Lattice Solution Research Articles

Thermodynamic behavior of binary mixtures of hard spheres: Semianalytical solutions on a Husimi lattice built with cubes.

We study binary mixtures of hard particles, which exclude up to their kth nearest neighbors (kNN) on the simple cubic lattice and have activities z_{k}. In the first model analyzed, point particles (0NN) are mixed with 1NN ones. The grand-canonical solution of this model on a Husimi lattice built with cubes unveils a phase diagram with a fluid and a solid phase separated by a continuous and a discontinuous transition line which meet at a tricritical point. A density anomaly, characterized by minima in isobaric curves of the total density of particles against z_{0} (or z_{1}), is also observed in this system. Overall, this scenario is identical to the one previously found for this model when defined on the square lattice. The second model investigated consists of the mixture of 1NN particles with 2NN ones. In this case, a very rich phase behavior is found in its Husimi lattice solution, with two solid phases-one associated with the ordering of 1NN particles (S1) and the other with the ordering of 2NN ones (S2)-beyond the fluid (F) phase. While the transitions between F-S2 and S1-S2 phases are always discontinuous, the F-S1 transition is continuous (discontinuous) for small (large) z_{2}. The critical and coexistence F-S1 lines meet at a tricritical point. Moreover, the coexistence F-S1,F-S2, and S1-S2 lines meet at a triple point. Density anomalies are absent in this case.

Investigation of the structural and magnetic properties of amorphous/nanocrystalline Fe70Ti10B20 (%at.) alloys by mechanical alloying

In this study, amorphous/nanocrystalline Fe 70 Ti 10 B 20 (at.%) alloys were synthesized by using elemental Fe, Ti and B powders under argon gas atmosphere via mechanical alloying method. The powders were ball milled at 20:1 ball-to-powder ratio and 500 rpm rotating speed up to 70 h. The phase and morphological properties of the synthesized powders were examined by using X-Ray diffractometer and scanning electron microscope with energy-dispersive X-ray spectroscope (SEM/EDX). The magnetic properties of powders were determined by vibrating sample magnetometer (VSM) at room temperature in the range of 0-20 kOe. It is determined that after 2.5 h of milling, Ti and B are started to dissolve in Fe lattice and Fe(TiB) solid solutions formed. XRD analysis revealed that the amorphous structures were formed after 30h of milling and the crystallite size and lattice strain were found ~ 8.2 nm and 1.16%, respectively. SEM images indicated that the particles were mostly agglomerated and the particles size distribution was in the range of 10-48 μm. Magnetic measurements revealed that after 70 h of milling, the saturation magnetization (M s ) and the coercivity (H c ) was about 94 emu/g and 117 Oe.

Liquid-Phase Synthesis and Properties of Constant Lattice Parameter AlGaInAsP Solid Solutions on Indium Phosphide Substrates

Heterophase equilibria in the AlxGayIn1– x– yAszP1 –z/InP heterosystem have been analyzed in terms of a simple solution model in the temperature range 813–1023 K. We have assessed the stability limits of AlGaInAsP solid solutions synthesized on indium phosphide substrates and investigated kinetic features of the crystallization of constant lattice parameter AlGaInAsP solid solutions on InP substrates. The crystallization rate of AlGaInAsP solid solutions on InP substrates has been shown to depend on the synthesis temperature, the molten zone composition and length, and the temperature gradient. In addition, it has been shown to decrease with increasing aluminum concentration, independent of the molten zone length, owing to the decrease in the diffusion coefficient in the liquid phase. We have obtained composition dependences of component distribution coefficients in the AlxGayIn1– x– yAszP1 –z /InP heterosystem and shown that raising the synthesis temperature leads to a reduction in KAl and KP. The effect of synthesis conditions on the structural and luminescence properties of AlGaInAsP solid solutions on InP has been examined.

XY model on interacting parallel planes with a soliton

We study two-coupled parallel planes with XY spins in each plane in the continuum limit with ferromagnetic coupling in the planes as well as between the planes. We find a soliton solution of the 2D sine-Poisson equation for the difference in the orientation angle of spins along a particular direction in the planes. If the planes are elastic then they will deform or move apart in the region of the soliton to reduce the magnetic energy. The balance between the gain in magnetic energy and the elastic energy cost determines the extent of deformation. We also discuss a soliton lattice solution and the associated periodic deformation of the planes.

