Self-consistent Electrostatic Field Research Articles

Analytic energy gradient for second-order Møller-Plesset perturbation theory based on the fragment molecular orbital method

The first derivative of the total energy with respect to nuclear coordinates (the energy gradient) in the fragment molecular orbital (FMO) method is applied to second order Møller-Plesset perturbation theory (MP2), resulting in the analytic derivative of the correlation energy in the external self-consistent electrostatic field. The completely analytic energy gradient equations are formulated at the FMO-MP2 level. Both for molecular clusters (H(2)O)(64) and a system with fragmentation across covalent bonds, a capped alanine decamer, the analytic FMO-MP2 energy gradients with the electrostatic dimer approximation are shown to be complete and accurate by comparing them with the corresponding numeric gradients. The developed gradient is parallelized with the parallel efficiency of about 97% on 32 Pentium4 nodes connected by Gigabit Ethernet.

A detailed study of collisionless explosion of single- and two-ion-species spherical nanoplasmas

The collisionless adiabatic expansion into vacuum of spherical plasma targets (clusters) composed of cold single- or multispecies ions and hot electrons is studied kinetically by numerical solving of the nonrelativistic equations of motion of plasma particles in the self-consistent electrostatic field. The expansion dynamics for the whole range of electron temperatures from much less than to much higher than the cluster Coulomb energy is described for various initial plasma density profiles and cluster structures. The explosion of two ion species heterogeneous (layered) and homogeneously mixed targets is studied in detail for the wide range of light ion concentration and kinematic parameter.

A nonlinear equation for ionic diffusion in a strong binary electrolyte

The problem of the one-dimensional electro-diffusion of ions in a strong binary electrolyte is considered. The mathematical description, known as the Poisson–Nernst–Planck (PNP) system, consists of a diffusion equation for each species augmented by transport owing to a self-consistent electrostatic field determined by the Poisson equation. This description is also relevant to other important problems in physics, such as electron and hole diffusion across semiconductor junctions and the diffusion of ions in plasmas. If concentrations do not vary appreciably over distances of the order of the Debye length, the Poisson equation can be replaced by the condition of local charge neutrality first introduced by Planck. It can then be shown that both species diffuse at the same rate with a common diffusivity that is intermediate between that of the slow and fast species (ambipolar diffusion). Here, we derive a more general theory by exploiting the ratio of the Debye length to a characteristic length scale as a small asymptotic parameter. It is shown that the concentration of either species may be described by a nonlinear partial differential equation that provides a better approximation than the classical linear equation for ambipolar diffusion, but reduces to it in the appropriate limit.

Ionization degree for strong evaporation of metals

Kinetic equations for ions and neutrals are numerically solved in the plasma sheath formed at a condensed phase when strong evaporation is taking place. The Boltzmann distribution is assumed for electrons. A weakly ionized vapor with the Debye length much shorter than the mean free path is considered. This is typical for laser evaporation of metals. Under these conditions, the sheath consists of a Knudsen layer and a thin charge separation layer between the Knudsen layer and the condensed phase. The self-consistent electrostatic field in the Knudsen layer is obtained from the quasineutrality condition. The potential barrier in the charge separation layer is determined by the charge balance. Kinetic boundary conditions for neutrals and charges are estimated by the detailed balance principle from the parameters of the saturated vapor. The transport of charges in the sheath is controlled by ions and depends on ion-neutral collisions and the self-consistent electrostatic field. Ionization degree in the vapor formed by strong evaporation increases with the Mach number and can attain values about 30% higher than the ionization degree in the saturated vapor. Two factors contribute to this increase. The first is the drop of the potential barrier in the charge separation layer and the second is the strengthening of the field in the Knudsen layer. The ionization equilibrium may be disturbed by a considerable excess of charges.

Numerical simulation of sheath plasma by the particle method taking account of coulomb collisions

We study the problem of the behavior of a plasma bounded longitudinally by an absorbing sheath. This model contains charged particles (electrons and ions) moving subject to a self-consistent electrostatic field. New particle pairs are generated in the region of a distributed source. As a numerical model we used the electrostatic “particle-in-cell” method supplemented by the Emmert model for a bulk source and the algorithm of binary Coulomb collisions using the Monte Carlo method. We give a mathematical statement of the problem. The computations were carried out using the direct implicit method with the “explicit limit” time step. The results of numerical simulation of this system are given. We consider the formation and evoluiton of potential structures (multiple weak nonmonotonic double layers). Five figures. Bibliography: 35 titles.

