Ion Velocity Distribution Function Research Articles

Kinetic theory of the positive column of a low-pressure discharge in a transverse magnetic field

The influence of a transverse magnetic field on the characteristics of the positive column of a planar low-pressure discharge is studied theoretically. The motion of magnetized electrons is described in the framework of a continuous-medium model, while the ion motion in the ambipolar electric field is described by means of a kinetic equation. Using mathematical transformations, the problem is reduced to a secondorder ordinary differential equation, from which the spatial distribution of the potential is found in an analytic form. The spatial distributions of the plasma density, mean plasma velocity, and electric potential are calculated, the ion velocity distribution function at the plasma boundary is found, and the electron energy as a function of the magnetic field is determined. It is shown that, as the magnetic field rises, the electron energy increases, the distributions of the plasma density and mean plasma velocity become asymmetric, the maximum of the plasma density is displaced in the direction of the Ampere force, and the ion flux in this direction becomes substantially larger than the counter-directed ion flux.

Ion Velocity Distribution Function Investigated Inside an Unstable Magnetized Plasma Exhibiting a Rotating Nonlinear Structure

The frequent situation where a strongly nonlinear rotating structure develops in a linear magnetized plasma column is investigated experimentally with emphasis on the ion velocity distribution function (IVDF). Most often, a mode m=2 appears exhibiting a large density and potential perturbation with angular frequency slightly above the ion cyclotron frequency. For the first time the spatiotemporal evolution of the IVDF is studied using time-resolved laser induced fluorescence to explore the ion's interaction with the nonlinear wave propagating inside the column and at the origin of plasma transport outside the limiter. The ion fluid exhibits an alternance from azimuthal to radial velocity due to the electric field inside the rotating structure. A fluid model also allows us to locally reconstruct the self-consistent electric field evolution which contradicts all existing theories.

Temporally resolved ion velocity distribution measurements in a radio-frequency plasma sheath

The ion velocity distribution function (IVDF) above and within a radio-frequency (RF) biased plasma sheath is studied experimentally with a pulsed laser-induced fluorescence diagnostic in an industrial plasma etch tool. Temporally resolved measurements taken at eight different phases of the 2.2 MHz bias waveform show that the ion dynamics vary dramatically throughout the RF cycle (the ratio of the average ion transit time through the sheath to the RF period is τion/τRF = 0.3). The position of the presheath/sheath edge is constant throughout the RF cycle and the time-averaged ion flux is conserved within the sheath region. The characteristic bimodal structure of the time-averaged ion distributions found in previous experiments is observed to arise from the time-dependent ion dynamics, in accord with existing theory. The large temporal variation of the IVDF has implications for the plasma chemistry and etching quality.

Sheath-Plasma Criterion of Fluid Theory with Finite Ion Temperature

with a polytropic index . The value of is important in relation to probe measurements, and one may set 1⁄4 3 for an adiabatic case (one-dimension), or 1⁄4 1 for an isothermal case. In fact, varies from three to unity owing to limited heat flow, since there is no heat flow in the adiabatic case and unhindered heat flow in the isothermal case. The value of for the sheath-plasma criterion has remained a problem. To derive the sheath-plasma criterion, equations applicable in a sheath region with space charge are used in general. However, there is another method of deriving the sheath-plasma criterion, in which equations applicable in a plasma or presheath region are used. Since quasi-neutrality is assumed in the equations in the second method, the equations become invalid in the sheath region owing to space charge, and a singularity gives a sheath-plasma boundary, where the ion drift speed is given by eq. (1) for Ti 1⁄4 0. The extension of the second method for Ti 61⁄4 0 is achieved by considering ion pressure, and shows the sheathplasma criterion indicating that the ion drift speed at the sheath-plasma boundary is given by eq. (2). As for the first method for Ti 61⁄4 0, the generalized sheath-plasma criterion of the ion velocity distribution function is obtained, although the details of the ion velocity distribution function are not determined. The purpose of this report is to find a solution of the equations in the second method for a finite ion temperature, which may vary in accordance with the polytropic approximation, because only the solution under the condition of Ti 1⁄4 0 or Ti 1⁄4 const: is known. Basic equations of the fluid theory treating a steady-state one-dimensional plasma with a uniform ion generation G (1⁄4 const:) are written as

