On the electrostatic potential due to a slab shaped and a semi-infinite NaCl-type lattice

Consider N layers of a NaCl-type ionic lattice such that in every layer one has an infinite square lattice of positive and negative unit point charges. We present formulae in which the electrostatic potential in an arbitrary field point is expressed as a sum of two rapidly converging lattice sums. For N→∞ we obtain formulae applicable for a semi-infinite lattice.     

Vacancies in binary alloys: II. Equilibrium concentration of vacancies in alloys with long-range order

General formulae are presented for the equilibrium concentration of vacancies in binary alloys having two types of lattice sites. The simple model used takes in account the pair interaction between nearest neighbouring atoms. The formulae derived in this paper yield in special cases identical results as the precedent theories.     

Observations on the Wilson fermions in the ϵ regime

We make several observations concerning the low quark mass region with Wilson fermions and how this is connected with the ? regime in the continuum. A transition from tiny cutoff effects to rather large discretization errors would take place in general with Wilson fermions if we lower the quark mass at finite lattice spacing. We argue that these two regions exhibit rather different behaviours concerning the coupling between cutoff effects and zero-modes. We interpolate between these two regimes adding to the continuum ? regime formulae, in the spirit of the Symanzik expansion, the relevant operators parametrising the leading cutoff effects. We compute the partition function, the chiral condensate, scalar and pseudo-scalar correlation functions. The final formulae can be used to fit lattice data to extract physical low energy constants, and to estimate systematic uncertainties coming from discretization errors. Moreover they suggest ways on how to remove these cutoff effects, the core of which are captured by the continuum zero modes.     

On two-point boundary correlations of the gl(1|1) supersymmetric vertex model

The gl(1|1) supersymmetric vertex model with domain wall boundary conditions (DWBC) on an N ×N square lattice is considered. We obtain the reduction formulae for the two-point boundary correlation functions of the model.     

Vector correlators in lattice QCD: Methods and applications

We discuss the calculation of the leading hadronic vacuum polarization in lattice QCD. Exploiting the excellent quality of the compiled experimental data for the e ⁺ e ⁻ → hadrons cross-section, we predict the outcome of large-volume lattice calculations at the physical pion mass, and design computational strategies for the lattice to have an impact on important phenomenological quantities such as the leading hadronic contribution to (g − 2) _μ and the running of the electromagnetic coupling constant. First, the R(s) ratio can be calculated directly on the lattice in the threshold region, and we provide the formulae to do so with twisted boundary conditions. Second, the current correlator projected onto zero spatial momentum, in a Euclidean time interval where it can be calculated accurately, provides a potentially critical test of the experimental R(s) ratio in the region that is most relevant for (g − 2) _μ . This observation can also be turned around: the vector correlator at intermediate distances can be used to determine the lattice spacing in fm, and we make a concrete proposal in this direction. Finally, we quantify the finite-size effects on the current correlator coming from low-energy two-pion states and provide a general parametrization of the vacuum polarization on the torus.     

重离子的束内散射

本文在Bjorken和Mtingwa束内散射理论的基础上进一步推导出了束内散射率的普遍表达式,并将该表达式应用于HIRFL—CSR的磁铁结构(lattice),计算并得到了束流横向发射度和纵向动量散度随时间变化的规律,结果表明,束内散射不会成为CSRlattice设计的障碍.     

Boundary Correlation Functions of the gl(1|1) Supersymmetric Vertex Model

The gl(1|1) supersymmetric vertex model with domain wall boundary conditions (DWBC) on an N×N square lattice is considered. We derive the reduction formulae for the one-point boundary correlation functions of the model. The determinant representation for the boundary correlation functions is also obtained.     

Topological charge of (lattice) gauge fields

Using recently derived explicit formulae for the 2- and 3-cochains in SU(2) gauge theory, we are able to integrate the Chern-Simons density analytically. We arrive — in SU(2) — at a local algebraic expression for the topological charge, which is the sum of local winding numbers associated with the corners (lattice points) of the cells covering the manifold plus contributions from possible isolated gauge singularities which manifest themselves as “vortices” in the 1-, 2- or 3-cochains. Among others we consider hypercubic geometry — i.e. covering the manifold by hypercubes — which is of particular interest to lattice Monte Carlo applications. Finally, we extend our results to SU(3) gauge theory.     

The triangular kagomé lattices revisited

The dimer problem, Ising spins and bond percolation on the triangular kagomé lattice have been studied extensively by physicists. In this paper, based on the fact the triangular kagomé lattice with toroidal boundary condition can be regarded as the line graph of 3.12.12 lattice with toroidal boundary condition, we derive the formulae of the number of spanning trees, the energy, and the Kirchhoff index of the triangular kagomé lattice with toroidal boundary condition.     

On the stochastic confinement in SU(2) lattice gauge theory

We extend the Olesen approach to confinement, originally proposed for SU(∞) gauge theory, to the SU(2) group. We perform Monte Carlo calculations of the spectral density, which describes the distribution of eigenvalues of the Wilson loop in the SU(2) lattice gauge theory (LGT), for square loops up to size 4 × 4. Our results indicate the onset of disorder in the 4-dimensional LGT so that at weak coupling confinement is due to non-abelian fluctuations of the gauge field. We describe the Monte Carlo data by formulae of the 2-dimensional LGT with some effective coupling constant. We formulate how this effective coupling constant should depend on the size of the loop in order that the Olesen hypothesis about the dimensional reduction (i.e. approximate reduction of the 4-dimensional LGT to the effective 2-dimensional LGT in the confinement region) would be held. Using the strong-coupling expansion where the Olesen hypothesis holds in fourteen orders, we perform quantitative estimates. The density of simple vortices up to size 4 × 4 is calculated. A connection between the definitions of spectral densities for the SU(∞) and SU(2) groups is considered. Explicit formulae for the spectral density in the 2-dimensional SU(2) LGT are derived.     

