Rhombic Lattice Research Articles

Complex rhombus lattice photonic crystals for broadband all-angle self-collimation

In this paper, we discuss two-dimensional photonic crystals of complexrhombus lattice structures to enable broadband, ‘all-angle’ light collimation.The proposed structures are made of complex elliptical dielectric rods (ε = 12) embedded in an air background. Based on the equi-frequency contours (EFCs) calculated bythe plane wave expansion method (PWEM), these structures show a wide range of open, nearlyflat EFCs, with the maximum angular deviation from the primary collimating direction less than2°. The broadest relative range of these open, nearly flat EFCs found is 17% of a chosencentral wavelength. For a sample element centered at 1550 nm, this range coverswavelengths from 1429 to 1695 nm; the devices can be scaled to cover alternativewavelength ranges. These structures could benefit applications that require a broadoperating wavelength range. Potential applications include collimating beam combiners,multiplexers and solar light collectors.

Organization of the lamin scaffold in the internal nuclear matrix of normal and transformed hepatocytes

Nuclear lamins are among the more abundant proteins making up the internal nuclear matrix, but very little is known about their structure in the nucleoplasm. Using immunoelectron microscopy, we demonstrate the organization of lamins in the nuclear matrix isolated from rat hepatocytes for the first time. Lamin epitopes are arrayed both in locally ordered clusters and in quasi-regular rows. Fourier filtering of the images demonstrates that the epitopes are placed at the nodes and halfway between the nodes of square or rhombic lattices that are about 50 nm on each side, as well as along rows at regular ∼25-nm intervals. In addition, we have compared this structure with that of the internal nuclear matrix isolated from persistent hepatocyte nodules. In transformed hepatocytes, the islands of lamin lattice are lost, and only a partial regularity in the rows of gold particles remains. We suggest that orthogonal lattice assembly might be an intrinsic property of lamin molecules, and that the disassembly may be triggered by simple molecular events such as phosphorylation.

Creating a Library of Complex Metallic Nanostructures via Harnessing Pattern Transformation of a Single PDMS Membrane

By harnessing the elastic instability in a single PDMS membrane consisting of a square lattice array of circular pores, we fabricated a library of complex nanostructures in Au with variable feature size, connectivity, and geometry, including arrays of diamond-plate patterns (or elliptic herringbones), compound structures of circular dots and elliptical lines, heartbeat waves, aligned ovals, and a rhombus lattice of holes and lines. This was achieved first by swelling the PDMS membrane, followed by convective assembly of nanoparticles on the membrane. By taking advantage of the unique 3-D topography of the nanoparticle film and its photoresist replica, we could gradually etch the photoresist film to vary the feature size and connectivity of the underlying Au patterns. Further, through a combination of mechanical stretching (at different strain levels and stretching angles) and solvent swelling of the same PDMS membrane, we created a richer library of complex patterns in Au without application of new masters.

Structural transitions in a crystalline bilayer: the case of Lennard-Jones and Gaussian coremodels

We study structural transitions in a system of interacting particlesarranged as a crystalline bilayer, as a function of the densityρ and the distanced between thelayers. As d is decreased a sequence of transitions involving triangular, rhombic, square and centredrectangular lattices is observed. The sequence of phases and the order of transitionsdepends on the nature of the interactions.

Crystalline structure and crystallization of stereoisomeric polyamides derived from arabinaric acid

The pair of stereoisomeric polyamides PA-6dAr and PA-6lAr, as well as the racemic stereocopolyamide PA-6dlAr was synthesized from hexamethylenediamine and 2,3,4-tri-O-methyl-arabinaric acid (d, l and their equimolar mixture, respectively). All these polyamides were aregic, thermally stable and display high crystallinity. The combined analysis by X-ray and electron diffraction revealed that the all three polyamides adopted the same crystal structure, which consists of a rhombic lattice with the chains in a highly contracted conformation capable of accommodating efficiently the sugar moiety in the space. The Avrami kinetic analysis revealed that copolyamide PA-6dlAr crystallized isothermally much slower than the optically homogeneous polyamides. A parallel study carried out on the racemic mixture of PA-6dAr and PA-6lAr evidenced that this mixture has a crystal structure and displays a thermal behavior similar to their separated components, and that its crystallization rate is intermediate between them and the racemic copolyamide PAdlAr.