Kinetic Models for Whistler Wave Scattering of Electrons in the Solar Corona and Wind

Kinetic models are necessary to describe the physical processes associated with non-Maxwellian velocity distribution functions (VDFs) of electrons or ions in the solar corona and wind. It is shown that pitch-angle scattering of electrons in the solar wind needs to be considered in kinetic solar wind models. Coulomb collisions are not efficient enough to provide this scattering, but resonant interaction with whistler waves is. A solar wind model for undisturbed fast wind is presented, and the influence of scattering on flare electron propagation is investigated. Furthermore, it is found that resonant interaction of electrons with whistler waves is capable of producing suprathermal tails of electron distributions even under quiet conditions without flare activity.

Heavy-ion flux enhancement in the vicinity of the Martian ionosphere during CIR passage: Mars Express ASPERA-3 observations

[1] A characteristic heavy-ion signature observed in the vicinity of the Martian ionosphere during passages of Corotating Interaction Region (CIR) structures in solar wind is reported. We analyzed data obtained by the IMA/ASPERA-3 onboard the Mars Express (MEX) from September to October 2007. We compared the solar wind velocity at Mars derived from a shifted Maxwellian fitting to the IMA data with time-shifted Advanced Composition Explorer satellite data taken at ∼1 AU to the Martian orbit. Using the derived solar wind velocity, we identified four CIR structures passing through Mars quasiperiodically. Coinciding with the CIR passages, the IMA observed heavy-ion flux enhancement in the vicinity of the Martian ionosphere. The heavy-ion energies reach ≥100 eV and sometimes up to approximately several kiloelectron volts. Observed ion velocity distribution functions show that they are mainly precipitating toward the Martian ionosphere. The flux of the precipitating ions is typically 105–106 (104–105) cm−2 s−1 for the energy range of 50–500 eV (≥500 eV) and it becomes by one order of magnitude higher in one event. While the flux level is consistent with a previous model prediction of sputtering ions, the intermittent occurrence of the heavy-ion precipitation differs from conventional expectation of constant precipitation. These results suggest that the efficiency of the sputtering process in the Martian atmospheric escape is highly variable with dynamic solar wind variations.

Formation of an anisotropic electron velocity distribution function and production of multicharged ions in the micropinch discharge plasma

Analysis of the results of polarimetric measurements of X-ray line radiation of multicharged ions in a Z-pinch discharge indicates that the formation of an anisotropic electron velocity distribution in the neck of the current channel and the generation of highly charged ions are separated in time. The generation of a fast electron beam in the longitudinal ohmic electric field in the stage of plasma compression in the neck results in the polarization of X-ray bremsstrahlung continuum. In the stage of expansion of the hot dense micropinch plasma, the radial electric field prevails, due to which X-ray line radiation of multicharged ions becomes linearly polarized.

Inertial electrostatic confinement and nuclear fusion in the interelectrode plasma of a nanosecond vacuum discharge. II: Particle-in-cell simulations

Results of particle-in-sell simulations of ion acceleration by using the KARAT code in a cylindrical geometry in the problem formulation corresponding to an actual experiment with a low-energy vacuum discharge with a hollow cathode are presented. The fundamental role of the formed virtual cathode is analyzed. The space-time dynamics of potential wells related to the formation of the virtual cathode is discussed. Quasi-steady potential wells (with a depth of ∼80% of the applied voltage) cause acceleration of deuterium ions to energies about the electron beam energy (∼50 keV). In the well, a quasi-isotropic velocity distribution function of fast ions forms. The results obtained are compared with available data on inertial electrostatic confinement fusion (IECF). In particular, similar correlations between the structure of potential wells and the neutron yield, as well as the scaling of the fusion power density, which increases with decreasing virtual cathode radius and increasing potential well depth, are considered. The chosen electrode configuration and potential well parameters provide power densities of nuclear DD fusion in a nanosecond vacuum discharge noticeably higher than those achieved in other similar IECF systems.

