Parallelogram Lattice Research Articles

Chiral active particles are sensitive reporters to environmental geometry

Chiral active particles (CAPs) are self-propelling particles that break time-reversal symmetry by orbiting or spinning, leading to intriguing behaviors. Here, we examined the dynamics of CAPs moving in 2D lattices of disk obstacles through active Brownian dynamics simulations and granular experiments with grass seeds. We find that the effective diffusivity of the CAPs is sensitive to the structure of the obstacle lattice, a feature absent in achiral active particles. We further studied the transport of CAPs in obstacle arrays under an external field and found a reentrant directional locking effect, which can be used to sort CAPs with different activities. Finally, we demonstrated that parallelogram lattices of obstacles without mirror symmetry can separate clockwise and counter-clockwise CAPs. The mechanisms of the above three novel phenomena are qualitatively explained. As such, our work provides a basis for designing chirality-based tools for single-cell diagnosis and separation, and active particle-based environmental sensors.

The [formula omitted]-vortex problem in a doubly periodic rectangular domain with constant background vorticity

We formulate the N point vortex problem in a doubly-periodic rectangle with a constant background vorticity using the hydrodynamic Green’s function. We derive an explicit formula for the hydrodynamic Green’s function and the velocities of the point vortices in terms of the Schottky–Klein prime function using a conformal mapping approach. The sum of vortex strengths can be arbitrary, and when it is zero we recover previous known results including the integrability of the two and three-vortex problems. A non-zero sum of vortex strengths leads to a constant background vorticity. We derive a Hamiltonian structure for the equations and show that the two-vortex problem is integrable, and classify all possible vortex motions. In addition, for general N, we obtain several fixed lattice equilibrium configurations including single-layered and double-layered equilibria. In the latter case, we also obtain lattice configurations with defects consisting of point vortices with inhomogeneous strengths. We find equilibria arranged in doubly-periodic rectangular and parallelogram lattices consisting of N=1,2,3,4 vortices per fundamental lattice.

Coverage-dependent study of nickel tetraphenyl-porphyrin on Au(332) and Au(788)

The coverage-dependent self-assembly of Nickel-tetraphenyl porphyins on Gold vicinal surfaces was studied by scanning tunneling microscopy. On Au(788) the molecular self-assembly is square-shaped, following the molecular symmetry. On Au(332) the molecules nucleate in a parallelogram lattice for monolayer coverage and in a squared lattice for multilayer coverage. Depending on the vicinal surface the molecules adopt either a saddle shaped or a more planar conformation. The experiments demonstrate that crystallographic direction, terrace sizes and coverage are important in establishing different assembly geometries.

Realization of a flat-band superprism on-chip from parallelogram lattice photonic crystals.

By optimizing the dispersion curve of a parallelogram-based 2D photonic crystal superprism for constant angular group velocity dispersion over a broad bandwidth, we designed a device capable of experimentally demonstrating linear dispersion from 1500 to 1600nm with clear separation of as many as eight channels, while maintaining a compact footprint.

The Brillouin zones and band gaps of a two-dimensional phononic crystal with parallelogram lattice structure

We present a detailed theoretical study on the acoustic band structure of two-dimensional (2D) phononic crystal. The 2D phononic crystal with parallelogram lattice structure is considered to be formed by rigid solid rods embedded in air. For the circular rods, some of the extrema of the acoustic bands appear in the usual high-symmetry points and, in contrast, we find that some of them are located in other specific lines. For the case of elliptic rods, our results indicate that it is necessary to study the whole first Brillouin zone to obtain rightly the band structure and corresponding band gaps. Furthermore, we evaluate the first and second band gaps using the plane wave expansion method and find that these gaps can be tuned by adjusting the side lengths ratio R, inclined angle θ and filling fraction F of the parallelogram lattice with circular rods. The results show that the largest value of the first band gap appears at θ=90° and F=0.7854. In contrast, the largest value of the second band gap is at θ=60° and F=0.9068. Our results indicate that the improvement of matching degree between scatterers and lattice pattern, rather than the reduction of structural symmetry, is mainly responsible for the enhancement of the band gaps in the 2D phononic crystal.