Application of the particle-in-cell method for numerical simulation of sheath plasma

We give a survey of papers on the numerical simulation of the sheath plasma using the particle-in-cell method. We study the problem of the behavior of a plasma bounded in the longitudinal direction of an absorbing wall. The model studied contains charged particles (electrons and ions) that move subject to a self-consistent electrostatic field. New pairs of particles are generated in the region of a distributed source. As a numerical model we use the electrostatic “particle-in-cell” method supplemented with the Emmert model for a bulk source and the algorithm of binary Coulomb collisions using a Monte Carlo method. Only electron-ion collisions are taken into account.

Application of the particle method for numerical simulation of sheath plasma taking account of coulomb electron-ion collisions

We study the problem of the behavior of a plasma constrained in the longitudinal direction of an absorbing sheath. The model in question contains charged particles (electrons and ions) than are moving under the action of a self-consistent electrostatic field. New particle pairs are generated in the region of a distributed source. As a numerical model we use the electrostatic “particle-in-cell method supplemented with the Emmert model for a three-dimensional source and the binary Coulomb collision algorithm with a Monte Carlo method. Only electron-ion collisions are considered here. The problem is formulated mathematically. We give a description of numerical methods of solving various problems, forming an algorithm for numerical simulation of the system in question. Computctions were carried out with a large step size in time applying the direct implicit method. We give the results of numerical simulation of the system using different numbers of particles per Debye length and different lengths of a disionless region.

Kinetic modeling of scrape-off layer plasmas

Electron transport along open field lines in the diverted scrape-off layer of a tokamak is studied numerically via a kinetic Fokker–Planck approach. The method allows calculation of the distribution function in a situation where large parallel temperature gradients are maintained by collisional relaxation and, at the same time, superthermal electrons stream freely from the midplane of the plasma to the target/sheath boundary. The method also allows calculation of the self-consistent electrostatic field associated with parallel gradients in the distribution function, as well as the potential drop across the target/sheath boundary, where the latter is calculated to enforce appropriate boundary conditions at the target, although the sheath itself is not resolved. The kinetic results are compared to classical fluid results for the case of a simple (nonradiative) divertor. The kinetic solutions exhibit an enhanced superthermal electron population in the vicinity of the target, which results in a larger sheath energy transmission factor, a lower bulk electron temperature, and a smaller sheath potential drop. The sheath potential largely determines the energy with which ions impact the target, thereby affecting the rate of target erosion. Ionization rates and radiation rates from impurities in the vicinity of the target also depend strongly on the local electron temperature and can be sensitive to superthermal tails.

Numerical Integration of Equations of Motion with a Self-Consistent Field given by an Implicit Equation

Abstract Numerical integration methods for equations of motion with the right hand side containing an implicit equation (e.g. an equation for the self-consistent electrostatic field in the system of polarizable molecules) are studied. Stable second-order predictors suitable for the Verlet (leap-frog) integrator and allowing one evaluation of the forces per integration step are proposed. Different strategies making the use of only partly stable higher-order predictors possible are discussed.

Numerical simulation of ECR plasma sources

The available electron cyclotron resonance plasma source has been simulated in two-dimensional configuration space (z, r) and three-dimensional velocity space (Vz, Vr, Vθ). Using a hybrid model, we treat ions as particles and electrons as fluid. The motion of ions and electrons is determined by external magnetic field and self-consistent electrostatic field. For a given magnetic distribution, the spatial profiles of electron temperature, plasma potential, plasma density and ionization rate have been obtained. The influences of microwave power, neutral gas pressure and magnetic field on plasma parameters have been studied. The comparisons between simulation and experimental results are made and the explanations for some physical phenomena are given.

An Operator Splitting Method for the Wigner–Poisson Problem

The Wigner–Poisson equation describes the quantum-mechanical motion of electrons in a self-consistent electrostatic field. The equation consists of a transport term and a non-linear pseudodifferential operator. In this paper we analyze an operator splitting method for the linear Wigner equation and the coupled Wigner–Poisson problem. For this semidiscretization in time, consistency and nonlinear stability are established in an $L^2 $-framework. We present a numerical example to illustrate the method.

