Equilateral Triangular Lattice Research Articles

Granular dynamics of inelastic spheres in Couette flow

A numerical program has been developed to simulate an assembly of inelastic, frictional hard spheres inside a control volume undergoing a steady-state rapid Couette flow induced by the top and bottom bumpy walls. The bumpy walls are made of hemispheric particles fixed onto flat plates. The flow particles can collide with the wall particles and the exposed flat areas of the walls. The macroscopic flow properties are found to depend on a number of material and geometric properties of the granules, the bumpy walls, and the control volume. These properties include the overall solids fraction of the system, the height of the shear gap, the wall-particle concentration, the wall-particle distribution, the diameter ratio of the wall particle to the flow particle, the coefficients of restitution, the friction coefficients, and the sticking tangential restitution coefficients between the flow particles, the wall particles, and the flat walls. A parametric study is undertaken to examine the effect of some of the interesting factors identified above. A new definition for the slip velocity yields positive values consistently, and it represents a significant improvement over the previous ones. By exposing the flat areas of the bumpy walls for collisions, the transfer of energy and momentum from the driving surfaces to the flow medium can be enhanced. Depending on the wall-particle distribution, there exist optimal wall-particle concentrations at which the stresses may be maximized or the slip velocities may be minimized. For hemispheric wall particles arranged in an equilateral triangular lattice, the optimal wall-particle area fraction for maximizing the stresses is about 0.44 while the one for minimizing the slip velocity is about 0.36. The simulation results also show that there exists for gravity-free Couette flow of inelastic, frictional spheres a critical solids fraction of about 0.5 beyond which the stresses are found to decrease with increasing solids concentration. In general, there is reasonable agreement between the simulation results for stresses and the experimental measurements.

Structural and magnetic properties of CuCl2 graphite intercalation compounds.

Structural and magnetic properties of stage-1, stage-2, and stage-3 ${\mathrm{CuCl}}_{2}$ graphite intercalation compounds (GIC's) were studied by means of x-ray, electron- and neutron-diffraction, dc magnetic susceptibility, and electron spin resonance (ESR) measurements. The ${\mathrm{Cu}}^{2+}$ ions form an isosceles triangular lattice with one short side and two longer sides. The in-plane dc magnetic susceptibility shows Curie-Weiss behavior above 150 K, a broad maximum around 62--65 K, indicative of low-dimensional magnetic correlations, and a Curie-type behavior below 20 K, attributable to paramagnetic inhomogeneities in the sample. The temperature and magnitude of the susceptibility maximum are more consistent with a two-dimensional Heisenberg antiferromagnet than a one-dimensional model, suggesting that the distortion from an equilateral triangular lattice is not an important factor in the magnetic behavior of ${\mathrm{CuCl}}_{2}$ GIC's. ESR measurements indicate that the local magnetic symmetry of ${\mathrm{Cu}}^{2+}$ spins is tetragonal. The angular dependence of the ESR linewidth at 4.2 K is explained by the combined effects of a canting of the tetragonal axis from the c axis and a weak anisotropy in the Land\'e g factor that favors spins to lie in the intercalate plane. No magnetic phase transition is observed from dc magnetic susceptibility down to 1.5 K and magnetic neutron scattering above 0.5 K.

A free‐space double‐grid diplexer for a millimeter‐wave radiometer

AbstractWe report on the design of a 45° incidence diplexer to be used in a millimeter‐wave radiometer. This equipment will be flown on a low‐orbiting satellite and should analyze the natural emissions of the Earth and the atmosphere in the bands around 90, 157, and 183 GHz. The diplexer has been realized with two grids of tripoles arranged in an equilateral triangular lattice and printed on both sides of a quartz slab. Experimental data are presented that are in good agreement with the theoretical predictions. © 1993 John Wiley & sons, Inc.

Plane-wave illumination of concentric-ring frequency-selective surfaces

In the paper we present the results of a modal analysis of the current distribution around the circumference of thin concentric rings arranged on both square and equilateral triangular lattices. The particular case of plane-wave illumination at 45° incidence is considered owing to its relevance to oblique-incidence diplexer systems. Measured and predicted transmission curves, together with computer curves illustrating the frequency dependence of various current-mode amplitudes, are plotted for selected concentric-ring structures, each with the same mean radius for the outer ring, but with different inner-ring radii. The results relate individual current modes to features in the observed transmission curves. They confirm that the copolar transmission and lower reflection bands can be modelled with only one current mode in each ring. Crosspolar effects are discussed for incidence on the square lattice in the planes ϕ = 0, 45° and 90°, where they are shown to be small for pure TE and TM.

Performance of two tripole arrays as frequency-selective surfaces

The current distribution along the arms of tripoles arranged on two equilateral triangular lattices has been computed and shown to have a simple cosinusoidal form. Calculated and measured transmission curves agree closely and give band spacings ≳2:1. The crosspolar performance in a 45°-incidence diplexer is lattice-dependent.

Balanced designs for observing intra-variety nearest-neighbour interactions

When an intra-variety competition experiment is performed using an arrangement of plants and spaces on a regular lattice of sites, it is desirable for some purposes for the design to be balanced. That is, equal numbers of plants are immediately surrounded by equal numbers of each of the possible numbers of spaces. For an equilateral triangular lattice this can be 0, 1, ...... 6 spaces, defining seven ‘treatments’ that can be applied to a plant. A set of nineteen distinct binary strings is presented, any of which can be used to generate an efficient balanced design which is easy to implement.