基于部分带隙光子晶体的微型宽带过渡波导

Partial bandgap characteristics of parallelogram lattice photonic crystals are proposed to suppress the radiation modes in a compact dielectric waveguide taper so as to obtain high transmittance in a large wavelength range. Band structure of the photonic crystals shows that there exists a partial bandgap. The photonic crystals with partial bandgap are then used as the cladding of a waveguide taper to reduce the radiation loss efficiently. In comparison with the conventional dielectric taper and the complete bandgap photonic crystal taper, the partial bandgap photonic crystal taper has a high transmittance of above 85% with a wide band of 170 nm.

Self-collimated waveguide bends and partial bandgap reflection of photonic crystals with parallelogram lattice

The photonic crystal structure with parallelogram lattice, capable of bending a self-collimated wave with free angles and partial bandgap reflection, is presented. The equifrequency contours show that the direction of the collimation wave can be turned by tuning the angle between the two basic vectors of the lattice. Acute, right, and obtuse angles of collimating waveguide bends have been realized by arc lattices of parallelogram photonic crystals. Moreover, partial bandgap reflection of the parallelogram lattice photonic crystals is validated from the equifrequency contours and the projected band structures. A waveguide taper based on this partial bandgap reflection is also designed and proved to have above 85% transmittance over a very wide operating bandwidth of 180 nm.

The orchard visibility problem and some variants

Imagine that you are standing at the center of a circular orchard, with trees centered at all of the lattice points except there is no tree at the origin itself (where you are standing). How large must the radius of the trees be in order to completely block your view (in every direction). Let R be the radius of the orchard, and r be the radius of the trees. It turns out that if R (the radius of the orchard) is an integer then you can see out if and only if r < 1 / R 2 + 1 . Allen [T.T. Allen, Polya's orchard problem, Amer. Math. Monthly 93 (1986) 98–104] attributes this problem to Polya [G. Polya, Zahlentheoretisches und wahrscheinlichkeitstheoretisches über die Sichtweite im Walde, Arch. Math. Phys. Ser. 2 27 (1918) 135–142] and solves it using trigonometric techniques. He generalizes the result to orchards whose radius is not an integer. We give an alternative proof of these results based on the Stern–Brocot wreath. We generalize the results to parallelogram lattices. We also consider the problem of what radius the trees need to have to block the view between some pair of trees. For parallelogram lattices, the ratio between the radius needed to block the view between all trees and the radius needed to block the view between some pair of trees asymptotically approaches 2 (as the radius of the orchard increases).

Critical points and dynamic systems with planar hexagonal symmetry

Abstract In this investigation, we detect and utilize critical points of functions with hexagonal symmetry in order to study their dynamics. The asymmetric unit in a parallelogram lattice is chosen as the initial searching region for a critical point set in a dynamic plane. The accelerated direct search algorithm is used within the parallelogram lattice to search for the critical points. Parameter space is separated into regions (chaotic, periodic or mixed) by the Ljapunov exponents of the critical points. Then the generalized Mandelbrot set (M-set), which is a cross-section of the parameter space, is constructed. Many chaotic attractors and filled-in Julia sets can be generated by using parameters from this kind of M-sets.

Study on self-collimated light-focusing device using the 2-D photonic crystal with a parallelogram lattice

This paper investigates the optical characteristics of a light-focusing device (LFD) consisting of two-dimensional (2-D) photonic crystals (PCs) configured in a parallelogram lattice structure. The LFD is composed of two jointed symmetric PCs that have a strong self-collimation effect. This self-collimation effect focuses the incident light wave. For a square lattice, however, in which the equi-frequency contours have fourfold symmetry (/spl pi//2 rotation symmetry), the band structure needed to realize an LFD was not obtained. The proposed solution was to consider the parallelogram lattice arrangement since its equi-frequency contours do not have /spl pi//2 rotation symmetry; a prototype LFD composed of 2-D PCs with parallelogram lattice structure exhibits desirable focusing performance. The authors also investigate the asymmetry of light wave transmission through the LFD. Finite-division time-domain (FDTD) simulations of the optical property of the LFD show that the LFD has distinct transmission asymmetry.

Elastic waves and homogenization in oblique periodic structures

A mathematical model has been constructed to describe elastic waves propagating in a two–dimensional solid containing a doubly periodic parallelogram array of circular holes. A multipole expansion method is employed which takes into account a coupling between shear and dilatational waves via the traction boundary conditions and determines the structure of the propagating modes. It is important that the homogenized elastic structure be anisotropic; this follows from analysis presented here. The algorithm has been implemented into a computer code, which was used to construct the dispersion diagrams and analyse the filtering properties of the composite structure. It is of particular interest to study the hexagonal and rhombic types of parallelogram lattices, which can be shown to exhibit phononic band–gaps.

