Ion-neutral Collisions Research Articles

Numerical Experiment on the Influence of Granulation-induced Waves on Solar Chromosphere Heating and Plasma Outflows in a Magnetic Arcade

This paper offers a fresh perspective on solar chromosphere heating and plasma outflows, focusing on the contribution of waves generated by solar granulation. Utilizing a 2.5D numerical experiment for the partially ionized lower solar atmosphere, we investigate the dissipation of these waves and their impact on plasma outflows and chromospheric heating via ion-neutral collisions. Employing the JOint ANalytical and Numerical Approach code, we adopt two-fluid model equations, examining partially ionized hydrogen plasma dynamics, including protons+electrons and neutrals, treated as two separate fluids that are coupled through ion-neutral collisions. Our investigation focuses on a quiet solar chromosphere region characterized by gravitational stratification and magnetic confinement by an initially set single magnetic arcade. The primary source of the waves is the solar convection beneath the photosphere. Our results demonstrate that ion-neutral collisions result in the dissipation of such waves, releasing thermal energy that heats the chromosphere plasma. Notably, this is accompanied by upward-directed plasma flows. Finally, we conclude that wave dissipation due to ion-neutral collisions in the two-fluid plasma model induces chromosphere heating and plasma outflows.

Dust–ion–acoustic damped solitary waves and shocks in laboratory and Saturn's E-ring magnetized nonthermal dusty plasmas with anisotropic ion pressure and dust-charge fluctuation

We study the oblique propagation of weakly nonlinear dust–ion–acoustic (DIA) solitary waves (SWs) and shocks in collisional magnetized nonthermal dusty plasmas that are relevant in laboratory and space (Saturn's E-ring) environments. We consider plasmas to be composed of q-nonextensive hot electrons, thermal positive ions, and immobile negatively charged dust grains immersed in a static magnetic field and take into account the effects of ion creation (source), ion loss (sink), ion–neutral and ion–dust collisions, anisotropic ion pressure, and dust-charge fluctuations on the evolution of small-amplitude SWs and shocks. The ion–neutral collision enhancement equilibrium dust-charge number is self-consistently determined using Newton–Raphson method. We found that in laboratory dusty plasmas with adiabatic dust-charge variation [i.e., when the dust charging frequency (νch) is much higher than the dust–plasma oscillation frequency (ωpd)], the DIA solitary waves (DIASWs) get damped by the effects of the ion–dust and ion–neutral collisions, whereas the ion creation and ion loss lead to the amplification of solitary waves, and they appear as only compressive types with positive potential. On the other hand, in Saturn's E-ring plasmas, where the collisional and ion creation or ion loss effects are insignificant, the non-adiabaticity of dust-charge variation can give rise to the evolution of either damped DIASWs or DIA shocks, depending on the smallness of the ratios νch/ωpd or ωpd/νch, respectively. Furthermore, two critical values of the nonextensive parameter q exist, below (or above) which, the DIASWs and shocks can appear as rarefactive (or compressive) types. The characteristics of DIASWs and shocks are also analyzed numerically for parameters relevant to the laboratory and Saturn's E-ring plasmas.

Influence of electrons on granulation-generated solar chromosphere heating and plasma outflows

Context. It is known that the solar atmosphere exhibits a varying degree of ionization through its different layers. The ionization degree directly depends on plasma temperature, that is, the lower the temperature, the lower the ionization degree. As a result, the plasma in the lower atmospheric layers (the photosphere and the chromosphere) is only partially ionized, which motivates the use of a three-fluid model. Aims. We consider, for the first time, the influence of electrons on granulation-generated solar chromosphere heating and plasma outflows. We attempt to detect variations in the ion temperature and plasma up- and downflows. Methods. We performed 2.5D numerical simulations of the generation and evolution of granulation-generated waves, flows, and other granulation-associated phenomena with an adapted JOANNA code. This code solves the simplified three-fluid equations for ions (protons) and electrons and neutrals (hydrogen atoms) that are coupled by collision forces. Results. Electron-neutral and electron–ion collisions provide extra heat in the low chromosphere and enhance plasma outflows in this region. The effect of electrons is small compared to ion–neutral collisions, which have a significantly greater effect on the heating process and the production of outflows. Ion–neutral collisions involve higher energy exchanges, making them the dominant mechanism over collisions with electrons. Conclusions. Electrons do not play a major role in heating and producing outflows, primarily because their mass is much lower compared to that of neutrals and ions, resulting in lower energy transfer during collisions.

