A thermodynamic lattice theory on melting curve and eutectic point of binary alloys. Application to fcc and bcc structure

A thermodynamic lattice theory has been developed for determination of the melting curves and eutectic points of binary alloys. Analytical expressions for the melting curves of binary alloys composed of constituent elements with the same structure have been derived from expressions for the ratio of root mean square fluctuation in atomic positions on the equilibrium lattice positions and the nearest neighbor distance. This melting curve provides information on Lindemann’s melting temperatures of binary alloys with respect to any proportion of constituent elements, as well as on their eutectic points. The theory has been applied to fcc and bcc structure. Numerical results for some binary alloys provide a good correspondence between the calculated and experimental phase diagrams, where the calculated results for Cu_1−x Ni _x agree well with the measured ones, and those for the other alloys are found to be in a reasonable agreement with experiment.     

Random walks on sparsely periodic and random lattices: I. Random walk properties from lattice bond enumeration

We have developed a new technique for calculating certain asymptotic random walk properties on sparsely periodic and related random lattices in two and three dimensions. This technique is based on an ansatz which relates the number of lattice bonds in “irreducible lattice fragments” to the number of steps along these bonds. We show that certain random walk properties can be calculated very simply on the basis of this ansatz and that they depend only on the density of bonds and not on the arrangement of the bonds within the lattice. The random walk properties calculated here (mean square displacements, number of distinct sites visited, probability of return to the origin) are in complete agreement with results obtained earlier via generating function techniques. A subsequent paper contains generating function calculations which verify a number of new results presented here, such as mean occupation frequency of lattice sites, and a proof of our basic assumption on the relation between the number of lattice bonds and random walk steps.     

Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian

We study the stationary points of what is known as the lattice Landau gauge fixing functional in one-dimensional compact U(1) lattice gauge theory, or as the Hamiltonian of the one-dimensional random phase XY model in statistical physics. An analytic solution of all stationary points is derived for lattices with an odd number of lattice sites and periodic boundary conditions. In the context of lattice gauge theory, these stationary points and their indices are used to compute the gauge fixing partition function, making reference in particular to the Neuberger problem. Interpreted as stationary points of the one-dimensional XY Hamiltonian, the solutions and their Hessian determinants allow us to evaluate a criterion which makes predictions on the existence of phase transitions and the corresponding critical energies in the thermodynamic limit.     

The O(n) Loop Model on the 3–12 Lattice

The partition function of the O(n) loop model on the honeycomb lattice is mapped to that of the O(n) loop model on the 3–12 lattice. Both models share the same operator content and thus critical exponents. The critical points are related via a simple transformation of variables. When n = 0 this gives the recently found exact value = 1.711041... for the connective constant of self-avoiding walks on the 3–12 lattice. The exact critical points are recovered for the Ising model on the 3–12 lattice and the dual asanoha lattice at n = 1.     

Exact variational methods and cluster-variation approximations

It is shown that for classical,d-dimensional lattice models with finite-range interactions the translation-invariant equilibrium states have the property that their mean entropy is completely determined by their restriction to a subset of the lattice that is infinite in (d–1) dimensions and has a width equal to the range of the interaction in the dth dimension. This property is used to show proper convergence toward the exact result for a hierarchy of approximations of the cluster-variation method that uses one-dimensionally increasing basis clusters in a two-dimensional lattice.     

On the universal behavior of two-dimensional Ising spin glasses

Ising spin glasses are studied, at zero temperature, on a hierarchical lattice as an approach to the square lattice. The stiffness exponent y, which governs the behavior of the interactions under changes of scale, is computed for several distinct continuous symmetric probability distributions for the couplings. All distributions considered lead to the same estimates, i.e., the exponent y is universal. Our results are compared with other estimates available for the two-dimensional Gaussian Ising spin glass.     

Correction to scaling for directed branched polymers (lattice animals)

Correction to scaling effects for the directed branched polymer (lattice animal) problem are calculated both from the field theoretic model due to Day and Lubensky (dynamics at the Yang-Lee edge singularity) and enumeration data. The universal correction to scaling exponent for the number of distinct animal configurations is estimated by numerical methods and field theoretic renormalization group results (=7–d expansion) extrapolated to exact results ford=2 andd=3; estimates of nonuniversal amplitudes are included.     

An algorithm for detecting abelian monopoles inSU 2-valued lattice gauge-Higgs systems

We present an algorithm which calculates the monopole number of anSU ₂-valued lattice gauge field, together with a lattice Higgs field, on a simplicial lattice of dimension ≧3. The calculation is gauge invariant. The expected value of the monopole density (for a fixed Higgs field) does not depend on the Higgs field. Partially supported by NSF grants DMS 8607168 and DMS 8907753 Partially supported by PSC-CUNY and by NSF grant DMS 8805485     

On the survival probability of a random walk in a finite lattice with a single trap

We consider the survival of a random walker in a finite lattice with periodic boundary conditions. The initial position of the random walker is uniformly distributed on the lattice with respect to the trap. We show that the survival of a random walker, U _n>, can be exactly related to the expected number of distinct sites visted on a trap-free lattice by U _n=1–S _n/N ^D (*) whereN ^D is the number of lattice points inD dimensions. We then analyze the behavior of S_n in any number of dimensions by using Tauberian methods. We find that at sufficiently long times S _n decays exponentially withn in all numbers of dimensions. InD = 1 and 2 dimensions there is an intermediate behavior which can be calculated and is valid forN ²N 1 whenD = 1 andN lnN n 1 whenD = 2. No such crossover exists when Z3. The form of (*) suggests that the single trap approximation is indeed a valid low-concentration limit for survival on an infinite lattice with a finite concentration of traps.     