Magnetic vortices in the Abelian Higgs model with derivative interactions

We study the properties of a single magnetic vortex and magnetic vortex lattices in a generalization of the Abelian Higgs model containing the simplest derivative interaction that preserves the [Formula: see text] gauge symmetry of the original model. The paper is motivated by the study of finite isospin chiral perturbation theory in a uniform, external magnetic field: since pions are Goldstone bosons of QCD (due to chiral symmetry breaking by the QCD vacuum), they interact through momentum-dependent terms. We find the asymptotic properties of single vortex solutions and compare them to the well-known solutions of the standard Abelian Higgs model. Furthermore, we study the vortex lattice solutions near the upper critical field using the method of “successive approximations,” which was originally used by Abrikosov in his seminal paper on type-II superconductors. We find the vortex lattice structure, which remains hexagonal as in the standard Abelian Higgs model, and condensation energy of the vortex lattices relative to the normal vacuum (in a uniform magnetic field).

Spin Asymmetries and Helicity Amplitudes in Photo-Production from Polarized Neutrons

Pseudoscalar meson photoproduction experiments lead the search for an expanded $$N^*$$ spectroscopy that is required by lattice solutions of QCD. Extensive data have been collected with proton targets that can potentially over-determine the production amplitude and allow a search for weak resonances. In contrast, data on photoproduction from neutrons is drastically limited. Both are needed since the $$\gamma p N^*$$ and $$\gamma n N^*$$ photo-couplings to $$I=1/2$$ states are different and provide complementary information on the mechanisms for resonance excitation. The main subtleties in experiments with an effective polarized neutron target are reviewed. The considerable impact of new polarization data on multipole analyses are discussed, using the example of recent $$\pi ^- p$$ measurements of the beam-target helicity asymmetry (E) from polarized $${ HD}$$ in CLAS at Jefferson Lab. New partial wave analyses incorporating these data have found significant changes in helicity amplitudes of some established resonances, and signatures of weak resonances that previously were observed only in hyperon decay channels.

Reprogramming Static Deformation Patterns in Mechanical Metamaterials.

This paper discusses an x-braced metamaterial lattice with the unusual property of exhibiting bandgaps in their deformation decay spectrum, and, hence, the capacity for reprogramming deformation patterns. The design of polarizing non-local lattice arising from the scenario of repeated zero eigenvalues of a system transfer matrix is also introduced. We develop a single mode fundamental solution for lattices with multiple degrees of freedom per node in the form of static Raleigh waves. These waves can be blocked at the material boundary when the solution is constructed with the polarization vectors of the bandgap. This single mode solution is used as a basis to build analytical displacement solutions for any applied essential and natural boundary condition. Subsequently, we address the bandgap design, leading to a comprehensive approach for predicting deformation pattern behavior within the interior of an x-braced plane lattice. Overall, we show that the stiffness parameter and unit-cell aspect ratio of the x-braced lattice can be tuned to completely block or filter static boundary deformations, and to reverse the dependence of deformation or strain energy decay parameter on the Raleigh wavenumber, a behavior known as the reverse Saint Venant’s edge effect (RSV). These findings could guide future research in engineering smart materials and structures with interesting functionalities, such as load pattern recognition, strain energy redistribution, and stress alleviation.

Truncated-unity parquet equations: Application to the repulsive Hubbard model

The parquet equations are a self-consistent set of equations for the effective two-particle vertex of an interacting many-fermion system. The application of these equations to bulk models is, however, demanding due to the complex emergent momentum and frequency structure of the vertex. Here, we show how a channel decomposition by means of truncated unities, which was developed in the context of the functional renormalization group to efficiently treat the momentum dependence, can be transferred to the parquet equations. This leads to a significantly reduced complexity and memory consumption scaling only linearly with the number of discrete momenta. We apply this technique to the half-filled repulsive Hubbard model on the square lattice and present approximate solutions for the channel-projected vertices and the full reducible vertex.