On equatorial spread F occurrence: A multi-dimensional quantitative assessment

The present study investigates the role of gravity wave induced seed perturbations in the occurrence of Equatorial Spread F (ESF) under the influence of the post sunset background conditions modulated by prevailing electrodynamics and neutral wind. Ionospheric foF2 data sets over geomagnetic equatorial station Trivandrum (8.5°N, 77°E and magnetic dip 0.68°N-corresponding to the period of study) corresponding to vernal and autumnal equinoctial periods encompassing high, low and moderate solar activity years, are used for the study. Meridional wind data is obtained either from ESA’s sun-synchronous satellite GOCE (Gravity field and steady-state Ocean Circulation Explorer) or derived using ionosonde h’F (base height of ionosphere at 2.5 MHz) data from Trivandrum (TVM- 8.5°N, 77°E and magnetic dip 0.68°N) and Sriharikota (SHAR−13.7°N, 80.2°E and magnetic dip 6.9°N-for period of study). This particular study is carried out for geomagnetically quiet days of Vernal Equinox (VE) and Autumnal Equinox (AE) seasons, which are most favoured for ESF occurrence over Indian longitudes. Considering thermospheric wind, ion-neutral collisions, and electric field effects in association with gravity wave seed, threshold curve is generated, which clearly demarcates ESF and NSF (Non spread F) days. Previous studies have addressed ESF variability in electrodynamical domain alone (wherein the layer is above a threshold level). The present study, for the first time, succeeds in demarcating ESF and NSF days by incorporating effects of electric field, neutral wind, collisional RT instability term, and gravity wave seed perturbations simultaneously irrespective of threshold height.

Numerical investigations of spatiotemporal dynamics of space-charge limited collisional sheaths

Electrostatic particle-in-cell (PIC) and direct simulation Monte Carlo (DSMC) methods are used to compare the plasma dynamics of collisionless with collisional emissive sheaths in partially ionized environments. Space-charge limited emissive sheaths submersed in a plasma with a density of ∼1017 m−3 are examined using a PIC-DSMC solver, CHAOS. Collisionless emissive sheaths with plasma domains sufficiently long (30 and 60 Debye lengths, λD) are subject to strong oscillations due to two-stream electron instability, whereas emissive sheaths in weakly collisional conditions with a short domain (15 λD) exhibit self-spike (sawtooth) oscillations in the plasma field due to the trapped charge-exchange (CEX) ion population within the virtual cathode (VC) region. The two-stream electron instability leads to strong temporal fluctuations in the total emission current, with maximum deviations of 60% and 100% from the time-averaged current for the long plasma domains, whereas CEX collisions cause strong spikes in the emission current if the domain size is short. Our PIC-DSMC simulations show for the first time that the interaction of the two types of instabilities causes the strength of the self-spike to be weakened due to the strong fluctuations caused by the two-stream instability when a sufficiently long computational domain with ion-neutral collisions is employed. By conducting a two-dimensional Fast Fourier Transform (FFT) on the collisional and collisionless sheaths with long domains, we show that the transient evolution of CEX entrapment in the VC increases frequency of sheath oscillations up to two times the ion-acoustic frequencies observed in the collisionless sheath. CEX collisions weaken the VC region and result in a total emission current more than that obtained from the collisionless case for the same domain length. With a more rarefied neutral environment of 1019 m−3 in the plasma sheath, the total emission current increases only 4% in comparison with 14% for one order of magnitude denser environment, within 20 μs. In addition, the spike period is tested with different neutral temperatures and densities. While we do not observe any self-spike in the more rarefied environment, the spike period increased from 5 to 7.5 μs when the neutral temperature is increased from 300 to 2000 K in the denser environment with the simulation time of 20 μs.