Ion acceleration in Ar–Xe and Ar–He plasmas. II. Ion velocity distribution functions

Ion velocity distribution functions (ivdfs) are investigated by laser induced fluorescence in Ar–Xe and Ar–He expanding helicon plasmas as a function of gas composition. In the case of Ar–Xe plasma, it was found that in the helicon source, both the Ar+ and Xe+ vdfs are unimodal. Their parallel speeds are subsonic and unaffected by changes in gas composition. At the end of the source, the argon ivdf shows a bimodal structure indicative of an electric double layer upstream of the measurement location. The fast argon ion component parallel velocity increases with Xe fraction from 6.7 to 8 km/s as the Xe fraction increases from 0% to 4%. In the expansion region, the bimodal character of Ar ivdf is maintained with a supersonic fast component reaching parallel speeds of 10.5 km/s. For all the studied plasma conditions and different spatial locations, the Xe+ vdf exhibits a unimodal structure with a maximum parallel flow velocity of 2.2 km/s at the end of the source. For Ar–He plasma, the Ar ivdf is bimodal with the fast ion component parallel velocity increasing from 5.2 to 7.8 km/s as the He fraction increases from 0% to 30%. For the same He fraction range, the slow argon ion population distribution changes from a single Gaussian to a wide distribution extending all the way from the speed of the fast population to 0 m/s.

Simultaneous two-dimensional laser-induced-fluorescence measurements of argon ions

Recent laser upgrades on the Hot Helicon Experiment at West Virginia University have enabled multiplexed simultaneous measurements of the ion velocity distribution function at a single location, expanding our capabilities in laser-induced fluorescence diagnostics. The laser output is split into two beams, each modulated with an optical chopper and injected perpendicular and parallel to the magnetic field. Light from the crossing point of the beams is transported to a narrow-band photomultiplier tube filtered at the fluorescence wavelength and monitored by two lock-in amplifiers, each referenced to one of the two chopper frequencies.

Kinetics of parametric instabilities of Alfvén waves: Evolution of ion distribution functions

Using numerical simulations in a hybrid regime, we studied the evolution of large‐amplitude Alfvén waves subject to modulational and decay instabilities, including the effects of ion kinetics. We considered both a monochromatic and incoherent spectrum of waves, different wave polarizations and amplitudes, and different plasma regimes, ranging from β < 1 to β > 1. We found in all cases that ion dynamics affects the instability evolution and saturation; as a feedback, wave‐particle interactions provide a nonlinear trapping of resonant particles that importantly change the properties of the ion velocity distribution functions. In particular, we observed a proton acceleration along the magnetic field and in some cases the formation of a parallel velocity beam traveling faster than the rest of the distribution. For the range of parameters used in our simulations, the fundamental ingredient in generating an ion beam is observed to be the parallel electric field carried by the density fluctuations driven by the ion‐acoustic modes generated by the parametric instabilities.

Fundamental aspects of substrate biasing: ion velocity distributions and nonlinear effects

Ion bombardment of the substrate is a significant parameter in plasma processing such as dry etching or thin film deposition. The ion bombardment is described by ion velocity distribution functions (IVDFs), which were here measured quantitatively at a sinusoidally and non-sinusoidally biased electrode. The electrode voltage was monitored and controlled in the frequency domain using fast Fourier transformation. IVDF measurements were performed by a floating retarding field analyzer. A full modulation of the IVDF by arbitrary bias waveforms is only achieved if sufficiently high sheath voltages are used. If the applied sheath voltages become too low, the IVDFs are only partly determined by the RF bias waveforms and the system response becomes nonlinear. An analytical sheath model is derived from the experimental data, which accounts for arbitrary bias waveforms as well as for collisional and nonlinear effects in the sheath. It is shown that a combined DC and RF biasing of the electrode is required to gain full control over the ion bombardment of the substrate.

