Solution of the Poisson equation for infinite and semi-infinite systems including near field corrections for charge densities of arbitrary shape

A method for solving the Poisson equation is presented for systems with three- or two-dimensional translational lattice symmetry, the latter applying to surfaces, interfaces or slabs. Special attention is given to the so-called near field correction (NFC), namely to a correction to the electrostatic potential arising from neighbouring (or near) cells, that is inherent, e.g. to the full-potential KKR method. The results of numerical tests presented serve to illustrate the effect of the NFC. Furthermore, the question of the convergence of `internal' angular momentum sums is addressed and discussed in detail.     

Radiation from a Line Source Placed in Two-dimensional Photonic Crystals

A novel formulation of radiation from a localized line source placed in two-dimensional photonic crystals consisting of layered periodic arrays of parallel circular cylinders is presented. The method employs the spectral domain approach. The spectral response of the photonic crystals to the line source excitation is calculated using the lattice sums, the T-matrix of a circular cylinder, and the generalized reflection and transmission matrices of the layered system. The far-zone radiated field is obtained using the conventional asymptotic method to the spectral response. The radiation patterns of the localized line source sandwiched by two photonic crystals are numerically studied.     

The Signed Loop Approach to the Ising Model: Foundations and Critical Point

The signed loop approach is a beautiful way to rigorously study the two-dimensional Ising model with no external field. In this paper, we explore the foundations of the method, including details that have so far been neglected or overlooked in the literature. We demonstrate how the method can be applied to the Ising model on the square lattice to derive explicit formal expressions for the free energy density and two-point functions in terms of sums over loops, valid all the way up to the self-dual point. As a corollary, it follows that the self-dual point is critical both for the behaviour of the free energy density, and for the decay of the two-point functions.     

Computing photonic band structures by Dirichlet-to-Neumann maps: The triangular lattice

An efficient semi-analytic method is developed for computing the band structures of two-dimensional photonic crystals which are triangular lattices of circular cylinders. The problem is formulated as an eigenvalue problem for a given frequency using the Dirichlet-to-Neumann (DtN) map of a hexagon unit cell. This is a linear eigenvalue problem even if the material is dispersive, where the eigenvalue depends on the Bloch wave vector. The DtN map is constructed from a cylindrical wave expansion, without using sophisticated lattice sums techniques. The eigenvalue problem can be efficiently solved by standard linear algebra programs, since it involves only matrices of relatively small size.     

Thermodynamic and transport properties of simple fluids using lattice sums: bulk phases and liquid-vapour interface

Molecular dynamics simulations in the canonical ensemble have been performed to obtain the thermodynamic and transport properties of the Lennard-Jones fluid. The dispersion interactions were calculated using lattice sums. This method makes it possible to simulate the full potential avoiding the inclusion of the long range corrections (LRC) during or at the end of simulations. In the calculation of dynamic properties in bulk phases and thermodynamic quantities of inhomogeneous systems where the interface is physically present, in general the LRC cannot easily be included. By using the lattice sums method, the results are independent of the truncation of the potential. In the liquid-vapour interface simulations it is not necessary to make any pre-judgments about the form of the LRC formula to calculate coexisting properties such as the surface tension. The lattice sums method has been applied to evaluate how well the full interaction can be calculated in the liquid phase and in the liquid-vapour interface. In the liquid phase the pressure, configurational energy, diffusion coefficient and shear viscosity were obtained. The results of the thermodynamic properties are compared with those obtained using the spherically truncated and shifted (STS) potential with the LRC added at the end of simulations, and excellent agreement is found. The transport properties are calculated on different system sizes for a state near the triple point. The diffusion coefficient using the lattice sums method increases with the number of molecules, and the results are higher than those of the STS model truncated at 2.5σ (STS2.5). The shear viscosity does not show any system size dependence for systems with more than 256 molecules, and the lattice sums results are essentially the same as those for the STS2.5. In the liquid-vapour equilibria the coexisting densities and vapour pressures for the full potential agree well with those obtained using the Gibbs ensemble and the NPT + test particle methods. The surface tension using lattice sums and truncation of forces at 2.5σ agrees well with STS results using large system sizes and cutoff distances.     