Landau quantization effects on damping Kawahara solitons in electron–positron–ion plasma in rotating ionized medium

For the dynamics of three-dimensional electron–positron–ion plasmas, a fluid quantum hydrodynamic model is proposed by considering Landau quantization effects in dense plasma. Ion–neutral collisions in the presence of the Coriolis force are also considered. The application of the reductive perturbation technique produces a wave evolution equation represented by a damped Korteweg–de Vries equation. This equation, however, is insufficient for describing waves in our system at very low dispersion coefficients. As a result, we considered the highest-order perturbation, which resulted in the damped Kawahara equation. The effects of the magnetic field, Landau quantization, the ratio of positron density to electron density, the ratio of positron density to ion density, and the direction cosine on linear dispersion laws as well as soliton and conoidal solutions of the damped Kawahara equation are explored. The understanding from this research can contribute to the broader field of astrophysics and aid in the interpretation of observational data from white dwarfs.

Extension of ion-neutral reactive collision model DNT+ to polar molecules based on average dipole orientation theory

The ion-neutral reactive collision model DNT+, which generates comprehensive ion-neutral collision cross section (CS) data sets for atoms and nonpolar molecules, has been extended to polar molecules. The extension is based on the average dipole orientation (ADO) theory, which adds the dipole moment to Langevin–Hassé CS. Furthermore, the ADO CS for short-range reactive collisions is covered with a rigid core to incorporate long-range elastic and charge-exchange collisions. The modified version of DNT+, i.e., DNT+DM, is applied to gas-phase H2O+–H2O and low-energy CF3+–CO collisions for its validation. The cross sections (CSs) for those collisions using DNT+DM show good agreement with literature data, proving that DNT+DM is valid to some extent. Help with ion swarm analyses and measurements is needed to make the predicted CSs more accurate.

Absolute rate coefficient measurements of the reactions of vibrationally cold HD+ and H3+ ions with neutral C atoms

Ion-neutral reactions are driving the formation of small molecules in the gas phase of interstellar clouds, where hydrogen molecules and their ions are by far the most important collision partners for any species in the astrochemical network. Here we present absolute rate coefficient measurements for the reactions HD++C→CH+/CD++D/H and H3++C→CH+/CH2++H2/H obtained using a recently commissioned ion-neutral collision setup at the Cryogenic Storage Ring. Our measurements with vibrationally cold ions result in significantly higher rate coefficients when compared with previous studies using internally excited ions, bringing them in better agreement with classical capture theories. Moreover, we have performed detailed quasiclassical trajectory (QCT) calculations for the HD++C reaction, using new potential energy surfaces. Our experimental results and the QCT calculations show very good agreement for the absolute cross section of the reactions, as well as for the isotope effect. These results have great potential relevance for the chemistry of the interstellar medium and the onset of organic chemistry in space. Published by the American Physical Society 2024

IDSimF: An Open-Source Framework for the Simulation of Molecular Ion Dynamics in Mass Spectrometry and Ion Mobility Spectrometry.

The development of mass spectrometric and ion mobility devices heavily depends on a comprehensive understanding of the behavior of ions within such systems. Therefore, numerical modeling of ion paths helps to optimize and verify existing devices, and contributes to the development of innovative ion optical systems and multipole geometries. This Article introduces IDSimF (Ion Dynamics Simulation Framework), an open-source simulation tool tailored to the nonrelativistic dynamics of molecular ions in mass and ion mobility spectrometry applications. Addressing limitations in existing software packages, as for example SIMION, OpenFOAM, and COMSOL, IDSimF offers a specialized platform for simulating ion trajectories in electric fields. IDSimF efficiently accounts for space charge effects and considers various ion-neutral collision models while handling chemical kinetics. The framework is highly modular with reduced user input configuration complexity and aims to support simulation efforts in development and optimization of in mass spectrometers.