Locally lattice sampling designs for isotropic random fields

For predicting $\int_G v(x)Z(x)dx$, where v is a fixed known function and Z is a stationary random field, a good sampling fesign should have a greater density of observations where v is relatively large in absolute value. Designs using this idea when $G = [0, 1]$ have been studied for some time. For G a region in two dimensions, very little is known about the statistical properties of cubature rules based on designs with varying density. This work proposes a class of designs that are locally parallelogram lattices but whose densities can vary. The asymptotic variance of the cubature error for these designs is obtained for a class of isotropic random fields and an asymptotically optimal sequence of cubature rules within this class is found. I conjecture that this sequence of cubature rules is asymptotically optimal with respect to all cubature rules.

Central charge of the parallelogram-lattice strong coupling Schwinger model.

We put forth a Fierz-transformed hopping expansion for strong coupling Wilson fermions. As an application, we show that the strong coupling Schwinger model on parallelogram lattices with nonbacktracking Wilson fermions span, as a function of the lattice skewness angle, the $\ensuremath{\Delta}=\ensuremath{-}1$ critical line of six-vertex models. This Fierz-transformed formulation also applies to backtracking Wilson fermions, which as we describe apparently correspond to richer systems. However, we have not been able to identify them with exactly solved models.

Structure of crystalline VDAC, the voltage-gated channel in the mitochondrial outer membrane

The voltage-gated, mitochondrial channel, VDAC, is formed by a 31-kDa outer-membrane polypeptide. Crystalline arrays of this channel, produced by phospholipase A2 treatment of Neurospora crassa mitochondrial outer membranes, consist of groups of 6 channels repeated on a parallelogram (“oblique”) lattice (a=13.3nm, b=11.5nm, γ = 109°). These membrane crystals are polymorphic, i.e. lateral contraction is triggered by a polyanion which also decreases VDAC’s gating potential. Projection images of unstained, frozen-hydrated VDAC arrays indicate that lattice contraction is accompanied by changes in the distribution of protein away from the hexameric repeat unit.

Cryo-Electron Microscopy and Correlation Averaging of Frozen-Hydrated, Polymorphic 2D Crystals of the Mitochondrial Outer Membrane Channel

Voltage-dependent, anion-selective channels (VDAC) are formed in the mitochondrial outer membrane (mitOM) by a 30-kDa polypeptide. These channels form ordered 2D arrays when mitOMs from Neurospora crassa are treated with soluble phospholipase A2. We obtain low-dose electron microscopic images of unstained specimens of VDAC crystals preserved in vitreous ice, using a Philips EM420 equipped with a Gatan cryo-transfer stage. We then use correlation analysis to compute average projections of the channel crystals. The procedure involves Fourier-filtration of a region within a crystal field to obtain a preliminary average that is subsequently cross-correlated with the entire crystal. Subregions are windowed from the crystal image at coordinates of peaks in the cross-correlation function (CCF, see Figures 1 and 2) and summed to form averages (Figure 3).The VDAC channel forms several different types of crystalline arrays in mitOMs. The polymorph first observed during phospholipase treatment is a parallelogram array (a=13 run, b=11.5 run, θ==109°) containing 6 water-filled pores per unit cell. Figure 1 shows the CCF of a sub-field of such an “oblique” array used to compute the correlation average of Figure 3A. With increased phospholipase treatment, other polymorphs are observed, often co-existing within the same crystal. For example, two distinct (but closely related) types of lattices occur in the field corresponding to the CCF of Figure 2: a “contracted” version of the parallelogram lattice (a=13 run, b=10 run, θ=99°), and a near-rectangular lattice (a=8.5 run, b=5 nm). The pattern of maxima in this CCF suggests that a third, near-hexagonal lattice (a=4.5 nm) may also be present. The correlation averages of Figures 3B-D were computed from polycrystalline fields, using peak coordinates in regions of CCFs corresponding to each of the three lattice types.

