Darwin Approximation Research Articles

Hamiltonian of a homogeneous two-component plasma

The Hamiltonian of one- and two-component plasmas is calculated in the negligible radiation Darwin approximation. Since the Hamiltonian is the phase space energy of the system its form indicates, according to statistical mechanics, the nature of the thermal equilibrium that plasmas strive to attain. The main issue is the length scale of the magnetic interaction energy. In the past a screening length lambda=1/square root of r(e)n], with n number density and r(e) classical electron radius, has been derived. We address the question whether the corresponding longer screening range obtained from the classical proton radius is physically relevant and the answer is affirmative. Starting from the Darwin Lagrangian it is nontrivial to find the Darwin Hamiltonian of a macroscopic system. For a homogeneous system we resolve the difficulty by temporarily approximating the particle number density by a smooth constant density. This leads to Yukawa-type screened vector potential. The nontrivial problem of finding the corresponding, divergence free, Coulomb gauge version is solved.

The Darwin Approximation of the Relativistic Vlasov-Maxwell System

We study the relativistic Vlasov-Maxwell system which describes large systems of particles interacting by means of their collectively generated forces. If the speed of light $c$ is considered as a parameter then it is known that in the Newtonian limit $c\to\infty$ the Vlasov-Poisson system is obtained. In this paper we determine the next order approximate system, which in the case of individual particles usually is called the Darwin approximation.

Higher-order relativistic corrections to the vibration–rotation levels of H 2S

Relativistic corrections beyond the simple one-electron mass–velocity–Darwin (MVD1) approximation to the ground-state electronic energy of H 2S are determined at over 250 geometries. The corrections considered include the two-electron Darwin, the Gaunt and Breit corrections, and the one-electron Lamb shift. Fitted correction surfaces are constructed and used with an accurate ab initio nonrelativistic Born–Oppenheimer potential, determined previously (J. Chem. Phys. 115 (2001) 1229), to calculate vibrational and rotational levels for H 2 32 S . The calculations suggest that one- and two-electron relativistic corrections have a noticable influence on the levels of H 2S. As for water, the effects considered have markedly different characteristics for the stretching and bending states.

Nonlinear perturbative electromagnetic (Darwin) particle simulation of high intensity beams

A multi-species perturbative nonlinear ( δf) electromagnetic particle simulation scheme has been developed for studying the propagation of intense charged particle beams in high-intensity accelerators and transport systems. The scheme is based on the Darwin approximation of Ampere's law, in which the transverse inductive electric field is neglected, resulting in the elimination of high-frequency transverse electromagnetic effects and, consequently, the associated numerical restrictions from the simulation. However, as noted previously, the presence of the time derivative of the vector potential in the equations of motion for the Darwin model can cause numerical instability. To circumvent this difficulty, we have adopted an approach by replacing the mechanical momentum, p z , in the direction of beam propagation, by the canonical momentum, P z = p z + qA z / c, as the phase-space variable. The resulting Ampere's law is then modified by the presence of an additional shielding term associated with the skin depth of the species. In order to minimize the numerical noise and to easily access both linear and nonlinear regimes for the physics of interest, we have also adopted the δf formalism for the Darwin model. The absence of unwanted high-frequency waves also enables us to use the adiabatic particle pushing scheme to compensate for the mass-ratio disparities for the various species of charge. The scheme is ideal for studying two-stream and filamentation instabilities, which may cause deterioration of the beam quality in the heavy ion fusion driver and fusion chamber.

A 3-D Darwin–EM Hybrid PIC Code for Ion Ring Studies

A new, 3-D electromagnetic (EM), hybrid, particle-in-cell (PIC) code, FLAME has been constructed to study low-frequency, large orbit plasmas in realistic cylindrical configurations. The stability and equilibrium of strong ion rings in magnetized plasmas are the first issues suitable for its application. In FLAME the EM-field is governed by Maxwell's equations in the quasi-neutral Darwin approximation (with displacement current neglected), the ion components are represented by discrete macro-particles, and the plasma electrons are modeled as a massless cold fluid. All physical quantities are expanded into finite Fourier series in the azimuthal (θ) direction. The discretization in the poloidal (r,z) plane is done by a finite-difference staggered grid method. The electron fluid equations include a finite scalar resistivity and macro-particles experience slowing-down collisions. A substantial reduction of computation time is achieved by enabling separate time advances of background and beam particle species in the time-averaged fields. FLAME has been optimized to run on parallel, MIMD systems, and has an object-oriented (C++) structure. The results of normal mode tests intended to verify the code ability to correctly model plasma phenomena are presented. We also investigate in 3-D the injection of a powerful annular ion beam into a plasma immersed in a magnetic cusp followed by an axially ramped applied magnetic field. A nonaxisymmetric perturbation is applied to the magnetic field and its effect on ion ring formation is analysed.

