Distribution Function Of Charged Particles Research Articles

On global weak solutions of the Vlasov-Poisson equations with external magnetic field

We consider the first mixed problem for the system of Vlasov-Poisson equations with a given external magnetic field in a bounded domain. This problem describes the kinetics of high-temperature plasma in controlled thermonuclear fusion plants and is considered with respect to unknown functions: electric field potential, distribution functions of positively charged ions and electrons. Additionally, we assumed that the distribution functions of charged particles satisfy the condition of mirror reflection from the boundary of the domain under consideration. We prove the existence of global weak solutions of such a problem.

On the Complex-Valued Distribution Function of Charged Particles in Magnetic Fields

In this work, we revisit Boltzmann’s distribution function, which, together with the Boltzmann equation, forms the basis for the kinetic theory of gases and solutions to problems in hydrodynamics. We show that magnetic fields may be included as an intrinsic constituent of the distribution function by theoretically motivating, deriving and analyzing its complex-valued version in its most general form. We then validate these considerations by using it to derive the equations of ideal magnetohydrodynamics, thus showing that our method, based on Boltzmann’s formalism, is suitable to describe the dynamics of charged particles in magnetic fields.

Charged particle dynamics and distribution functions in low pressure dual-frequency capacitively coupled plasmas operated at low frequencies and high voltages

In high aspect ratio (HAR) dielectric plasma etching, dual-frequency capacitively coupled radio-frequency plasmas operated at low pressures of 1 Pa or less are used. Such plasma sources are often driven by a voltage waveform that includes a low-frequency component in the range of hundreds of kHz with a voltage amplitude of 10 kV and more to generate highly energetic vertical ion bombardment at the wafer. In such discharges, the energetic positive ions can overcome the repelling potential created by positive wall charges inside the etch features, which allows high aspect ratios to be reached. In order to increase the plasma density a high-frequency driving component at several 10 MHz is typically applied simultaneously. Under such discharge conditions, the boundary surfaces are bombarded by extremely energetic particles, of which the consequences are poorly understood. We investigate the charged particle dynamics and distribution functions in this strongly non-local regime in argon discharges by particle-in-cell simulations. By including a complex implementation of plasma-surface interactions, electron induced secondary electron emission (δ-electrons) is found to have a strong effect on the ionization dynamics and the plasma density. Due to the high ion energies at the electrodes, very high yields of the ion induced secondary electron emission (γ-electrons) are found. However, unlike in classical capacitive plasmas, these γ-electrons do not cause significant ionization directly, since upon acceleration in the high voltage sheaths, these electrons are too energetic to ionize the neutral gas efficiently. These γ- and δ-electrons as well as electrons created in the plasma bulk and accelerated towards the electrodes to high energies by reversed electric fields during the local sheath collapse are found to induce the emission of a high number of δ-electrons, when they hit boundary surfaces. This regime is understood fundamentally based on the following approach: first, dual-frequency discharges with identical electrode materials are studied at different pressures and high-frequency driving voltages. Second, the effects of using electrodes made of different materials and characterized by different secondary electron emission coefficients are studied. The electron dynamics and charged particle distribution functions at boundary surfaces are determined including discharge asymmetries generated by using different materials at the powered and grounded electrodes.

Towards a better estimation of energy and species of primary cosmic rays in the knee region with the Tibet hybrid experiment: Utilization of refined EAS lateral distributions

A new hybrid experiment (YAC-II+Tibet-III+MD) located at Yangbajing has started to improve the capability of explicit measurements of the cosmic-ray components (P, He, CNO, Fe, etc) at the knee energy region since 2014. Considering the different features of air shower development for different primary cosmic-ray nuclei, using a full Monte Carlo simulation, we modified the lateral distribution functions for proton-induced, helium-induced and iron-induced air showers, respectively. The results show that the air shower size of different nuclei obtained by the modified Nishimura-Kamata-Greisen function is consistent with the true shower size within 5% systematic errors. Furthermore, we studied the Extensive Air Shower age parameter determination from the lateral distribution functions of charged particles near the air shower core with the Tibet hybrid experiment. The derived age parameter can be well used to estimate the particle type of the incident primary cosmic rays.

