The hexagonal versus the square lattice

We establish Schmutz Schaller's conjecture that the hexagonal lattice is `better' than the square lattice. Schmutz Schaller (Bulletin of the AMS 35 (1998), p. 201), motivated by considerations from hyperbolic geometry, conjectured that in dimensions 2 to 8 the best known lattice sphere packings have `maximal lengths' and goes on to write: "In dimension 2 the conjecture means in particular that the hexagonal lattice is `better' than the square lattice. More precisely, let 0<h_1<h_2<... be the positive integers, listed in ascending order, which can be written as h_i=x^2+3y^2 for integers x and y. Let 0<q_1<q_2<... be the positive integers, listed in ascending order, which can be written as q_i=x^2+y^2 for integers x and y. Then the conjecture is that q_i<=h_i for i=1,2,3,..." Our proof requires computational prime number theory in combination with methods from a preprint of the first author (to appear in Math. Comp.), arXiv:math.NT/0112100.Comment: 24 pages, 6 figures, 2 table

The 5f localization/delocalization in square and hexagonal americium monolayers: A FP-LAPW electronic structure study

The electronic and geometrical properties of bulk americium and square and hexagonal americium monolayers have been studied with the full-potential linearized augmented plane wave (FP-LAPW) method. The effects of several common approximations are examined: (1) non-spin polarization (NSP) vs. spin polarization (SP); (2) scalar-relativity (no spin-orbit coupling (NSO)) vs. full-relativity (i.e., with spin-orbit (SO) coupling included); (3) local-density approximation (LDA) vs. generalized-gradient approximation (GGA). Our results indicate that both spin polarization and spin orbit coupling play important roles in determining the geometrical and electronic properties of americium bulk and monolayers. A compression of both americium square and hexagonal monolayers compared to the americium bulk is also observed. In general, the LDA is found to underestimate the equilibrium lattice constant and give a larger total energy compared to the GGA calculations. While spin orbit coupling shows a similar effect on both square and hexagonal monolayer calculations regardless of the model, GGA versus LDA, an unusual spin polarization effect on both square and hexagonal monolayers is found in the LDA results as compared with the GGA results. The 5f delocalization transition of americium is employed to explain our observed unusual spin polarization effect. In addition, our results at the LDA level of theory indicate a possible 5f delocalization could happen in the americium surface within the same Am II (fcc crystal structure) phase, unlike the usually reported americium 5f delocalization which is associated with crystal structure change. The similarities and dissimilarities between the properties of an Am monolayer and a Pu monolayer are discussed in detail.Comment: 22 pages, 8 figure

Scaling prediction for self-avoiding polygons revisited

We analyse new exact enumeration data for self-avoiding polygons, counted by perimeter and area on the square, triangular and hexagonal lattices. In extending earlier analyses, we focus on the perimeter moments in the vicinity of the bicritical point. We also consider the shape of the critical curve near the bicritical point, which describes the crossover to the branched polymer phase. Our recently conjectured expression for the scaling function of rooted self-avoiding polygons is further supported. For (unrooted) self-avoiding polygons, the analysis reveals the presence of an additional additive term with a new universal amplitude. We conjecture the exact value of this amplitude.Comment: 17 pages, 3 figure

Piezoelectric mechanism of orientation of a bilayer Wigner crystal in a GaAs matrix

A mechanism for orientation of bilayer classical Wigner crystals in a piezoelectric medium is considered. For the GaAs system the piezoelectric correction to the electrostatic interaction between electrons is calculated. It is shown that taking into account the correction due to the piezoelectric effect leads to a dependence of the total energy of the electron crystal on its orientation with respect to the crystallographic axes of the surrounding matrix. A generalization of Ewald's method is obtained for calculating the anisotropic interaction between electrons in a Wigner crystal. The method is used to calculate the energy of bilayer Wigner crystals in electron layers parallel to the crystallographic planes (001), (0-11), and (111) as a function of their orientation and the distance between layers, and the energetically most favorable orientation for all types of electron lattices in a bilayer system is found. It is shown that phase transitions between structures with different lattice symmetry in a Wigner crystal can be accompanied by a change of its orientation.Comment: 11 pages, 4 eps figures include

Design of two-dimensional particle assemblies using isotropic pair interactions with an attractive well

Using ground-state and relative-entropy based inverse design strategies, isotropic interactions with an attractive well are determined to stabilize and promote as- sembly of particles into two-dimensional square, honeycomb, and kagome lattices. The design rules inferred from these results are discussed and validated in the dis- covery of interactions that favor assembly of the highly open truncated-square and truncated-hexagonal lattices.Comment: 11 pages, 5 figures and supplemental materia

Lyapunov spectral analysis of a nonequilibrium Ising-like transition

By simulating a nonequilibrium coupled map lattice that undergoes an Ising-like phase transition, we show that the Lyapunov spectrum and related dynamical quantities such as the dimension correlation length~$\xi_\delta$ are insensitive to the onset of long-range ferromagnetic order. As a function of lattice coupling constant~$g$ and for certain lattice maps, the Lyapunov dimension density and other dynamical order parameters go through a minimum. The occurrence of this minimum as a function of~$g$ depends on the number of nearest neighbors of a lattice point but not on the lattice symmetry, on the lattice dimensionality or on the position of the Ising-like transition. In one-space dimension, the spatial correlation length associated with magnitude fluctuations and the length~$\xi_\delta$ are approximately equal, with both varying linearly with the radius of the lattice coupling.Comment: 29 pages of text plus 15 figures, uses REVTeX macros. Submitted to Phys. Rev. E

Via Hexagons to Squares in Ferrofluids: Experiments on Hysteretic Surface Transformations under Variation of the Normal Magnetic Field

We report on different surface patterns on magnetic liquids following the Rosensweig instability. We compare the bifurcation from the flat surface to a hexagonal array of spikes with the transition to squares at higher fields. From a radioscopic mapping of the surface topography we extract amplitudes and wavelengths. For the hexagon--square transition, which is complex because of coexisting domains, we tailor a set of order parameters like peak--to--peak distance, circularity, angular correlation function and pattern specific amplitudes from Fourier space. These measures enable us to quantify the smooth hysteretic transition. Voronoi diagrams indicate a pinning of the domains. Thus the smoothness of the transition is roughness on a small scale.Comment: 17 pages, 14 figure