Construction of the weakly-relativistic Fokker-Planck kinetic equation in the Darwin approximation

The explicit form of the weakly-relativistic Vlasov and the Fokker-Planck collision operators for a multispecies plasma are evaluated by using a systematic expansion in β_=v_/c and retaining corrections up to the second order. Properties of these operators are investigated.

Darwin magnetic interaction energy and its macroscopic consequences.

In metals and plasmas the Coulomb interaction between mobile charged particles is screened. The main long-range interaction between the particles is then the magnetic interaction. When radiation is negligible the simplest way to study this interaction is to use the Darwin approximation. In this way one retains a conservative finite degree of freedom problem. We review the derivation of the Darwin Lagrangian and present careful derivations of the corresponding Hamiltonian in various limits. Our results go beyond those of previous authors in several respects. We point out some consequences of the magnetic interaction energy for the dynamics of charged particles with screened Coulomb interaction. Applications to metallic conduction electrons and to plasmas are considered. \textcopyright{} 1996 The American Physical Society.

A finite element formulation of the Darwin PIC model for use on unstructured grids

In this paper we introduce a new formulation of the Darwin approximation of Maxwell's equations and discuss its domain of applicability. We describe our finite element implementation of this model, allowing the use of unstructured grids, and its coupling with a PIC method for the particles.

Collisionless reconnection in two‐dimensional magnetotail equilibria

A two‐dimensional (x, z) particle simulation model based on the Darwin approximation to Maxwell's equations is developed for studying collisionless reconnection in the magnetotail. The particles and fields are initialized in accord with a general equilibrium configuration which includes a pressure gradient along the tail axis and tail flaring. The model is used to investigate a number of theoretical issues regarding the spontaneous ion tearing instability under the assumption that the electron dynamics are unimportant. It is demonstrated both numerically and analytically that in a thin current sheet with ρ i0 ∼ λ (ρ i0 is the ion Larmor radius based on the lobe field and λ is the characteristic thickness of the current sheet) the growth rates in the absence of a normal field component are much smaller than expected based upon the analytic theory for a thick sheet (ρi0 ≪ λ). For such a thin current sheet the presence of a normal field Bz on axis of even a few percent strongly inhibits the growth of the instability. This result is not altered by the addition of a constant By component smaller than the lobe field. It is demonstrated further that the transition to stability occurs when the cyclotron frequency based on Bz equals the growth rate of the Bz = 0 tearing mode. This requires typically a normal field of the order of 6% of the lobe field. If a sufficiently large external perturbation of the lobe magnetic field reduces the normal field on axis below the stabilization threshold over a significant fraction of a tearing mode wavelength, then one can recover the rapid instability of the one‐dimensional neutral sheet.

Computational Plasma Physics: With Applications of Fusion and Astrophysics

"Computational Plasma Physics: With Applications of Fusion and Astrophysics." Fusion Technology, 17(2), pp. 362–363 Additional informationNotes on contributorsDennis W. HewettDennis W. Hewett has worked in computational plasma physics starting with his PhD research at the University of Kansas in 1973 under Thomas P. Armstrong. While at Los Alamos from 1974 to 1982, he worked on a variety of problems in magnetic fusion research, including writing and using finite- and zero-electron mass hybrid models as well as other PIC procedures. He was also involved in developing new techniques for finding Vlasov equilibria and for solving systems of linearized equations that arise in resistive magnetohydrodynamics. His move to inertia I confinement fusion (ICF) at Lawrence Livermore National Laboratory resulted in a shift in emphasis for applications while still maintaining a strong interest in computational methods. He implemented a two-dimensional direct-implicit electromagnetic PIC algorithm that has lead to new developments in magnetic reconnection. His recent interests include considering new ways to use the radiationless or Darwin approximation of Maxwell’s equations and applications in particle sources and acceleration, particularly as they apply to heavy-ion-driven ICF.