A two-dimensional, axisymmetric model of a magnetized glow discharge plasma

A two-dimensional, axisymmetric model of a bounded, partially ionized, magnetized glow discharge plasma has been developed. The model treats positive ions as particles and electrons as a fluid in a hybrid configuration. The results reported are directed towards simulating an electron cyclotron resonance (ECR), microwave-sustained plasma. Microwave power profiles in the plasma are assumed, and the resulting electron and ion transport in the applied magnetic and self-consistent electrostatic field is calculated. Sheaths under typical operating conditions are very thin and the authors apply an analytic sheath model to avoid integration in the sheaths. Typical results are presented for a common ECR reactor geometry and conditions, and comparisons with experimental data are made when possible.

Analysis of velocity diffusion of electrons with Vlasov-Poisson simulations

Wave-induced velocity diffusion is studied with a one-dimensional relativistic Vlasov-Poisson code. The diffusion coefficient is determined by following test electrons in the self-consistent electrostatic field formed by a narrow spectrum of electron plasma waves. The diffusion coefficient is found to be slightly larger than the quasilinear value at intermediate values of Chirikov's overlap parameter. The largest deviation is about 20-30%. At higher values of the overlap parameter, the diffusion is slower than quasilinear for a small number of field modes, but faster than quasilinear for a large number of the modes.

On the multiple‐minima problem in the conformational analysis of polypeptides. V. Application of the self‐consistent electrostatic field and the electrostatically driven monte carlo methods to bovine pancreatic trypsin inhibitor

In connection with the accompanying paper to test various models for the hydration of polypeptides, we have explored a limited portion of the conformational energy hyperspace of the small protein bovine pancreatic trypsin inhibitor (BPTI) with the aid of two search methods developed in this laboratory. A series of low-energy conformations was obtained as a result of this study. These conformations constitute a set of local minima in the conformational energy space of the molecule as described by the ECEPP/2 (Empirical Conformational Energy Program for Peptides) potential energy function, without the inclusion of hydration. Five different initial conformations were used in this exploration: the first corresponds to an energy-refined structure based on the crystallographic coordinates (4PTI) provided by Deisenhofer and Steigemann and reported previously by Meirovitch and Scheraga. The remaining four initial conformations were obtained by using a Variable-Target-Function procedure, applied to the experimental Cartesian coordinates (5PTI) reported by Wlodawer et al. The self-consistent electrostatic field (SCEF) and the electrostatically driven Monte Carlo (EDMC) methods were used to search the conformational space. The SCEF and EDMC methodologies assume that a polypeptide or protein molecule is driven toward the native structure mainly by the action of the electrostatic interactions. Application of these methodologies led to a set of conformations (up to 50 kcal/mol lower than the starting ones) with ECEPP/2 energies lower than any of those that we had previously found. Application of both methods to the initial conformation generated from 4PTI led to a series of low-energy conformations exhibiting similar rms deviations with respect to the experimental data (4PTI) as did the starting conformation. However, statistical analysis of the runs that had started from the conformations generated by using the variable-target-function procedure (and applying the EDMC method) indicated that the rms deviations of the atomic positions of the new low-energy conformations tended to increase as the energy improved, when compared with the X-ray data from which the starting conformations had been generated. The structures with the lowest energies also had radii of gyration smaller than the experimentally observed one. These results indicated a need to include hydration in the potential function, and provided the conformations used in the accompanying paper to test various hydration models.

Interfacing the finite-element method with the method of characteristics in self-consistent electrostatic field models

Two methods are described for interfacing the finite-element method, used to calculate the electric field structure for a known charge distribution, with the method of characteristics, used to calculate the charge distribution for a known electric field structure, so as to obtain self-consistent descriptions of electrostatic systems involving two-way source-field coupling. Both methods circumvent difficulties which have limited application of the finite-element method of characteristics methodology in the past. One method involves automatic retriangulation, based upon the Delaunay algorithm, on each source-field iteration, using finite-element nodes lying directly upon characteristic lines. The other method involves projection of characteristic lines back through a fixed finite-element mesh to their initiation points on the injection electrode. Both methods make use of second-order potential variations within finite elements and thus involve curved characteristic trajectories within each element. Results of each method are compared with those of previous work on self consistent source-field coupling.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The effect of the time-dependent self-consistent electrostatic field on gyrotron operation

The time-dependent self-consistent electrostatic field is shown to have a deleterious effect on gyrotron operation. As the electron beam density increases, the nonlinear efficiency is degraded seriously by the self-electrostatic field. Time-dependent multimode simulations demonstrate that at sufficiently large beam densities the electron cyclotron instability is quenched and the oscillation will not start. Contrary to previous investigations, electrostatic effects do not necessarily increase the linear growth rate of the electromagnetic cavity mode and, depending on the beam density, electrostatic effects can actually stabilize the mode. Typically, however, considerations of linear theory are not important in an overmoded, open resonator gyrotron because the system evolves nonlinearly into a state consisting of a single mode which is linearly stable, but nonlinearly the most efficient mode.