Collective Thomson Scattering of 77 GHz High‐Power ECRH Beam in LHD

AbstractThe measurement of the ion velocity distribution is especially important in reactor relevant plasmas. The collective Thomson scattering (CTS) is one of the most promising methods for evaluating the ion velocity distribution function. In spite of its potential, this method had long been suffered from the absence of adequate power sources. Recent development in the higher power, and higher frequency range of the gyrotron and the transmission technique enabled to realize the measurement of not only the bulk but also high energy component of the ion velocity distribution function. One of the newly installed 77 GHz gyrotrons in LHD that can generate more than 1 MW over 5 s and is connected to one of the Gaussian mirror beam launching antenna sets is used as a probing beam for CTS. A heterodyne receiver system of a fundamental mixer with fixed frequency local oscillator was installed on the upstream of the transmission line. The probe beam is 100% power modulated at 50 Hz to separate the scattering component from background ECE. Signals that can be attributed to the CTS were obtained and the analysis method of these data are developed (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Fulfillment of the kinetic Bohm criterion in a quasineutral particle-in-cell model

Quasineutral particle-in-cell models of ions must fulfill the kinetic Bohm criterion, in its inequality form, at the domain boundary in order to match correctly with solutions of the Debye sheaths tied to the walls. The simple, fluid form of the Bohm criterion is shown to be a bad approximation of the exact, kinetic form when the ion velocity distribution function has a significant dispersion and involves different charge numbers. The fulfillment of the Bohm criterion is measured by a weighting algorithm at the boundary, but linear weighting algorithms have difficulties to reproduce the nonlinear behavior around the sheath edge. A surface weighting algorithm with an extended temporal weighting is proposed and shown to behave better than the standard volumetric weighting. Still, this must be supplemented by a forcing algorithm of the kinetic Bohm criterion. This postulates a small potential fall in a supplementary, thin, transition layer. The electron-wall interaction is shown to be of little relevance in the fulfillment of the Bohm criterion.

Identification of prominence ejecta by the proton distribution function and magnetic fine structure in interplanetary coronal mass ejections in the inner heliosphere

This work presents in situ solar wind observations of three magnetic clouds (MCs) that contain cold high‐density material when Helios 2 was located at 0.3 AU on 9 May 1979, 0.5 AU on 30 March 1976, and 0.7 AU on 24 December 1978. In the cold high‐density regions embedded in the interplanetary coronal mass ejections we find (1) that the number density of protons is higher than in other regions inside the magnetic cloud, (2) the possible existence of He+, (3) that the thermal velocity distribution functions are more isotropic and appear to be colder than in the other regions of the MC, and the proton temperature is lower than that of the ambient plasma, and (4) that the associated magnetic field configuration can for all three MC events be identified as a flux rope. This cold high‐density region is located at the polarity inversion line in the center of the bipolar structure of the MC magnetic field (consistent with previous solar observation work that found that a prominence lies over the neutral line of the related bipolar solar magnetic field). Specifically, for the first magnetic cloud event on 8 May 1979, a coronal mass ejection (CME) was related to an eruptive prominence previously reported as a result of the observation of Solwind (P78‐1). Therefore, we identify the cold and dense region in the MC as the prominence material. It is the first time that prominence ejecta were identified by both the plasma and magnetic field features inside 1 AU, and it is also the first time that the thermal ion velocity distribution functions were used to investigate the microstate of the prominence material. Moreover, from our three cases, we also found that this material tended to fall behind the magnetic cloud and become smaller as it propagated farther away from the Sun, which confirms speculations in previous work. Overall, our in situ observations are consistent with three‐part CME models.