The K-shell ionisation of atoms by relativistic α particles

The K-shell ionisation of atoms by fast alpha particles is investigated using the Moller relativistic modification of the Born approximation. The incident and scattered alpha particles were described by relativistic plane waves. The Darwin approximation was used for the description of the K-shell atomic electrons and the ejected electron. The total cross sections were calculated as functions of alpha -particle impact energy for Sn, Ag, Au, Pb and compared with the available experimental data and calculations. The results obtained are in good accordance with the experimental data of Dost et al (1981). The influence of spin change of the K-shell electron on the total cross section was also studied. The results show that the collisions involving spin change of the atomic electron could be significant at higher energies of the incident alpha particles, especially for atoms with greater nuclear charge Z.

Particle simulation of plasma processes in the earth's magnetotail

An electromagnetic code in the Darwin approximation is used to model process in the current sheet of the Earth's magnetotail that may lead to the generation of electrostatic field responsible for auroral electron acceleration. It is shown that the Hamiltonian formulation has advantages over the more commonly used Lagrangian formulation of the Darwin approximation. This paper describes the specific techniques that are needed, to include the magnetic field normal to the current sheet, the presence of a convection electric field and the continuous inflow of particles from the lobe. Simulations incorporating these new features reveal new effects that maybe of significant importance to the understanding of auroral and geomagnetic storm phenomena.

Interaction of the plasma sheet with the lobes of the Earth's magnetotail

Simulations with a two‐dimensional electromagnetic particle code in the Darwin approximation are used to investigate mechanisms for the generation of potential differences across magnetic field lines in the magnetotail region earthward of the neutral line. Two mechanisms are found. One is the electrostatic ion cyclotron wave generated by counterstreaming ion beams. Another and more powerful process depends on the generation of magnetic fields, By, normal to the noon‐midnight meridian plane of the magnetosphere. Currents carried primarily by electrons generate a By. This causes the magnetic field to loop out of the meridian plane in a direction such that electrons lose potential energy and gain kinetic energy in the convection electric field. Much of the kinetic energy goes into motion in the x direction in the region of the neutral sheet. The current carried by the electrons is in a direction to intensify the By field. The electrons are also compressed, resulting in a negative space charge in the region of the electron current jet. This process therefore generates electrostatic potential variations across magnetic field lines. Contrary to prior expectations, there is no evidence that tearing mode instabilities or conducting boundaries play any prominent role in the generation of the potentials.

Hybrid simulation of ion beams in background plasma

We present a 2 1 2 - dimensional ( r, z, v r, v 0, v z; ϱ ϱφ =0 ) quasineutral hybrid code suitable for studying a broad class of plasma and beam-plasma configurations. Ions are represented by particles and electrons by an inertialess thermal fluid which obeys a generalized Ohm's law. Fields are solved in the quasineutral Darwin approximation. Several collisional and atomic processes are also included in the model. The code has been applied to the problem of propagation of intense ion beams through preformed z-pinch plasma channels, a scenario of interest for inertial confinement fusion devices.

Numerical determination of injector designs for high beam quality

The performance of a free electron laser strongly depends on the electron beam quality or brightness. The electron beam is transported into the free electron laser after it has been accelerated to the desired energy. Typically the maximum beam brightness produced by an accelerator is constrained by the beam brightness delivered by the accelerator injector. Thus it is important to design the accelerator injector to yield the required electron beam brightness. The DPC (Darwin Particle Code) computer code has been written to numerically model accelerator injectors. DPC solves for the transport of a beam from emission through acceleration up to the full energy of the injector. The relativistic force equation is solved to determine particle orbits. Field equations are solved for self-consistent electric and magnetic fields in the Darwin approximation. DPC has been used to investigate the beam quality consequences of AK (anode-cathode) gap, accelerating stress, electrode configuration and axial magnetic field profile.

Enhancing the brightness of high current electron guns

Concepts such as the two-beam accelerator offer the possibility of translating pulsed power technology into a form useful to the design of high luminosity accelerators for high-energy physics applications. Realization of the promise of these concepts will require the design of electron guns which are optimized with respect to beam brightness at current levels of approximately 1 kA. Because high luminosity implies accelerator operation at high repetition rates, the high-current beam source must be designed so that the beam does not intercept the electrodes. In our investigations of electron gun configurations, we have found that the brightness of a given source is set by practical design choices such as peak voltage, cathode type, gun electrode geometry, and focusing field topology. To investigate the sensitivity of beam brightness to these factors in a manner suitable for modelling transient phenomena at the beam head, we have developed a Darwin approximation particle code, DPC. The main component in our experimental program is a readily modified electron gun that allows us to test many candidate cathode materials, types, and electrode geometries at field stresses up to 1 MV/cm. We have also developed several diagnostics suitable for measuring the brightness of intense, low-emittance beams.