A simple calculation for the average number of steps to trapping in lattice random walks

We show that the asymptotic results for the average number of steps to trapping at an irreversible trapping site on aD-dimensional finite lattice can be obtained from the generating function for random walks on aninfinite perfect lattice. This introduces a significant simplification into such calculations. An interesting corollary of these calculations is the conclusion that a random walker traverses, on the average, all the distinct nontrapping lattice sites before arriving on the trapping site.This work was supported in part by NSF Grants MPS72-04363-A03 and CHE75-20624.     

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.     

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     

晶格各向同性应变对闪锌矿结构CrS和CrSe的半金属性和磁性的影响

采用基于密度泛函理论的全势能线性缀加平面波方法,对闪锌矿结构CrS和CrSe的电子结构进行自旋极化计算.闪锌矿相CrS和CrSe处于平衡晶格常数时为半金属性,它们的分子自旋磁矩都为4.00μ_B. 使晶体相对于平衡晶格在±10%的范围内发生各向同性应变,并计算闪锌矿相CrS和CrSe的电子结构.计算结果表明,闪锌矿相CrS和CrSe相对于平衡晶格的各向同性应变分别为-1%–10%和-4%–10%时仍然保持半金属铁磁性,并且分子总磁矩都稳定于4.00μ_B. 关键词： 各向同性应变 电子结构 半金属性 磁性     

Ranges and profiles of distribution of low-energy ions channeling in metal and semiconductor single crystals

In the present work peculiarities of trajectories and energy losses, ranges and profiles of distribution of low-energy different-mass ions channeling in thin single crystals of metals and semiconductors have been thoroughly studied by computer simulation in binary collision approximation. The character of oscillations of channeled-ion trajectories depending on their energies, aiming points from the axis of a channel, kind of interaction potential, crystal lattice type and temperature has been determined. It has been found that, in the case of light ions even at low energy, the main contribution to energy loss is made by inelastic energy losses, whereas for heavy ions, already at E < 10 keV, elastic energy losses exceed inelastic ones. Profiles of the distribution of channeled ions have been calculated depending on crystal lattice type, kind of ions and their energy.     

Electrical resistivity study on SmSe1−x As x under pressure

Electrical resistivity measurements under pressure and lattice parameter study on SmSe_1−x As _x are reported here. The estimate d lattice parameter and valence is calculated for the same and they are found to be in good agreement with the experimental data. The electrical conductivity increases with the increase of As concentration. An erratum to this article is available at .     

Monomer-dimer model on a scale-free small-world network

The explicit determination of the number of monomer-dimer arrangements on a network is a theoretical challenge, and exact solutions to monomer-dimer problem are available only for few limiting graphs with a single monomer on the boundary, e.g., rectangular lattice and quartic lattice; however, analytical research (even numerical result) for monomer-dimer problem on scale-free small-world networks is still missing despite the fact that a vast variety of real systems display simultaneously scale-free and small-world structures. In this paper, we address the monomer-dimer problem defined on a scale-free small-world network and obtain the exact formula for the number of all possible monomer-dimer arrangements on the network, based on which we also determine the asymptotic growth constant of the number of monomer-dimer arrangements in the network. We show that the obtained asymptotic growth constant is much less than its counterparts corresponding to two-dimensional lattice and Sierpinski fractal having the same average degree as the studied network, which indicates from another aspect that scale-free networks have a fundamentally distinct architecture as opposed to regular lattices and fractals without power-law behavior.     

Lattice dynamical models of adaptive spatio-temporal phenomena

We describe the rich spectrum of spatio-temporal phenomena emerging from a class of models incorporating adaptive dynamics on a lattice of nonlinear (typically chaotic) elements. The investigation is based on extensive numerical simulations which reveal many novel dynamical phases, ranging from spatio-temporal fixed points and cycles of all orders, to parameter regimes displaying marked scaling properties (as manifest in distinct 1/f spectral characteristics and power law distributions of spatial quantities).     

Critical Points and Supersymmetric Vacua, III: String/M Models

A fundamental problem in contemporary string/M theory is to count the number of inequivalent vacua satisfying constraints in a string theory model. This article contains the first rigorous results on the number and distribution of supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau 3-fold X with flux. In particular, complete proofs of the counting formulas in Ashok-Douglas [AD] and Denef-Douglas [DD1] are given, together with van der Corput style remainder estimates.Supersymmetric vacua are critical points of certain holomorphic sections (flux superpotentials) of a line bundle over the moduli space of complex structures on X × T ² with respect to the Weil-Petersson connection. Flux superpotentials form a lattice of full rank in a 2 b ₃(X)-dimensional real subspace . We show that the density of critical points in for this lattice of sections is well approximated by Gaussian measures of the kind studied in [DSZ1,DSZ2,AD,DD1].Research partially supported by DOE grant DE-FG02-96ER40959 (first author) and NSF grants DMS-0100474 (second author) and DMS-0302518 (third author).     

On Equilibrium Distribution of a Reversible Growth Model

We study a probabilistic model of interacting spins indexed by elements of a finite subset of the d-dimensional integer lattice, d≥1. Conditions of time reversibility are examined. It is shown that the model equilibrium distribution converges to a limit distribution as the indexing set expands to the whole lattice. The occupied site percolation problem is solved for the limit distribution. Two models with similar dynamics are also discussed.