Harnack Inequalities and Discrete—Continuous Error Estimates for a Chain of Atoms with Two—Body Interactions

We consider deformations in ℝ³ of an infinite linear chain of atoms where each atom interacts with all others through a two-body potential. We compute the effect of an external force applied to the chain. At equilibrium, the positions of the particles satisfy an Euler–Lagrange equation. For large classes of potentials, we prove that every solution is well approximated by the solution of a continuous model when applied forces and displacements of the atoms are small. We establish an error estimate between the discrete and the continuous solution based on a Harnack lemma of independent interest. Finally we apply our results to some Lennard-Jones potentials.     

Identification of gradient elasticity parameters based on interatomic interaction potentials accounting for modified Lorentz-Berthelot rules

The identification algorithm developed here for scale parameters of gradient elasticity combines solutions for a deformed heterogeneous composite fragment in a continuous one-dimensional model and for a diatomic chain in a discrete atomistic model. For the identification, the models are taken equivalent and the effective stiffnesses of equivalent composite fragments are compared. In the discrete model, only the nearest neighbor atoms interact and the interaction between dissimilar atoms are determined by a modified Lorentz-Berthelot rule. As a result, the effective stiffness of the discrete composite represented as a nonuniform atomic chain is found. The continuous model is a gradient one and takes into account nonlocal effects in the volume and adhesive properties of phase boundaries. The problem of determining the effective stiffness of a composite fragment is solved analytically in the one-dimensional approximation. The study is aimed to develop a procedure of identifying the scale parameters of gradient theories such that the parameters would be independent of the choice of potentials used in discrete modeling. On the example of modeling using the Morse and Lennard-Jones potentials, we propose an identification methodology invariant with respect to the choice of potentials. It is shown that the invariance is provided if the potentials in discrete modeling are coincided in the vicinity of equilibrium points. It is demonstrated that for unambiguous determination of the scale parameters of gradient elasticity, it suffices to use the simplest two-parameter potentials approximating any other potentials subject to equal equilibrium bond distances and equal second derivatives at the equilibrium point (i.e., force constants). An example of identifying the gradient elasticity parameters is presented for a two-phase W-Si composite.     

碱金属原子多极极化率的解析计算及其应用

通过考虑碱金属原子中的价电子在模型势中的运动, 给出了碱金属原子激发态(包括分立谱和连续谱)的波函数, 导出了碱金属原子的多极动态极化率的计算式中所涉及的矩阵元的解析表达式, 实现了碱金属原子多极动态极化率的解析计算. 作为应用, 利用范德瓦尔斯相互作用系数与动态极化率之间的积分关系, 计算了异核基态碱金属原子间的三体相互作用系数, 将计算结果与此前用稳定变分方法得到的结果进行了比较, 结果显示两者是一致的. 此工作为后续研究激发态碱金属原子间的相互作用奠定了基础.     

Density and thermodynamics of hydrogen adsorbed inside narrow carbon nanotubes

A model is proposed for calculating the thermodynamic functions and the equilibrium density of a one-dimensional chain of molecules (atoms) adsorbed inside a narrow nanotube. The model considers both the interaction between introduced atoms (molecules) and their interaction with the nanotube walls. The quantum-mechanical effects resulting in discrete energy levels of a particle and in its smeared position between neighbors are taken into account. In calculating the free energy at a nonzero temperature, the phonon contribution and the particle transitions to excited levels are considered. The model is applied to calculate the thermodynamic parameters of adsorbed hydrogen molecules inside extremely narrow single-wall carbon nanotubes of the (3,3) and (6,0) type. It is shown that external pressure gives rise to a sequence of first-order phase transitions, which change the density of adsorbed hydrogen molecules.     

Adsorption of cesium atoms at structural defects on sapphire surfaces

Results are presented from an experimental study of the adsorption of cesium atoms on sapphire surfaces and their photostimulated desorption. The adsorbed atoms are found to form chains on the surface which are localized near one-dimensional structural defects of the surface. One-dimensional adsorption is analyzed theoretically with different assumptions regarding the mobility of adsorbed atoms along a chain. A comparison of the theories with experimental data favors localized adsorption described in terms of a one-dimensional lattice gas model. The energy of adsorption for an isolated atom on a linear surface structural defect is 0.58 eV, while the energy of attraction between neighboring atoms in the chain is 26 meV. Zh. éksp. Teor. Fiz. 112, 362–370 (July 1997)     