Energetic Ion Moments and Polytropic Index in Saturn's Magnetosphere using Cassini/MIMI Measurements: A Simple Model Based on κ‐Distribution Functions

AbstractMoments of the charged particle distribution function provide a compact way of studying the transport, acceleration, and interactions of plasma and energetic particles in the magnetosphere. We employ κ‐distributions to describe the energy spectra of H+ and O+, based on >20 keV measurements by the three detectors of Cassini's Magnetospheric Imaging Instrument, covering the time period from DOY 183/2004 to 016/2016, 5 < L < 20. From the analytical spectra we calculate the equatorial distributions of energetic ion moments inside Saturn's magnetosphere and then focus on the distributions of the characteristic energy (Ec=IE/In), temperature, and κ‐index of these ions. A semiempirical model is utilized to simulate the equatorial ion moments in both local time and L‐shell, allowing the derivation of the polytropic index (Γ) for both H+ and O+. Primary results are as follows: (a) The ∼9 < L < 20 region corresponds to a local equatorial acceleration region, where subadiabatic transport of H+ (Γ∼1.25) and quasi‐isothermal behavior of O+ (Γ∼0.95) dominate the ion energetics; (b) energetic ions are heavily depleted in the inner magnetospheric regions, and their behavior appears to be quasi‐isothermal (Γ<1); (c) the (quasi‐) periodic energetic ion injections in the outer parts of Saturn's magnetosphere (especially beyond 17–18 RS) produce durable signatures in the energetic ion moments; (d) the plasma sheet does not seem to have a ground thermodynamic state, but the extended neutral gas distribution at Saturn provides an effective cooling mechanism that does not allow the plasma sheet to behave adiabatically.

New Possibilities of Probe Detection of Anisotropic Charged-Particle Distribution Functions in an Arbitrary-Symmetry Plasma

We have analyzed the precision and systematic errors of the familiar method of a plane single-ended probe for measuring the anisotropic distribution functions for electrons and ions in plasma. Analytic relations that connect the plasma parameters and the required number of terms in a series are obtained under the conditions when anisotropy is due to the presence of an electric field in the plasma. It has been shown that ten terms of the series are usually sufficient to adequately describe the ion distribution function, except for the case of strong fields. For strongly anisotropic charged particle distribution functions, the spline interpolation technique for experimental data is proposed and tested, which substantially reduces the systematic error for a preset number of terms in the series. It has been shown for abundant actual plasma objects with the mirror symmetry that the number of required orientations of a plane probe for the experimental distribution function can be reduced by half compared to the most general case of the absence of any symmetry for a fixed number of terms in the Legendre series, while for the same number of orientations, the number of terms in the series, which are determined from these measurements, doubled accordingly.

Numerical experiments on charging of a spherical body in a plasma with Maxwellian distributions of charged particles

New results of numerical simulation of collisionless plasma perturbation caused by a sphere absorbing electrons and ions are presented. Consideration is given to nonstationary phenomena accompanying the process of charging as well as to plasma steady state reached at long times. Corresponding asymptotic values of charges of the sphere and trapped-ion cloud around it have been found along with self-consistent electric field pattern depending on parameters of the unperturbed plasma. It is established that contribution of the trapped ions to screening of the charged sphere can be quite significant, so that the screening becomes essentially nonlinear in nature. A simple interconnection between the sphere radius, electron and ion Debye lengths has been revealed as the condition for maximum trapped-ion effect. Kinetic structure of the space charge induced in the plasma is discussed with relation to the specific form of the unperturbed charged particle distribution functions.

PLASMA TECHNOLOGIES FOR THE PROTECTION OF RADIO ELECTRONIC MEANS FROM EXPOSURE TO HIGH–POWER ELECTROMAGNETIC RADIATIONS WITH ULTRASHORT PULSE DURATION

The urgency of the improvement of methods and means of protection of radio electronic means (REM) due to the development and application of high–power electromagnetic radiation (EMR) generators with ultrashort pulses duration (UPD) is substantiated. It is pointed out that it is possible to solve the problem of REM protection by the complex application of plasma technologies using gaseous and modified solid–state media. By analyzing the known achievements in the field of developing effective methods and creating protection facilities for REM, a number of unresolved problems in the field of creation of plasma protection technologies have been determined. The technique of solving the formulated problems is presented and the main relations are obtained to determine the expediency of using the proposed technology in the interests of the REM integrated protection from the powerful EMR UPD. A relation is made for the breakdown criterion in a gaseous plasma medium that relates the value of the breakdown field to the concentration of charged particles determined by the ionization source. The structure of a solid–state plasma medium is described, which can be used as a protective shield. The order of finding the REM screening coefficient is shown, based on the determination of the distribution function of charged particles for finding the main macroscopic properties of plasma by solving the kinetic equation of the Lenard–Balescu equation. A relation is given for the damping coefficient of an electromagnetic wave in solid–state plasma. Numerical estimates are presented in the form of graphs showing the possibility of using plasma technology to protect the REM from a powerful EMR through possible channels of penetration. The discussion of the obtained results is presented and it is indicated on the possibility and prospects of using the proposed technology for REM protection, especially with limitations on the weight dimensions of protection devices