Binary crystals in two-dimensional two-component Yukawa mixtures

The zero-temperature phase diagram of binary mixtures of like-charge particles interacting via a screened Coulomb pair potential is calculated as a function of composition and charge ratio. The potential energy obtained by a Lekner summation is minimized among a variety of candidate two-dimensional crystals. A wealth of different stable crystal structures is identified including A, B, AB(2), A(2)B, and AB(4) structures [A (B) particles correspond to large (small) charge.] Their elementary cells consist of triangular, square, or rhombic lattices of the A particles with a basis comprising various structures of A and B particles. For small charge asymmetry there are no intermediate crystals besides the pure A and B triangular crystals. The predicted structures are detectable in experiments on confined mixtures of like-charge colloids or dusty plasma sheets.

Instability of Square Vortex Lattice ind-Wave Superconductors is due to Paramagnetic Depairing

The effects of paramagnetic depairing on structural transitions between vortex lattices of a quasi-two-dimensional d-wave superconductor are examined. We find that, when the Maki parameter alphaM is of order unity, a square lattice induced by a d-wave pairing is destabilized with increasing fields, and that a reentrant rhombic lattice occurs in higher fields. Further, a weak Fermi surface anisotropy competitive with the pairing symmetry induces another structural transition near Hc2. These results are consistent with the structure changes of the vortex lattice in CeCoIn5 in H parallel c determined from recent neutron scattering data.

Colloidal Crystal Growth at Externally Imposed Nucleation Clusters

We study the conditions under which and how an imposed cluster of fixed colloidal particles at prescribed positions triggers crystal nucleation from a metastable colloidal fluid. Dynamical density functional theory of freezing and Brownian dynamics simulations are applied to a two-dimensional colloidal system with dipolar interactions. The externally imposed nucleation clusters involve colloidal particles either on a rhombic lattice or along two linear arrays separated by a gap. Crystal growth occurs after the peaks of the nucleation cluster have first relaxed to a cutout of the stable bulk crystal.

Research on Three Types of Rhombus Lattice Photonic Band Structures

The gap-midgap ratio for three types of rhombus lattice photonic band structures are analyzed by plane wave expansion method, which is confirmed by the HFSS simulation. Firstly, general wave vectors in the first Briliouin zone are derived. Secondly, the gap-midgap ratios as a function of filling factor and background material are investigated, respectively. These results would provide theoretical instruction for designing arbitrary bend waveguides or other optical integrated devices using photonic crystal with a rhombus lattice.

二维菱形晶格光子带隙研究

The relative band gap for a rhombus lattice photonic crystal is studied by plane wave expansion method and high frequency structure simulator (HFSS) simulation. General wave vectors in the first Briliouin zone are derived. The relative band gap as a function of air-filling factor and background material is investigated, respectively, and the nature of photonic band gap for different lattice angles is analyzed by the distribution of electric energy. These results would provide theoretical instruction for designing optical integrated devices using photonic crystal with a rhombus lattice.

Lane formation in oppositely charged colloids driven by an electric field: Chaining and two-dimensional crystallization

A binary mixture of oppositely charged colloids which is driven by an external electric field is studied by extensive Brownian dynamics computer simulations, ignoring hydrodynamic interactions. The particle interaction is modeled via a screened Coulomb potential together with a steric repulsion. A strong electric field leads to lane formation of oppositely driven lanes. Each lane comprises particles of the same charge. A nonequilibrium "phase diagram" classifying different steady states is obtained as a function of the colloidal volume fraction and the Coulomb coupling. Different steady states are characterized by structural correlations perpendicular and parallel to the applied field. We find a variety of different phases involving lane chains at small volume fraction and low screening, and lanes with two-dimensional crystalline order perpendicular to the field at high volume fraction. The lateral crystalline order can be a square, triangular, or rhombic lattice. In between there is a lateral network structure. These predictions can be verified in real-space experiments on oppositely charged colloids.