Elimination of electromagnetic radiation in plasma simulation: The Darwin or magneto inductive approximation

For many astrophysica1 and magnetic fusion applications, the purely electromagnetic modes generated by real as well as simulation “plasma” fluctuations are a source of high frequency radiation that is often irrelevant to the physics of interest. Unfortunately, a numerical CFL stability limit prevents either making c infinite or Δt large while using the usual explicit Maxwell’s equations for the fields. A modification of Maxwell’s equations, which provides implicitly the field components, circumvents this problem. The solution is to neglect retardation effects so that the electromagnetic propagation speed is effectively infinite. The purely electromagnetic modes in this limit evolve “instantly” to a time-asymptotic configuration about the macroscopic plasma configuration at each new time level. The Darwin or magneto inductive approximation effectively provides infinite propagation speeds for purely electromagnetic modes by converting Maxwell’s equations from hyperbolic to elliptic in character. In practice, this is accomplished by neglecting the solenoidal part of the displacement current. The elimination of the CFL time step constraint more than offsets the substantially more complicated field solution that is required. The details of a numerical implementation of this model will be presented. Numerical examples will be given and extentions of the Darwin field solution to other plasma models also will be considered.

Brightness limitations in multi-kiloampere electron beam sources

Heuristic relationships such as the Lawson-Penner criterion, used to scale free electron laser (FEL) amplifier gain and efficiency over orders of magnitude in beam current and brightness, have no fundamental basis. The brightness of a given source is set by practical design choices such as peak voltage, cathode type, gun electrode geometry, and focusing field topology. The design of low emittance, high current electron guns has received considerable attention at Livermore over the past few years. The measured brightnesses of the experimental test accelerator (ETA) and advanced test accelerator (ATA) guns are less than predicted with the EBQ [1] gun design code; this discrepancy is due to plasma effects from the present cold, plasma cathode in the code. The EBQ code is well suited to exploring the current limits of gridless relativistic Pierce columns with moderate current density (< 50 A/cm 2) at the cathode. As EBQ uses a steady-state calculation it is not amenable for study of transient phenomena at the beam head. For this purpose, a Darwin approximation code, DPC, has been written. The main component in our experimental cathode development effort is a readily modified electron gun that will allow us to test many candidate cathode materials, types and electrode geometries at field stresses up to 1 MV/cm.

Numerical simulation of ion beam propagation in Z-pinch plasma channels

Propagation of intense ion beams in preformed z-pinch plasma channels, a scenario of interest for inertial confinement fusion devices, has been studied with a 2½-dimensional (r, z, vr, vθ , vz; ∂/∂θ = 0) hybrid simulation model. In the code, ions are represented by particles and electrons by an inertialess thermal fluid which obeys a generalized Ohm's law. Fields are solved in the quasi-neutral Darwin approximation. Several collisional and atomic processes are included. The simulations of this paper are performed with collisional effects scaled down by a factor of ten. The results are then extrapolated to estimate that propagation efficiencies in the 70% range are possible using 5–7 MeV, 1–2 MA proton beams with initial divergences in the range of 1.5° to 7° in a 4-m hydrogen channel. Current neutralization in excess of 99% is found, and no gross axisymmetric instabilities are observed.

The K-shell ionisation of atoms by relativistic protons

The K-shell ionisation is investigated using an extension of the first-order time-dependent perturbation-theory treatment of Moller (1932) taking Dirac plane waves for the description of the incident and scattered protons and the Darwin approximation for the relativistic wavefunctions of the K-shell atomic electrons and the ejected electron. The differential cross sections delta 2 sigma / delta k delta q and d sigma /dk, where k is the wavenumber of the ejected electron and hq is the momentum change of the proton, are calculated. The double differential cross section for collisions in which a change of spin of the atomic electron occurs is less sharply peaked for small q, and has a broader overall distribution with respect to q, than the double differential cross section for collisions without spin change, although being much less in magnitude than the latter for small q. This leads to total K-shell ionisation cross sections sigma with significant contributions coming from collisions with spin change of the atomic electron for large atomic numbers at high energies of impact.