Quasi-classical estimates of the lattice constant and band gap of a crystal: Two-dimensional boron nitride

The quasi-classical method for estimating the structural and energy parameters of crystals is justified. In the initial approximation, the charge-density and potential distributions are represented by step functions. As a result, the energy of the ground state of the crystal and the matrix elements of the secular equation determining its electronic structure can be expressed through finite sums. This approach makes it possible to avoid uncontrollable errors introduced in the truncation of infinite series, and the error of the approach itself does not exceed a few percent. The lattice constant a=2.64 Å and the band gap E _g =6.22 eV for two-dimensional boron nitride are determined by the quasi-classical method.     

Two-Dimensional Breather Lattice Solutions and Compact-Like Discrete Breathers and Their Stability in Discrete Two-Dimensional Monatomic β-FPU Lattice

We restrict our attention to the discrete two-dimensional monatomic β-FPU lattice. We look for two- dimensional breather lattice solutions and two-dimensional compact-like discrete breathers by using trying method and analyze their stability by using Aubry＇s linearly stable theory. We obtain the conditions of existence and stability of two-dimensional breather lattice solutions and two-dimensional compact-like discrete breathers in the discrete two- dimensional monatomic β-FPU lattice.     

Rigorous Mode Analysis of Coupled Cavity Waveguides in Two-Dimensional Photonic Crystals*

A new approach for modal analysis of coupled cavity waveguides (CCW) in two-dimensional photonic crystals is presented. The mode propagation constants and the mode field profiles can be accurately derived by a simple matrix calculation, using a one-dimensional lattice sums, a T-matrix of an isolated circular cylinder, and generalized reflection matrices. Numerical examples have confirmed that the convergence of numerical solutions is very fast and the accuracy is very high.* Supported by the 21^st Century COE Program “Reconstruction of Social Insfrastructure Related to Information Science and Electrical Engineerings”.     

Dimer problem solved by Vdovichenko's method

The dimer problem on a two-dimensional lattice is reduced to a problem of random walks on the lattice, and then the latter problem is solved by the method which Vdovichenko developed in order to derive an exact expression for the partition function of the Ising model on a two-dimensional lattice. The result for the generating function is identical to the usual result for the general lattice. For loose-packed lattices such as the square and the honeycomb lattice on a torus, it takes the form of a linear combination of four determinants, where no problem of determining signs occurs.     

Rapid evaluation of oscillating lattice sums

We have generalized a method for the rapid evaluation of oscillating lattice sums for layered geometries, based on the Ewald's summation technique. Different types of long range interactions as well as different lattice symmetries can be considered, in particular modulated magnetic structures, periodic domain patterns, or periodic lattice distortions. As examples the demagnetizing energies for different thin films systems and magnetic excitations in a single atomic layer are calculated.     

应用格子Boltzmann模型模拟一类二维偏微分方程

针对一类含源的二维非线性偏微分方程, 通过Chapman-Enskog展开技术和多尺度分析提出了带修正项的简单格子Boltzmann模型. 用模型模拟了几类二维偏微分方程, 数值模拟结果与精确解相符合. 成功将格子Boltzmann方法应用到二维偏微分方程的数值求解中. 关键词： 二维非线性偏微分方程 格子Boltzmann模型 Chapman-Enskog多尺度展开     

二维棋盘格子复式晶格的完全光子带隙研究

设计了一种棋盘格子复式晶格的二维光子晶体：在二维正方形格子中，把截面为正方形的柱子旋转45°，同时在每个原胞中心引入一个圆形截面的柱子构成的光子晶体结构. 用平面波展开计算棋盘格子复式晶格的完全光子带隙，结果表明：棋盘格子复式晶格的完全光子带隙的Δω/ω比值几乎是普通棋盘格子的5倍，完全光子带隙的个数也增加. 与其他复式结构相比较，发现其最佳的Δω/ω比值是一类粗锐复合结构光子晶体的2.1倍. 关键词： 二维光子晶体 复式晶格 完全光子带隙     