Collisionless distribution function of charged particles ensemble in a tokamak magnetic configuration with magnetic island

The collisionless distribution function of charged particle ensemble in the magnetic field of tokamak with a magnetic island is calculated. The calculation is based on the solution of the kinetic equation with source together with three-dimensional numerical calculations of charged particle trajectories. It is shown that in case of an inhomogeneous source trajectory, motion of trapped particles leads to anisotropization of the initially isotropic distribution of particle ensemble. The absence of contribution from the passing particles decreases the efficiency of spontaneous generation of a non-induction current in the magnetic island in comparison with the bootstrap effect in the system of nested magnetic surfaces.

Electrostatic analyzer with a 3‐D instantaneous field of view for fast measurements of plasma distribution functions in space

AbstractWe describe the concept and properties of a new electrostatic optic which aims to provide a 2π sr instantaneous field of view to characterize space plasmas. It consists of a set of concentric toroidal electrodes that form a number of independent energy‐selective channels. Charged particles are deflected toward a common imaging planar detector. The full 3‐D distribution function of charged particles is obtained through a single energy sweep. Angle and energy resolution of the optics depends on the number of toroidal electrodes, on their radii of curvature, on their spacing, and on the angular aperture of the channels. We present the performances, as derived from numerical simulations, of an initial implementation of this concept that would fit the need of many space plasma physics applications. The proposed instrument has 192 entrance windows corresponding to eight polar channels each with 24 azimuthal sectors. The initial version of this 3‐D plasma analyzer may cover energies from a few eV up to 30 keV, typically with a channel‐dependent energy resolution varying from 10% to 7%. The angular acceptance varies with the direction of the incident particle from 3° to 12°. With a total geometric factor of two sensor heads reaching ~0.23 cm2 sr eV/eV, this “donut” shape analyzer has enough sensitivity to allow very fast measurements of plasma distribution functions in most terrestrial and planetary environments on three‐axis stabilized as well as on spinning satellites.

Kinetic models of sub-ion cylindrical magnetic hole

Magnetic holes are magnetoplasma structures very similar to the classical θ–pinch. They are widely observed in the space plasma and identified by the substantial magnetic field depressions on scales from magnetohydrodynamic range to electron scales. In this paper, we develop the kinetic models of cylindrically symmetric magnetic holes with sub-ion scales using two types of charged particle distribution functions (both current-carrying and background plasma populations are included). We demonstrate that developed magnetic holes have configurations very similar to those revealed in the recent spacecraft observations in the Earth magnetosphere: both localized electron currents and strong radial electric fields are found at the magnetic hole boundary. We demonstrate that for realistic plasma parameters, the inclusion of ion currents into the model produces magnetic holes with double-scale magnetic field profile. We find that the magnetic hole depth (amplitude of the magnetic field depression) depends on the magnetic hole typical radius. Possible applications of developed models are discussed.

Computer simulation of the jet of rarefied plasma outflowing from a nozzle of a plasma thruster

Physical, mathematical and numerical models for the problem of the outflow of a jet of rarefied plasma from the nozzle of a plasma thruster of the cylindrical and rectangular shape have been considered. The main results of computation experiments have been presented: distribution functions of charged particles in the plasma flow, velocity and concentration fields of charged particles, and isolines of the potential of the self-consistent electric field.

Acceleration of ions and nano dust at a comet in the solar wind

A quasi-neutral hybrid simulation of the interaction of the solar wind with the atmosphere of a comet is used together with a test particle simulation of cometary ions and dust to compute trajectories and velocity distribution functions of charged particles, starting outside the diamagnetic cavity at 150km cometocentric distance. The simulations are run with parameters suited to make predictions for comet 67P/Churyumov–Gerasimenko when it is at a heliocentric distance of 1.45AU. It is found that the shape of the ion trajectories depends on the location of the source, and that a velocity distribution that is observed at a given point in space is influenced by the spatial structure of the source. Charged dust grains with radii in the 1–10nm range are accelerated from the nucleus to a distance of 2.9×104km in between 15min and 2h approximately. Dust particles smaller than 10nm in radius are accelerated to speeds over 10km/s.