Propagation of solitons in a two-dimensional nonlinear square lattice

We investigate the existence of solitary waves in a nonlinear square spring–mass lattice. In the lattice, the masses interact with their neighbors through linear springs, and are connected to the ground by a nonlinear spring whose force is expressed as a polynomial function of the masses out-of-plane displacement. The low-order Taylor series expansions of the discrete equations lead to a continuum representation that holds in the long wavelength limit. Under this assumption, solitary wave solutions are sought within the long wavelength approximation, and the subsequent application of multiple scales to the resulting nonlinear continuum equations. The study focuses on weak nonlinearities of the ground stiffness and reveals the existence of 3 types of solitons, namely a ‘bright’, a ‘dark’, and a ‘vortex’ soliton. These solitons result from the balance of dispersive and nonlinear effects in the lattice, setting aside other relevant phenomena in 2D waves such as diffraction that may lead to a field that does not change during propagation in nonlinear media. For equal constants of the in-plane springs, the governing equation reduces to the Klein–Gordon type, for which bright and dark solitons replicate solutions for one-dimensional lattices. However, unequal constants of the in-plane springs aligned with the two principal lattice directions lead to conditions in which the soliton propagation direction, defined by the group velocity, differs from the wave vector direction, which is unique to two-dimensional assemblies. Furthermore, vortex solitons are obtained for isotropic lattices, which shows similarities with results previously found in optics, thermal media and quantum plasmas. The paper describes the main parameters defining the existence of these solitary waves, and verifies the analytical predictions through numerical simulations. Results show the validity of obtained solutions and illustrate the main characteristics of the solitary waves found in the considered nonlinear mechanical lattice. The study provides an analysis of the physics of waves in nonlinear systems, and may lead to novel designs of devices that can be used for high-performance waveguides.

Hysteresis and Phase Transitions in a Lattice Regularization of an Ill-Posed Forward–Backward Diffusion Equation

We consider a lattice regularization for an ill-posed diffusion equation with trilinear constitutive law and study the dynamics of phase interfaces in the parabolic scaling limit. Our main result guarantees for a certain class of single-interface initial data that the lattice solutions satisfy asymptotically a free boundary problem with hysteretic Stefan condition. The key challenge in the proof is to control the microscopic fluctuations that are inevitably produced by the backward diffusion when a particle passes the spinodal region.

New exact solutions for a discrete electrical lattice using the analytical methods

This paper retrieves soliton solutions to an equation in nonlinear electrical transmission lines using the semi-inverse variational principle method (SIVPM), the $\exp(-\Omega(\xi))$ -expansion method (EEM) and the improved $\tan(\phi/2)$ -expansion method (ITEM), with the aid of the symbolic computation package Maple. As a result, the SIVPM, EEM and ITEM methods are successfully employed and some new exact solitary wave solutions are acquired in terms of kink-singular soliton solution, hyperbolic solution, trigonometric solution, dark and bright soliton solutions. All solutions have been verified back into their corresponding equations with the aid of the Maple package program. We depicted the physical explanation of the extracted solutions with the choice of different parameters by plotting some 2D and 3D illustrations. Finally, we show that the used methods are robust and more efficient than other methods. More importantly, the solutions found in this work can have significant applications in telecommunication systems where solitons are used to codify data.

On Abrikosov Lattice Solutions of the Ginzburg-Landau Equations

We prove existence of Abrikosov vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, with multiple magnetic flux quanta per fundamental cell. We also revisit the existence proof for the Abrikosov vortex lattices, streamlining some arguments and providing some essential details missing in earlier proofs for a single magnetic flux quantum per a fundamental cell.

On stability of Abrikosov vortex lattices

The Ginzburg–Landau equations play a key role in superconductivity and particle physics. They inspired many imitations in other areas of physics. These equations have two remarkable classes of solutions – vortices and (Abrikosov) vortex lattices. For the standard cylindrical geometry, the existence theory for these solutions, as well as the stability theory of vortices are well developed. The latter is done within the context of the time-dependent Ginzburg–Landau equations – the Gorkov–Eliashberg–Schmid equations of superconductivity – and the abelian Higgs model of particle physics.We study stability of Abrikosov vortex lattices under finite energy perturbations satisfying a natural parity condition (both defined precisely in the text) for the dynamics given by the Gorkov–Eliashberg–Schmid equations. For magnetic fields close to the second critical magnetic field and for arbitrary lattice shapes, we prove that there exist two functions on the space of lattices, such that Abrikosov vortex lattice solutions are asymptotically stable, provided the superconductor is of Type II and these functions are positive, and unstable, for superconductors of Type I, or if one of these functions is negative.

Entire solutions for bistable lattice differential equations with obstacles

We consider scalar lattice dierential equations posed on square lattices in two space dimensions. Under certain natural conditions we show that wave-like solutions exist when obstacles (characterized by \holes) are present in the lattice. Our work generalizes to the discrete spatial setting the results obtained in [9] for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.