Collective electronically excited states of a uniform chain of atoms

Electronically excited states of finite uniform chains of atoms were considered taking into account the influence of the continuous energy spectrum. Traditional quantum-chemical methods for calculating two-electron transitions between neighboring chain atoms were combined with the asymptotic theory of interactions between excited atoms and neutral particles and the mathematical apparatus of the theory of multiple scattering for taking into account intercenter transitions in an ensemble of interacting centers. Recurrence equations for describing energy zones containing symmetrical and antisymmetric excited state levels of chains with an arbitrary length were obtained. Depending on system parameters, different modes of the distribution of the electron density of collective excited states were possible. At a certain ratio between level shifts and exchange integral values, excited states with a uniform electron density distribution over all chain nodes could form for certain solutions. This was a fortuitous circumstance caused by the influence of the continuous spectrum. Such states appeared at small principal quantum number n values, they were similar to one-electron excitations of the type of Frenkel excitons, when an electron was localized near its Coulomb center. These conditions were rapidly disturbed as n increased, and one-electron excitations of a linear molecule were formed in the system (that is, limiting excitations of the type of Wannier-Mott excitons did not form).     

Exactly soluble peierls models

Exactly soluble models for electrons on a deformable discrete chain are found. They include typical continuous Peierls models as special limits. The chain structure, electronic spectrum and thermodynamic functions in the ground state are found. The region of a nearly half filled band corresponding to the polyacetylene model is described in detail.     

Delay of Spatial Quantum Correlations in Magnetic Nanostructures

Simultaneous quantum correlations between two spins in magnetic nanostructures are considered in the model of a linear chain of a finite number of atoms with exchange interaction between electron spins of neighboring atoms in the framework of the Heisenberg ferromagnetism theory. We assume that in the initial state, the spins of all chain atoms except the first two are oriented along the same direction. The spins of the first two atoms are flipped. Due to the exchange interaction, this initial state generates a spin flip wave along the chain. The expressions obtained for nonstationary quantum amplitudes of the flip probability waves for an even number of spins can be used for calculating quantum correlations between two spins separated by a large distance in a chain. Numerical calculations of the spin correlator reveal that the correlation between two spins in the chain occurs with a delay on the order of the time of propagation of the exchange interaction along the spin chain. After the delay, the spin correlation amplitude abruptly increases followed by subsequent oscillatory temporal behavior.     

Stretching of semiflexible polymers with elastic bonds

A semiflexible harmonic chain model with extensible bonds is introduced and applied to the stretching of semiflexible polymers or filaments. The semiflexible harmonic chain model allows to study effects from bending rigidity, bond extension, discrete chain structure, and finite length of a semiflexible polymer in a unified manner. The interplay between bond extension and external force can be described by an effective inextensible chain with increased stretching force, which leads to apparently reduced persistence lengths in force-extension relations. We obtain force-extension relations for strong- and weak-stretching regimes which include the effects of extensible bonds, discrete chain structure, and finite polymer length. We discuss the associated characteristic force scales and calculate the crossover behaviour of the force-extension curves. Strong stretching is governed by the discrete chain structure and the bond extensibility. The linear response for weak stretching depends on the relative size of the contour length and the persistence length which affects the behaviour of very rigid filaments such as F-actin. The results for the force-extension relations are corroborated by transfer matrix and variational calculations.PACS: 87.15.-v Biomolecules: structure and physical properties - 87.15.Aa Theory and modeling; computer simulation - 87.15.La Mechanical properties     

Stationary dark localized modes: discrete nonlinear Schrödinger equations

Various kinds of stationary dark localized modes in discrete nonlinear Schr?dinger equations are considered. A criterion for the existence of such excitations is introduced and an estimation of a localization region is provided. The results are illustrated in examples of the deformable discrete nonlinear Schr?dinger equation, of the model of Frenkel excitons in a chain of two-level atoms, and of the model of a one-dimensional Heisenberg ferromagnetic in the stationary phase approximation. The three models display essentially different properties. It is shown that at an arbitrary amplitude of the background it is impossible to reach strong localization of dark modes. In the meantime, in the model of Frenkel excitons, exact dark compacton solutions are found.     

Dicke quantum spin glass of atoms and photons

Recent studies of strongly interacting atoms and photons in optical cavities have rekindled interest in the Dicke model of atomic qubits coupled to discrete photon cavity modes. We study the multimode Dicke model with variable atom-photon couplings. We argue that a quantum spin-glass phase can appear, with a random linear combination of the cavity modes superradiant. We compute atomic and photon spectral response functions across this quantum phase transition, both of which should be accessible in experiments.     