Comparison of fast ion collective Thomson scattering measurements at ASDEX Upgrade with numerical simulations

Collective Thomson scattering (CTS) experiments were carried out at ASDEX Upgrade to measure the one-dimensional velocity distribution functions of fast ion populations. These measurements are compared with simulations using the codes TRANSP/NUBEAM and ASCOT for two different neutral beam injection (NBI) configurations: two NBI sources and only one NBI source. The measured CTS spectra as well as the inferred one-dimensional fast ion velocity distribution functions are clearly asymmetric as a consequence of the anisotropy of the beam ion populations and the selected geometry of the experiment. As expected, the one-beam configuration can clearly be distinguished from the two-beam configuration. The fast ion population is smaller and the asymmetry is less pronounced for the one-beam configuration. Salient features of the numerical simulation results agree with the CTS measurements while quantitative discrepancies in absolute values and gradients are found.

Estimation of the Effective Collective Thomson Scattering Volume Using a Gaussian Beam in the LHD ECRH System

Collective Thomson scattering (CTS) is a promising candidate for direct measurement of the velocity distribution function of ions in fusion devices. The relevant frequency for a CTS probing beam source is in the millimeter- to submillimeter-wave range. The scattering volume of CTS is an important parameter for determining the intensity of the scattered power and the spatial resolution of the measurement. The conventional method of estimating the scattering volume is to simply calculate the geometrical volume of the probing and receiving beams. This calculation is valid only when the beams are diffraction free. The effective scattering volume is defined and applied to a CTS system using strongly focused beams and its validity is examined experimentally.

Collective Thomson Scattering Study Using Gyrotron in LHD

The collective Thomson scattering (CTS) is one of the most promising methods for evaluating the ion velocity distribution function. The study of CTS diagnostic has been started utilizing the gyrotron and antenna/transmission systems installed in LHD for high power local electron heating. One of the high power gyrotrons at 77 GHz is selected as a probing power source and a set of highly focused antenna system is used for the probing and receiving antenna. The specific feature of the system, receiver design are described and preliminary data obtained in for ECRH plasma in LHD are discussed.

Empirical electron cross-field mobility in a Hall effect thruster

Electron transport across the magnetic field in Hall effect thrusters is still an open question. Models have so far assumed 1∕B2 or 1∕B scaling laws for the “anomalous” electron mobility, adjusted to reproduce the integrated performance parameters of the thruster. We show that models based on such mobility laws predict very different ion velocity distribution functions (IVDF) than measured by laser induced fluorescence (LIF). A fixed spatial mobility profile, obtained by analysis of improved LIF measurements, leads to much better model predictions of thruster performance and IVDF than 1∕B2 or 1∕B mobility laws for discharge voltages in the 500–700V range.

Extension of the Bissell–Johnson plasma-sheath model for application to fusion-relevant and general plasmas

This article presents an approach to solving a special Fredholm-type integral equation of the first kind with a particular kernel containing a modified Bessel function for applications in plasma physics. From the physical point of view, the problem was defined by Bissell and Johnson (B&J) [Phys. Fluids 30, 779 (1987)] as a task to find the potential profile and the ion velocity distribution function in a plane-parallel discharge with a Maxwellian ion source. The B&J model is a generalization of the well-known Tonks–Langmuir (T&L) [Phys. Rev. 34, 876 (1929)] discharge model characterized by a “cold” ion source. Unlike the T&L model, which can be readily solved analytically, attempts to solve the B&J model with a “warm” ion source have been done only numerically. However, the validity of numerical solutions up to date remains constrained to a rather limited range of a crucial independent parameter of the B&J integral equation, which mathematically is the width of a Gaussian distribution and physically represents the ion temperature. It was solved only for moderately warm ion sources. This paper presents the exact numerical solution of the B&J model, which is valid without any restriction regarding the above-mentioned parameter. It is shown that the ion temperature is very different from the temperature of the ion source. The new results with high-temperature ion sources are not only of particular importance for understanding and describing the plasma-sheath boundary in fusion plasmas, but are of considerable interest for discharge problems in general. The eigenvalue of the problem, found analytically by Harrison and Thompson [Proc. Phys. Soc. 74, 145 (1959)] for the particular case of a cold ion source, is here extended to arbitrary ion-source temperatures.