Generalized Bethe ansatz equations for Hofstadter problem

The problem of diagonalization of the quantum mechanical Hamiltonian, governing dynamics of an electron on a two-dimensional triangular or square lattice in external uniform magnetic field, applied perpendicularly to the lattice plane, the flux through lattice cell, divided by the elementary quantum flux, being a rational number, is reduced to the generalized Bethe ansatz like equations on the high genus algebraic curve. Our formulae for the trigonometric case, where the genus of the curve vanishes, contain as a particular case a recent result of Wiegmann and Zabrodin.Supported by the Russian Academy of Sciences and Academy of Finland     

Study of Layout Principle and Simulated Calculation for a Resonant Extraction System

Based on the general theory and related technique of 1/3 integer resonant extraction, main principle of layout for momentum-amplitude selection method is studied in detail and some simple analytic formulae are given. The design of slow extraction system for Beijing Proton Therapy Synchrotron (BPTS) is then given as an example by considering its lattice. The results are also compared with simulated calculation. Study shows that an extraction beam for BPTS with extraction efficiency above 98% and momentum spread of 0.11 % can be obtained.     

共振引出系统的布局研究和模拟计算

在介绍1/3整数共振引出基本理论和相关技术的同时,通过简明的解析公式,详细研究和分析了共振引出系统的布局原理.以一台小型医用质子同步加速器的磁铁聚焦结构为基础,给出了设计实例,并与计算机模拟结果进行了比较.     

Experimental tests of the dielectric electronegativity approach to the problem of heterojunction band discontinuities

We tested Van Vechten's recent suggestion for improving the accuracy in predicting heterojunction band discontinuities from semiempirical ionization potential formulae. The results provide some interesting suggestions on the dependence of the ionization potential on the lattice parameter.     

Resonance switching in a large sonic crystal cavity

In this work we present an experimental realization of resonance switching inside a large, relative to the lattice constant, sonic crystal (SC) cavity. The resonances arise from a temporary trapping of the acoustical energy between two sonic crystal slabs when the frequency of the wave falls within a partial band-gap region (thus, the propagation inside the sonic crystal is inhibited for normal incidence and a total reflection occurs). We show that by modifying the geometry of the sonic crystal lattice, from square to centered rectangular, it is possible to switch some resonances on and off, for a certain frequency region corresponding to the second band-gap for normal incidence. We also study the generalization of this phenomenon for the same lattice and band-gap region, as a function of the length of the cavity and the filling fraction the lattice.     

An exact analytic treatment of an ordered binary alloy in a Bethe lattice including first and second nearest neighbour interactions

Exact analytic formulae are obtained for the electronic structure of an ordered binary alloy in a Bethe lattice described by a tight-binding hamiltonian that includes nearest neighbour and second nearest neighbour interactions. Some interesting results are obtained and discussed in the text.     

Homogenization of Three-Dimensional Periodic Solid-Solid Elastic Composites

We develop an approach to homogenize three-dimensional periodic solid-solid elastic composites with cubic lattice at low frequencies, by using plane wave expansion and perturbation theory with respect to the long wavelength limit. Based on the fact that the two shear waves propagating along lattice axis are degenerated, we derive formulae for effective velocities parallel and normal to the lattice axis, from which three independent effective elastic moduli are calculated, respectively. Theoretical results, which take into account the multiple scattering and the structure of the periodic medium, are in good agreement with the previous isotropic theory at high-symmetry directions.     

The interaction of light with a semiinfinite dielectric as a phonon problem; the generalized Snellius law and Fresnel formulae

Using Ewald's formulae it is possible to evaluate the electromagnetic field generated by a mechanical plane wave propagating through a semiinfinite dipole crystal. The problem of refraction and reflection of an external electromagnetic wave on a semiinfinite dipole crystal is thus reduced to the evaluation of the forced vibrations of this crystal excited by the incident electromagnetic wave. Applying this approach, the generalized Snellius law and generalized Fresnel formulae can be deduced. In the limit of long waves these general formulae change into the usual classical expressions. The Ewald extinction theorem follows directly from the lattice dynamics of our system.     

Some empirical relations for electron mobility in II–VI compound semiconductors

Electron mobility has been calculated in a number of binary II–VI compound semiconductors using a displaced Maxwellian distribution function and taking the various scattering mechanisms into consideration at different lattice temperatures and for various amounts of ionized impurity concentrations. It is observed that the low field mobility values can be expressed by a cubic power relationship with lattice temperature and with ionized impurity concentration using a least mean square fit technique with an accuracy better than 5 per cent. Similarly, the field dependence of mobility can also be expressed as a power series of the applied electric field. It is suggested that these equations can be profitably used for a quick estimation of mobility values as a check on experiments and also as sufficiently accurate formulae for simulation and modelling purposes.     

A Fourier (k-) space design approach for controllable photonic band and localization states in aperiodic lattices

In this paper we present a systematic study of photonic bandgap engineering using aperiodic lattices (ALs). Up to now ALs have tended to be defined by specific formulae (e.g. Fibonacci, Cantor), and theories have neglected other useful ALs along with the vast majority of non-useful (random) ALs. Here, we present a practical and efficient Fourier space-based general theory to identify all those ALs having useful band properties, which are characterized by well-defined Fourier (i.e. lattice momentum) components. Direct control of field localization comes via control of the Parseval strength competition between the different Fourier components characterizing a lattice. Real-space optimization of ALs tends to be computationally demanding. However, via our Fourier space-based simulated annealing inverse optimization algorithm, we efficiently tailor the relative strength of the AL Fourier components for precise control of photonic band and localization properties.