Discrete breathers in a two-dimensional Morse lattice with an on-site harmonic potential

Two-dimensional discrete breathers in a two-dimensional Morse lattice with on-site harmonic potentials are investigated. Under the harmonic approximation, the linear dispersion relations for the triangular and the square lattices are discussed. The existence of discrete breathers in a two-dimensional Morse lattice with on-site harmonic potentials is proved by using local inharmonic approximation and the numerical method. The localization and amplitude of two-dimensional discrete breathers correlate closely to the Morse parameter a and the on-site parameter κ.     

Resistance Calculation of Pentagonal Lattice Structure of Resistors

In this study, the effective resistance between any two lattice sites in a two-dimensional pentagonal lattice structure of identical resistors is determined by means of the lattice Green’s function method. Some numerical results of the resistance for small separations between lattice sites are presented.     

Cluster variation method for lattice models

A noncombinatorial variation method for calculation of thermodynamic and correlation functions in a wide class of lattice models is developed. The method is illustrated on the model of a two-dimensional lattice gas consisting ofM kinds of molecules with pairwise interaction.     

New Form for the Coulomb Potential in Complex Crystals. Part III

The present paper continues our works [1, 2] devoted to the construction of the Coulomb potential at an arbitrary point of the unit cell of crystals with a close-packed hexagonal (CPH) lattice by the Green's function method. Convergence of the real space and reciprocal lattice sums in the Green's function is investigated and the optimal convergence parameter is chosen. The method is used to construct the Coulomb potential in Ti, Co, and Zn metals. Calculations are performed for four directions. Nonmonotonic behavior of nuclear and electron components of the potential in the cell is established. The crystal potential is also calculated for these metals with allowance for the exchange interaction in the Slater approximation.     

Localized self-trapping in two-dimensional molecular lattice with interaction between Wannier-Mott excitons and phonon lattice

We investigate the interactions of lattice phonons with Wannier-Mott exciton, the exciton that has a large radius in two-dimensional molecular lattice, by the method of continuum limit approximation, and obtain that the self-trapping can also appear in two-dimensional molecular lattice with a harmonic and nonlinear potential. The exciton effect on molecular lattice does not distort the molecular lattice but only makes it localized and the localization can also react, again through phonon coupling, to trap the energy and prevents its dispersion.     

Exponentially convergent lattice sums

For any oblique incidence and arbitrarily high order, lattice sums for one-dimensional gratings can be expressed in terms of exponentially convergent series. The scattering Green's function can be efficiently evaluated also in the grating plane. Numerical implementation of the method is 200 times faster than for the previous best result.     

Gleichgewichtsbedingungen in der Gittertheorie

The equilibrium conditions of an infinite crystal lattice result in symmetry relations of lattice sums. This was first shown byKun Huang who obtained these relations by comparing the linear theory of elasticity with the corresponding limit of lattice theory, a procedure which has been criticized by some authors. It is shown here how one can obtainKun Huang's results by using only trivial invariance properties of the potential energy. The same method is applied to get equilibrium relations for higher order (anharmonic) terms of lattice theory which are symmetry relations of tensors used in non linear elastic theory.     

Discrete gap breathers in a two-dimensional diatomic face-centered square lattice

In this paper we study the existence and stability of two-dimensional discrete gap breathers in a two-dimensional diatomic face-centered square lattice consisting of alternating light and heavy atoms, with on-site potential and coupling potential. This study is focused on two-dimensional breathers with their frequency in the gap that separates the acoustic and optical bands of the phonon spectrum. We demonstrate the possibility of the existence of two-dimensional gap breathers by using a numerical method. Six types of two-dimensional gap breathers are obtained, i.e., symmetric, mirror-symmetric and asymmetric, whether the center of the breather is on a light or a heavy atom. The difference between one-dimensional discrete gap breathers and two-dimensional discrete gap breathers is also discussed. We use Aubry's theory to analyze the stability of discrete gap breathers in the two-dimensional diatomic face-centered square lattice.