Bump competition and lattice solutions in two-dimensional neural fields

Some forms of competition among activity bumps in a two-dimensional neural field are studied. First, threshold dynamics is included and rivalry evolutions are considered. The relations between parameters and dominance durations can match experimental observations about ageing. Next, the threshold dynamics is omitted from the model and we focus on the properties of the steady-state. From noisy inputs, hexagonal grids are formed by a symmetry-breaking process. Particular issues about solution existence and stability conditions are considered. We speculate that they affect the possibility of producing basis grids which may be combined to form feature maps.

A lattice-based changeable threshold multi-secret sharing scheme and its application to threshold cryptography

In this paper, we propose a threshold increasing algorithm for a (t; n) latticebased Threshold Multi-Stage Secret Sharing (TMSSS) scheme. To realize the changeability feature, we use the zero addition protocol to construct a new (t0; n) TMSSS scheme. Therefore, the new scheme enjoys the signi cant feature of threshold changeability along with the inherited features of being multi-stage, multi-use, and veri able derived fromour previously proposed lattice-based TMSSS scheme. Furthermore, we use the improved TMSSS scheme to propose a threshold decryption algorithm for the Learning With Error (LWE) based public key encryption scheme based on the study of Lindner and Peikert. For threshold decryption, each authorized subset of participants decrypts the ciphertext partially and sends the result to the combiner. The combiner can decrypt the ciphertext using the partial decryptions. The security of both schemes is based on hardness of lattice problems, i.e. LWE and Inhomogeneous Small Integer Solution (ISIS) problems, which are believed to resist against the quantum algorithms. The proposed schemes are eficient, especially on the participants' side, making them suitable for the applications in which the participants have limited processing capacities.

Dark and gray solitons of ( $$2+1$$ 2 + 1 )-dimensional nonlocal nonlinear media with periodic response function

By using the standard symmetry reduction method, some exact analytical solutions including gray solitons and gray soliton lattice solutions are derived for the (\(2+1\))-dimensional nonlinear optical media with periodic nonlocal response. Furthermore, dark/gray soliton solutions and dark soliton lattice solutions are found by means of hyperbolic function expansion method and elliptic function expansion method for the nonlocal nonlinear system, respectively. It is found that two critical points exist for soliton solutions, and the switching dynamics of solitons may be described by the critical points.

A note on the possible existence of an instanton-like self-dual solution to lattice Euclidean gravity

The self-dual solution to lattice Euclidean gravity is constructed. In contrast to the well known Eguchi-Hanson solution to continuous Euclidean Gravity, the lattice solution is asymptotically globally Euclidean, i.e., the boundary of the space as r −→ ∞ is S3 = SU(2).

Soliton solutions and traveling wave solutions for a discrete electrical lattice with nonlinear dispersion through the generalized Riccati equation mapping method

In this paper, through the generalized Riccati equation mapping method, we investigate soliton solutions in the upper and lower forbidden band gab of the Salerno equation describing nonlinear discrete electrical lattice. As a result, we obtain various hyperbolic and trigonometric functions solutions and for some appropriated parameters we obtain exact solutions including kink, antikink, breathers, and dark and bright solitons. The obtained solutions are useful for the signal transmission through the electrical lattice.

The energetics of prenucleation clusters in lattice solutions.

According to classical nucleation theory, nucleation from solution involves the formation of small atomic clusters. Most formulations of classical nucleation use continuum "droplet" approximations to describe the properties of these clusters. However, the discrete atomic nature of very small clusters may cause deviations from these approximations. Here, we present a self-consistent framework for describing the nature of these deviations. We use our framework to investigate the formation of "polycube" atomic clusters on a cubic lattice, for which we have used combinatoric data to calculate the thermodynamic properties of clusters with 17 atoms or less. We show that the classical continuum droplet model emerges as a natural approach to describe the free energy of small clusters, but with a size-dependent surface tension. However, this formulation only arises if an appropriate "site-normalized" definition is adopted for the free energy of formation. These results are independently confirmed through the use of Monte Carlo calculations. Our results show that clusters formed from sparingly soluble materials (μM solubility range) tend to adopt compact configurations that minimize the solvent-solute interaction energy. As a consequence, there are distinct minima in the cluster-size-energy landscape that correspond to especially compact configurations. Conversely, highly soluble materials (1M) form clusters with expanded configurations that maximize configurational entropy. The effective surface tension of these clusters tends to smoothly and systematically decrease as the cluster size increases. However, materials with intermediate solubility (1 mM) are found to have a balanced behavior, with cluster energies that follow the classical "droplet" scaling laws remarkably well.