and is shown to be in excellent agreement with fully explicit, non-linear, dynamic finite element simulations for inclined struts. This simplified dynamics model is used to quantify the relationships between quasi-static responses, loading frequencies and energy dissipation during cycling loading. A key finding is that effective damping during cyclic loading is dramatically increased by non-linear behavior, even when the corresponding quasi-static result exhibits zero hysteresis. The implications for structured foams and the design of lightweight structural dampers is briefly discussed.

Fundamental Domains for Rhombic Lattices with Dihedral Symmetry of Order 8

We show by construction that every rhombic lattice $\Gamma$ in $\mathbb{R}^{2}$ has a fundamental domain whose symmetry group contains the point group of $\Gamma$ as a subgroup of index $2$. This solves the last open case of a question raised in a preprint by the authors on fundamental domains for planar lattices whose symmetry groups properly contain the point groups of the lattices.

The Influence of Molybdenum Equivalent on the Anisotropy of Thermal Expansion of Titanium Martensite Lattice

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Two-dimensional graphyne-like carbon nitrides: Moderate band gaps, high carrier mobility, high flexibility and type-II band alignment

Two-dimensional (2D) carbon nitrides are drawing increasing interests due to the structural diversities and unique electronic properties, superiors to the pure 2D carbon materials. Motivated by the recent experimental progresses, we investigated from first-principles a new family of 2D carbon nitrides composed of sp- and sp2-hybridized carbon atoms and with kagome, rhombic and hexagonal lattices. The kagome and rhombic lattices exhibit moderate electronic band gaps (0.98–3.34 eV), high carrier mobility in the order of 3 × 105 cm2V−1s−1, high flexibility with critical strain up to 44%–47%, and type-II band alignment which are desired for applications in electronic devices and solar cells. Our findings are expected to offer theoretical bases for synthesis and applications of these novel 2D carbon nitrides and promote the following experimental exploration.