Effect of defects in regular nanosystems on interference processes at the reemission of attosecond electromagnetic pulses

The effect of defects in nanostructured targets on interference spectra at the reemission of attosecond electromagnetic pulses has been considered. General expressions have been obtained for calculations of spectral distributions for one-, two-, and three-dimensional multiatomic nanosystems consisting of identical complex atoms with defects such as bends, vacancies, and breaks. Changes in interference spectra by a linear chain with several removed atoms (chain with breaks) and by a linear chain with a bend have been calculated as examples allowing a simple analytical representation. Generalization to two- and three-dimensional nanosystems has been developed.     

DISCRETE MULTISCALE ANALYSIS: A BIATOMIC LATTICE SYSTEM

We discuss a discrete approach to the multiscale reductive perturbative method and apply it to a biatomic chain with a nonlinear interaction between the atoms. This system is important to describe the time evolution of localized solitonic excitations.

We require that also the reduced equation be discrete. To do so coherently we need to discretize the time variable to be able to get asymptotic discrete waves and carry out a discrete multiscale expansion around them. Our resulting nonlinear equation will be a kind of discrete Nonlinear Schrödinger equation. If we make its continuum limit, we obtain the standard Nonlinear Schrödinger differential equation.     

Encapsulation of methane molecules into carbon nanotubes

Methane gas (CH₄) is a chemical compound comprising a carbon atom surrounded by four hydrogen atoms, and carbon nanotubes have been proposed as possible molecular containers for the storage of such gases. In this paper, we investigate the interaction energy between a CH₄ molecule and a carbon nanotube using two different models for the CH₄ molecule, the first discrete and the second continuous. In the first model, we consider the total interaction as the sum of the individual interactions between each atom of the molecule and the nanotube. We first determine the interaction energy by assuming that the carbon atom and one of the hydrogen atoms lie on the axis of the tube with the other three hydrogen atoms offset from the axis. Symmetry is assumed with regard to the arrangement of the three hydrogen atoms surrounding the carbon atom on the axis. We then rotate the atomic position into 100 discrete orientations and determine the average interaction energy from all orientations. In the second model, we approximate the CH₄ molecule by assuming that the four hydrogen atoms are smeared over a spherical surface of a certain radius with the carbon atom located at the center of the sphere. The total interaction energy between the CH₄ molecule and the carbon nanotube for this model is calculated as the sum of the individual interaction energies between both the carbon atom and the spherical surface and the carbon nanotube. These models are analyzed to determine the dimensions of the particular nanotubes which will readily suck-up CH₄ molecules. Our results determine the minimum and maximum interaction energies required for CH₄ encapsulation in different tube sizes, and establish the second model of the CH₄ molecule as a simple and elegant model which might be exploited for other problems.     

Zur Behandlung kollektiver Strahlungseffekte vieler Atome

A quantization procedure using spherical electromagnetic waves with different central points is developed and used to describe the interaction of the field with a system of many atoms. For two atoms and for an infinite linear chain built by identical equidistant atoms the Hamiltonian is simplified by a transformation which in the latter case has some analogy to spin wave theory.     

Two-step spin conversion and other effects in the atom-phonon coupling model

We study an atom-phonon coupling model introduced recently for spin-conversion phenomenon. The originality of this model, performed on a linear chain of atoms, is that the elastic force constant values of the spring linking two atoms depends on their electronic states. This leads to introduce naturally in the chain long- and short-range interactions, which appear respectively like a Zeeman and an exchange interactions. The exchange-like interaction can be ferro-, antiferro- or equal to zero. The effects of long-range interactions have already been studied. Here we study those of the short-range interaction. Some parts of the chain phase diagram are analysed and the main features of the experimental behaviours of spin conversion compounds are qualitatively reproduced.Received: 2 February 2004, Published online: 29 June 2004PACS: 63.20.Kr Phonon-electron and phonon-phonon interactions - 63.50. + x Vibrational states in disordered systems - 64.60.-i General studies of phase transitions     

Entanglement reciprocation using three level atoms in a lambda configuration

We propose a scheme in which entanglement can be transferred from atoms (discrete variables) to entangled states of cavity fields (continuous variables). The cavities play the role of a kind of quantum memory for entanglement, in such a way that it is possible to retrieve it back to the atoms. In our method, two three level atoms in a lambda configuration, previously entangled, are set to interact with single mode cavity fields prepared in coherent states. During the process, one e-bit of entanglement may be deposited in the cavities in an efficient way. We also show that the stored entanglement may be transferred back to flying atoms.