Cross polarisation of electromagnetic waves transmitted through waveguide dichroic plates

Dichroic or frequency-sensitive surfaces modify the cross-polar performance of feeds transmitting through them. In the paper we investigate the influence of two flat waveguide dichroic plates 1 cm thick and 15 cm in diameter, placed 4 cm in front of a simple, linearly-polarised, choked waveguide feed. Measured cross-polar levels, mainly in the 45° planes, are compared with the levels predicted by a modal analysis in the frequency range 19–26 GHz. The results agree well. With these particular plates the cross-polar levels over most of the frequency range are lower when the guides are arranged on a close-packed equilateral triangular lattice than when on the more-symmetrical square lattice. Ten wave guide modes are included in the analysis, and their relative importance is discussed. Two cumulative radiation patterns are illustrated, comparing the development of the farfield radiation patterns for the two lattices as the contributions of successive guide modes are summed.

Geometry of waveguides in a dichroic secondary mirror

Dichroic Cassegrain mirrors have been constructed from waveguide stacked on a variety of lattices. Their performance in the transmission mode is discussed. Circular guide on an equilateral triangular lattice gave the best results. To improve the copolar patterns the structure was loaded with dielectric.

Ginzburg-Landau theory of the transition at Hc(2)in thin film superconductors

An investigation is made, based on Ginzburg-Landau theory of the transition of a thin film of type II superconductor in a strong perpendicular magnetic field. By considering the fourth-order terms in the GL free energy expression a reasonable description can be obtained of the conductivity through the transition, which reproduces the equilibrium flux-flow conductivity for H >Hc(2). Also considered is the possibility that the geometrical structure of the Abrikosov lattice may fluctuate away from the equilateral triangular lattice close to Hc.

Robust Estimation of Density for a Two-Dimensional Point Process

A method is proposed for obtaining estimators of the density of a two-dimensional point process which are reasonably unbiased for Poisson, square lattice and equilateral triangular lattice processes. A simple fairly robust estimator is discussed.

Flux-Line Lattices of Pure Type II Superconductors with Anisotropic Fermi Surface

The free energy appropriate for a pure single crystal type II superconductor with an anisotropic Fermi surface is calculated near the upper critical field H 0 , at low tempera· tures. It is shown that various flux-line lattices can occur depending on the magnitude of the anisotropy and the direction of the applied magnetic field relative to the crystal axis. § I. Introduction In a previous paper1> devoted to the analysis near T 0 , we have shown qualitatively that in a superconductor with an anisotropic Fermi surface of cubic symmetry, an equilateral triangular flux-line lattice is stable just below the transi­ tion temperature, but other lattice structures occur depending on the magnitude of the anisotropy and the direction of the applied magnetic field relative to the crystal axis at low temperatures. Furthermore, the anisotropy removes the degeneracy for the direction of the array of the flux-line. In fact, experiments2> show many structures of the flux-line lattice (FLL) and a certain correlation between its array and the crystal axes. The FLL ar­ rangement of PbTl, for example, is qualitatively consistent with the results of I. That is, a square FLL exists in the magnetic field parallel to the four-fold crystal symmetry axis ( [1.0.0] direction), while the FLL remains to be an equilateral triangular when the field is parallel to the three-fold crystal symmetry axis ( [1.1.1] direction). In this paper we investigate FLL near the upper critical field at low tem­ peratures extending the work of I because the anisotropy has a large effect on the properties of a superconductor at those temperature region. In the following we use the unit system h=kB=c=l.

Transmission through a Conducting Screen Perforated Periodically with Apertures

A general solution to the problem of determining first the aperture field distribution and then the transmission and reflection coefficients of an infinite planar conducting sheet perforated periodically with apertures has been formulated. The excitation is considered to be a plane wave incident at any arbitrary angle. The aperture dimensions and array element spacings were assumed to be comparable with the wavelength of the incident electromagnetic field. The solution given can include the effect of a dielectric slab used to support the thin conducting sheet. The solution is obtained by matching the tangential field components at the surface of the screen. The resulting integral equation is solved by the method of moments which reduces the integral equation to a system of linear algebraic equations that can be solved with the use of a digital computer. Accurate results for both the magnitude and phase of the aperture field distribution and the transmission coefficients for the propagating modes are determined explicitly for a specific example of slots arranged in an equilateral triangular lattice. The balance of power flow between the reflected and the transmitted waves has been checked with satisfactory results. The solution can be applied to the problem of scattering from a conducting screen with periodic apertures and to the complementary problem of scattering from a set of conducting plates by the use of Babinet's principle.

Bulk Solution of Ginzburg-Landau Equations for Type II Superconductors: Upper Critical Field Region

A solution is described of the Ginzburg---Landau equations of the Abrikosov type for a homogeneous type II superconductor just below the upper critical field. This solution is characterized by magnetic field maxima corresponding to an equilateral triangular lattice and a value $\ensuremath{\beta}=1.16$ of Abrikosov's parameter. Abrikosov's square-lattice solution with $\ensuremath{\beta}=1.18$ has a higher free energy and is unstable with respect to the triangular lattice solution.

Pressure Drop for Parallel Flow Through Rod Bundles

Abstract Friction factors for flow parallel to a rod bundle have been determined experimentally in a Reynolds-number range from approximately 5000 to 100,000. Three rod arrays were used; namely, a 1.20 pitch-to-rod diameter ratio on a square lattice, and a 1.12 pitch-to-rod diameter ratio on both square and equilateral triangular lattices. Entrance plus exit-loss coefficients, and joint loss coefficients between axially aligned rod bundles also were measured experimentally for tube-sheet end connections. For the square arrays, the effect of tube-sheet flow area on these loss coefficients was investigated. Results are presented in graphical form.