X-ray study of oxygen monolayers on graphite

Oxygen monolayers on graphite have been investigated in the temperature region between 15 and 53 K by X-ray diffraction. The monolayer δ phase with a low 2D density consists of a centered parallelogram lattice, with the molecular axes parallel to the graphite surface. The monolayer ζ phase with a higher 2D density forms a simple triangular lattice with the molecular axes perpendicular to the graphite surface; the model of two mutually incommensurate triangular layers for the ζ phase proposed by previous X-ray investigators is inappropriate in monolayer region. The coverages in the phase boundaries proposed by the present study are nearly consistent to those reported by a neutron scattering study, but considerably lower than those reported by previous X-ray and heat-capacity studies. This discrepancy reveals that in those X-ray and heat-capacity experiments a considerable amount of oxygen nucleated as bulk O2 in the cell without being adsorbed on the graphite surface, because a nonequilibrium state occurred during the cooling of the sample. A comparison of the coverages in the phase boundaries with the monolayer capacities calculated from the lattice vectors of the 2D solids suggests that a few percent occupancy of molecules in the second layer is needed during the δ-ζ transition of the first layer of O2 on graphite. The δ phase melted into the fluid I, while the melting of the ζ and δ + ζ coexistence phases occurred in two stages: from the solids into the fluid II and then into the fluid I.

Three-dimensional structure of mitochondrial outer-membrane channels from fungus and liver

The mitochondrial outer membrane (OM) contains a class of 30-kD polypeptides called VDAC, for the voltage-dependent, anion-selective channel activity which they induce in phospholipid membranes. VDAC occurs in extended periodic arrays on the outer membranes of mitochondria of the fungus Neurospora crassa. The arrays are polymorphic, varying from a parallelogram lattice (Fig. 1a) with six negative-stain-accumulating sites per unit cell (a=12.6 nm, b=11.1 nm, θ=109°) to a rectangular lattice (Fig. 1b) with two densely staining loci per unit cell (a=8.5 nm, b=5.0 nm). We have calculated the three-dimensional structure of the oblique OM array embedded in uranyl acetate. Images were recorded from a crystalline membrane which was tilted relative to the 100keV electron beam from +40° to -40° in 10° intervals. A common crystalline area in each image was Fourier-filtered by a quasi-optical procedure using reflections out to 1/(3.0 nm). Following rotational and translational alignment, the images were used to reconstruct the OM array in three dimensions by a weighted back-projection procedure.

X-ray study of molecular oxygen adsorbed on graphite

We have performed a detailed X-ray diffraction study of O 2 adsorbed on UCAR-ZYX and Le Carbon Lorraine vermicular exfoliated graphite between 15 and 50 K. At least four phases of physisorbed oxygen are found. The monolayer δ phase consists of a centered parallelogram lattice, with the molecular axes parallel to the graphite surface. The data are consistent with a triple point at 26 K. The melting transition at a coverage of one monolayer appears to be first order. At higher coverages the molecules undergo a lying-down to standing-up transition; the higher coverage ζ phase froms an approximately triangular lattice with the molecular axes perpendicular to the graphite surface. Satellite peaks around the (1, 0) Bragg peak indicate, however, that this cannot be a simple triangular lattice; possible explanations include successively incommensurate layers or a sinusoidal density modulation. For coverages in the two-layer region the ζ phase modulation peaks disappear at 37 K, and at 40 K the adsorbed oxygen appears to undergo a first order melting transition into a fluid phase. With increasing coverage, the 2D X-ray diffraction profiles and phase boundaries do not connect smoothly onto those of the 3D α and β phases. At low temperatures ( T < 30 K) the ζ phase always coexists with bulk crystallites; for temperatures near the 2D melting transition the 3D peaks are not observable. These data, taken together with the heat capacity results, suggest a wetting transition with only the bilayer lamellar phase or bulk O 2 being stable at low temperatures.

Ideal efficiency of a radiating element in an infinite array

The efficiency of a radiating element in an infinite planar array has been defined as the ratio of the power radiated to the power available from the element. This efficiency is a measure of the strength of mutual coupling effects and also of the average reflection coefficient magnitude at a typical element in the active array as the elements are phased over a prescribed range. An upper bound to this efficiency, which is only a function of the array geometry, exists and corresponds to an array of antennas with certain ideal characteristics. A technique for evaluating this ideal or maximum efficiency of an element in an arbitrary infinite is presented. While this efficiency is readily evaluated for some special configurations, the computation for any general space becomes straightforward in terms of the formalism. The formalism is based on the lattice and a transformed (Cartesian) version of the space and reciprocal lattices. The required integral is numerically equal to the area common to a unit square and an ellipse in this transformed lattice. Results are given for various rhombic and parallelogram lattices which include the square and the equilateral triangle as special cases. It is noted that for parallelograms of aspect ratio greater than about 4 to 1, the efficiency is nearly the same as for the limiting linear array.