The Q,G-method in physics of collisionless plasma

An analytic method for analyzing nonstationary processes in collisionless one-dimensional bounded plasma—the Q,G-method—is proposed. Formulas for the distribution function of charged particles escaping from the boundary and moving without collisions in a nonstationary electric field of arbitrary form are obtained. Specific features of the distribution function are revealed and analyzed.

Numerical Modeling of the Time Evolution of Super-Small-Scale Irregularities in the Near-Earth Rarefied Plasma

The time evolution of the magnetic field aligned super-small-scale irregularities in the concentration of charged particles, existing in the near-Earth rarefied plasma, is studied with the help of the model simulation. A new version of the two-dimensional mathematical model, developed earlier in the Polar Geophysical Institute, is utilized to investigate the temporal history of the irregularity with circular cross section, created initially in the near-Earth plasma. The utilized model is based on a numerical solution of the Vlasov-Poisson system of equations, with the Vlasov equations describing the distribution functions of charged particles and the Poisson equation governing the self-consistent electric field. The results of simulation indicate that the mobility of the positive ions ought to influence essentially on the time evolution of the super-small-scale irregularities in the concentration of charged particles, existing in the near-Earth rarefied plasma.

Orbit-based representation of equilibrium distribution functions for low-noise initialization of kinetic simulations of toroidal plasmas

This work deals with the initial loading of phase space markers for global gyrokinetic particle-in-cell (PIC) simulations of plasmas that are magnetically confined in a toroidally axisymmetric configuration. A method is presented, which allows to prepare a marker distribution that is independent of time. This is achieved by discretizing the phase space along lines of constants of motion, which allows to load markers on the toroidal surfaces of unperturbed guiding center orbits. On each orbit surface, markers are distributed uniformly in time, so their distribution represents the compressible motion of physical particles. This method allows to initialize global PIC codes with an accurate equilibrium distribution function for charged particles, taking into account prompt losses to the wall. It facilitates simulations with lower noise levels and minimal noise–signal correlation; especially, in the linear regime. The problem considered is the representation of energetic ions in tokamaks, which are characterized by large drifts across magnetic surfaces.

Eigenmodes of quasi-static magnetic islands in current sheet

As observation have shown, magnetic islands often appear before and/or after the onset of magnetic reconnections in the current sheets, and they also appear in the current sheets in the solar corona, Earth’s magnetotail, and Earth’s magnetopause. Thus, the existence of magnetic islands can affect the initial conditions in magnetic reconnection. In this paper, we propose a model of quasi-static magnetic island eigenmodes in the current sheet. This model analytically describes the magnetic field structures in the quasi-static case, which will provide a possible approach to reconstructing the magnetic structures in the current sheet via observation data. This model is self-consistent in the kinetic theory. Also, the distribution function of charged particles in the magnetic island can be calculated.

Lateral distribution function of charged particles in EAS near the axis

High-precision measurements of the lateral distribution function (LDF) of charged particles near the axis of extensive air showers (EASes) were performed with the CARPET air shower array of the Baksan Neutrino Observatory for several ranges of Ne, including the knee region. For the sake of comparison with the experiment, calculations for primary protons and iron nuclei were made using the CORSIKA code (the QGSJET01C model of interaction). The measured experimental LDF is compared with our calculations.

Kinetic Theory for Distribution Functions of Wave-Particle Interactions in Plasmas

The evolution of a charged particle distribution function under the influence of coherent electromagnetic waves in a plasma is determined from kinetic theory. For coherent waves, the dynamical phase space of particles is an inhomogeneous mix of chaotic and regular orbits. The persistence of long time correlations between the particle motion and the phase of the waves invalidates any simplifying Markovian or statistical assumptions--the basis for usual quasilinear theories. The generalized formalism in this Letter leads to a hierarchy of evolution equations for the reduced distribution function. The evolution operators, in contrast to the quasilinear theories, are time dependent and nonsingular and include the rich phase space dynamics of particles interacting with coherent waves.

The distribution functions of charged particles in the vicinity of a cylindrical body in a flow of collisionless plasma in magnetic field

The distribution functions are investigated of charged particles in the wall region in the vicinity of a cylindrical body placed in a flow of collisionless plasma. For comparison with the studies by other authors, the calculation is first performed for the case of quiescent plasma in the absence of magnetic field. Subsequent calculations consistently include the directional velocity of plasma and external magnetic field. The basic features of obtained distribution functions are revealed and explained.