Computer experiments of an electron beam injection into space plasma.

A computer experiment is carried out to study the dynamic behavior of plasma particles in response to self-consistent electromagnetic and electrostatic fields produced by an artificial injection of a dense electron beam into the ambient plasma. A two-and-one-half dimensional electromagnetic particle code (KEMPO) is used for this purpose. The electron beam is assumed to be injected along the external magnetic field, B0, and to have a spatially finite extent in directions both parallel and perpendicular to the beam motion at t=0. Most of the beam electrons are decelerated, and this deceleration is accompanied by an elongation and splitting of the beam. However, a fraction of the electrons in the beam is not decelerated but instead is accelerated in both positive and negative directions along B0. A substantial heating of beam electrons is also observed. Associated with the acceleration and thermalization of the beam electrons, various kinds of electromagnetic and electrostatic waves are excited through wave-particle interactions. On a time scale of ion motion, ions are attracted toward the center of the moving beam, forming an ion density peak, which, in turn, causes a strong acceleration of electrons. A BGK type quasi-steady state is reached and maintained for a finite time.

Thermodynamic limit in a charged plasma in equilibrium

We study the existence conditions for the thermodynamic limit in the chain of BBGKY equations for the equilibrium correlation functions of a charged plasma. It is shown that in order for the thermodynamic limit to exist the charge of the plasma cannot increase faster than the surface area of the plasma. When this condition is satisfied the equilibrium correlation functions of the charged plasma are asymptotically identical to the correlation functions of a neutral plasma in a self-consistent electrostatic field, which depends only on the one-particle correlation functions. For a plasma which is uniform everywhere except in a thin surface region, this field is found in explicit form. For a plasma occupying an infinite half-space, the problem is equivalent to a neutral plasma near a charged wall.

Multidipole confinement of argon and potassium plasmas

Multidipole confinement of an argon discharge plasma and a potassium plasma produced by surface ionization was investigated. In both cases, a large volume of uniform density quiescent plasma was produced in the central region of low magnetic field strength. Measurements of electron and ion densities and space potentials suggest that self-consistent electrostatic fields resulted in crossfield (to B) ion confinement and that the cusp leak widths were determined by a noncollisional mechanism (e.g., Bohm diffusion). A comparison of the nondischarge potassium plasmas produced with and without multidipole confinement indicates that the ion confinement time was improved by a factor of ∼15 by the multidipole magnetic field.

Neighbourstrand interactions in one-dimensional crystal orbital calculations on organometallic polymers-the tetrathiosquarato nickel(II) system

The band structure of tetrathiosquarato nickel(II), TTSqNi 1 , is studied by means of crystal orbital (CO) calculations based on a semi-empirical INDO Hamiltonian under the inclusion of neighbourstrand interactions. The interchain coupling is determined in the framework of a self-consistent electrostatic field approximation where the band structure of the reference stack is influenced due to the potential of the neighbouring polymer backbones. Intrachain polarization and electrostatic Coulomb energies are considered in the present CO model. The dispersion of the reference strand as well as charge redistributions and charge shifts are investigated as a function of the number of the neighbouring chains considered in the tight-binding formalism, the mutual orientation of the various backbones in the “real solid” and the strength of the external field. Charge shifts, modifications in the “covalency” of the different bonds, alterations in the ε( k) curves and density of states distribution as response to the external field are discussed. It is shown that symmetry broken solutions (charge density waves, CDWs) are encountered in the CO formalism if the strength of the external potential exceeds a critical value. This charge condensation is however subject to all limitations given by the mean-field approximation. Necessary condition for CDW solutions are mutual displacements between the TTSqNi backbones in the direction on the stacking axis. General rules on the importance and the consequences of interstrand interactions in low-dimensional materials with organometallic fragments as molecular building blocks are formulated.