Symmetry of nanotubes rolled up from arbitrary two-dimensional lattices along an arbitrary chiral vector

The line group describing full symmetry of a nanotube rolled up from an arbitrary layer along an arbitrary chiral vector is found. A helical axis and pure rotations are always present, while mirror and glide planes appear only for special chiral vectors in rhombic or rectangular lattices. Nanotubes are not translationally periodical unless the unit cell of the layer satisfies very specific conditions. Physical consequences, including the incommensurability of carbon nanotubes, are discussed.

Elastic properties of anisotropic domain wall lattices

Interest in the elastic properties of regular lattices constructed from domain walls has recently been motivated by cosmological applications as solid dark energy. This work investigates the particularly simple examples of triangular, hexagonal, and square lattices in two dimensions and a variety of more complicated lattices in three dimensions which have cubic symmetry. The relevant rigidity coefficients are computed taking into account nonaffine perturbations where necessary, and these are used to evaluate the propagation velocity for any macroscopic scale perturbation mode. Using this information we assess the stability of the various configurations. It is found that triangular lattices are isotropic and stable, whereas hexagonal lattices are unstable. It is argued that the simple orthonormal cases of a square in two dimensions and the cube in three are stable, except to perturbations of infinite extent. We also find that the more complicated case of a rhombic dodecahedral lattice is stable, except to the existence of transverse modes in certain directions, whereas a lattice formed from truncated octahedra is unstable.

Distorted vortex lattice in a tetrahedral superconductor

The equilibrium shape and the orientation of the vortex lattice are studied for an s-wave tetrahedral superconductor in the vicinity of the upper critical field. The phase diagram, which includes transitions between rhombic and rectangular lattices, is constructed in the parameter space of the Ginzburg-Landau functional. The developed theory is applied to the heavy-fermion superconductor PrOs4Sb12. In a wide range of parameters, the shape of the vortex lattice is only weakly dependent on the temperature. The neutron scattering measurements of the vortex lattice in PrOs4Sb12 can be explained by particularities of the tetrahedral symmetry group and are further supported by analysis of the appropriate band structure calculations.

Effects of injected atomic coherence in broad-area lasers

We analyze the effect of injected atomic coherence on transverse patterns of a broad area laser by means of the semiclassical two-level Maxwell-Bloch equations. A single longitudinal mode is considered. The injected atomic coherence forces a spatially homogeneous profile to appear and locks the field phase to a single value. Above a pump threshold value a very rich scenario of patterns is developed. Near threshold we find stationary patterns such as rhombic and hexagonal lattices. Well above threshold nonstationary patterns such as complex highly ordered vortex lattices traveling along the cross section, and nearly traveling waves appear.

TEM observations of Al 72Ni 12Co 16 decagonal quasicrystal subjected to high-temperature indentation

The effect of high-temperature indentation testing on the microstructural change of a single-grained Al 72Ni 12Co 16 decagonal quasicrystal ( d-QC) has been investigated by transmission electron diffraction and microscopy (TEM) technique. A definite surface normal to the 10-fold axis of the d-QC was subjected to Vickers indentation test at an elevated temperature of 800 °C with a load of 1 kgf. The TEM investigation of this sample has evidently shown that there are specific types of defects densely distributed in the indented-region, such as grain-boundaries, dislocations and phason-related defects. Furthermore, high-resolution electron microscopy observations have revealed the widespread distribution of nanocrystalline domains in the indented-region, where groups of decagonal atom clusters have a periodic arrangement to form a rhombic lattice. On the basis of these observations, possible deformation mechanisms of the d-QC during high-temperature indentation will be discussed.

Patterns of synchrony in lattice dynamical systems

From the point of view of coupled systems developed by Stewart, Golubitsky and Pivato, lattice differential equations consist of choosing a phase space Rk for each point in a lattice, and a system of differential equations on each of these spaces Rk such that the whole system is translation invariant. The architecture of a lattice differential equation specifies the sites that are coupled to each other (nearest neighbour coupling (NN) is a standard example). A polydiagonal is a finite-dimensional subspace of phase space obtained by setting coordinates in different phase spaces as equal. There is a colouring of the network associated with each polydiagonal obtained by colouring any two cells that have equal coordinates with the same colour. A pattern of synchrony is a colouring associated with a polydiagonal that is flow-invariant for every lattice differential equation with a given architecture. We prove that every pattern of synchrony for a fixed architecture in planar lattice differential equations is spatially doubly-periodic, assuming that the couplings are sufficiently extensive. For example, nearest and next nearest neighbour couplings are needed for square and hexagonal couplings, but a third level of coupling is needed for the corresponding result to hold in rhombic and primitive cubic lattices. On planar lattices this result is known to fail if the network architecture consists only of NN. The techniques we develop to prove spatial periodicity and finiteness can be applied to other lattices as well.

Clifford algebra approach to the coincidence problem for planar lattices

The problem of coincidences of planar lattices is analyzed using Clifford algebra. It is shown that an arbitrary coincidence isometry can be decomposed as a product of coincidence reflections and this allows planar coincidence lattices to be characterized algebraically. The cases of square, rectangular and rhombic lattices are worked out in detail. One of the aims of this work is to show the potential usefulness of Clifford algebra in crystallography. The power of Clifford algebra for expressing geometric ideas is exploited here and the procedure presented can be generalized to higher dimensions.

A mesoscopic model of a two-dimensional solid state structural transformation: statics anddynamics

We study the equilibrium properties of a system of particles in two dimensions, interactingvia pair and three-body potentials. This system undergoes a structural transition from asquare to a rhombic lattice and thus constitutes a simple model for a generic tetragonal toorthorhombic transition. We aim at an intermediate level of description lyingin between that of coarse grained elastic strain Hamiltonians and microscopicab initio approaches. We obtain macroscopic thermodynamic properties and thephase diagram at zero and finite temperatures as a function of the density andthe relative strengths of the pair and three-body energies using lattice sums,an approximate ‘cell model’ theory and molecular dynamics simulations in theNV T ensemble. In addition, we study the dynamics of nucleation following a quench from thesquare to the triangular phase (Rao and Sengupta 2003 Phys. Rev. Lett. 91 045502). As in realsolids, the final microstructure depends sensitively on the depth of the quench—a shallowquench results in an equilibrium ferrite while a deep quench gives rise to a metastabletwinned martensite. We find, in accordance with experiments, that the twinned martensiteis associated with a diffusionless transformation. We propose that this model solidmay be used as a test bed for studies of the statics and dynamics of structuraltransitions.

Rhombic Periodicity Lattice in Bifurcational Symmetry Breaking Problems

AbstractGeneral theory of symmetry breaking problems in branching theory is presented in the works [1–3]. Such problems are invariant relative to Euclidean space motions group and their solution which is invariant relatively this group is the rest state or uniform linear motion. At the stability loss cell structure solutions arise, which are invariant relative to the group of the definite period shifts along the definite directions, passing mutually under discrete subgroup transformations that is defined by the symmetry of elementary cell of the periodicity. Thus the Euclidean space motions group is changed by the symmetry of a certain crystallographic group.In this article for the case of 4–dimensional degeneracy of the linearized Fredholm operator abstract bifurcational symmetry breaking problems both for stationary and Andronov‐Hopf bifurcation with planar rhombic periodicity lattice are considered. Applications to hydrodynamical problems are given.