Influence of a transverse magnetic field on wakefield oscillations around a charged dust grain in complex plasma

In the presence of ion streaming, the potential around dust particles immersed in plasma becomes anisotropic. In this scenario, the repulsive Debye–Hückel potential is superimposed with an attractive wake potential. This work presents an analytical study of the complex behavior of such a wake potential in the presence of a magnetic field (oriented transversely to the ion flow) and ion-neutral collisions using linear response formalism, both in subsonic and supersonic regimes. The amplitude and periodicity of this potential are found to be controlled by the interplay among ion streaming velocity, ion cyclotron frequency, and ion-neutral collision frequency. Due to the tunable nature of this potential, it is possible to control the crystal formation, phase transitions, and transport properties of dusty plasma by adjusting the external magnetic field. The study also reveals that the wake potential almost disappears in a collision-dominant regime.

Tearing-mediated Reconnection in Magnetohydrodynamic Poorly Ionized Plasmas. I. Onset and Linear Evolution

In high-Lundquist-number plasmas, reconnection proceeds via the onset of tearing, followed by a nonlinear phase during which plasmoids continuously form, merge, and are ejected from the current sheet (CS). This process is understood in fully ionized, magnetohydrodynamic plasmas. However, many plasma environments, such as star-forming molecular clouds and the solar chromosphere, are poorly ionized. We use theory and computation to study tearing-mediated reconnection in such poorly ionized systems. In this paper, we focus on the onset and linear evolution of this process. In poorly ionized plasmas, magnetic nulls on scales below v A,n0/ν ni0, with v A,n0 the neutral Alfvén speed and ν ni0 the neutral–ion collision frequency, will self-sharpen via ambipolar diffusion. This sharpening occurs at an increasing rate, inhibiting the onset of reconnection. Once the CS becomes thin enough, however, ions decouple from neutrals and the thinning of the CS slows, allowing the onset of tearing in a time of order νni0−1 . We find that the wavelength and growth rate of the mode that first disrupts the forming sheet can be predicted from a poorly ionized tearing dispersion relation; as the plasma recombination rate increases and ionization fraction decreases, the growth rate becomes an increasing multiple of ν ni0 and the wavelength becomes a decreasing fraction of v A,n0/ν ni0.

Existence of Electrostatic Ion Cyclotron Waves in a Laboratory Created E Region Ionospheric‐Like Plasma

AbstractMolecular ions are relatively cold in the E region ionosphere; however, they can upwell to the magnetosphere during geomagnetically active times. Resonance between electrostatic ion cyclotron (EIC) waves is a potential pathway to energize molecular ions. In this work, the E region ionospheric plasma was modeled in the laboratory, and EIC waves were excited by a nonuniform field‐aligned current. The EIC wave was excited even when the ion neutral collision frequency is much higher than the ion cyclotron frequency, and the fundamental frequency was observed to be below the ion cyclotron frequency. In addition, the wave dispersion of the collisional EIC wave was calculated, which shows a consistent trend with experimental results as the collisions increasing. Therefore, this work suggests that EIC waves can be excited in the E region ionospheric‐like plasma, which can support the explanation of the energization of molecular ions in the E region ionosphere.

Alfvén pulse driven spicule-like jets in the presence of thermal conduction and ion-neutral collision in two-fluid regime.

We present the formation of quasi-periodic cool spicule-like jets in the solar atmosphere using 2.5-D numerical simulation in two-fluid regime (ions+neutrals) under the presence of thermal conduction and ion-neutral collision. The nonlinear, impulsive Alfvénic perturbations at the top of the photosphere trigger field aligned magnetoacoustic perturbations due to ponderomotive force. The transport of energy from Alfvén pulse to such vertical velocity perturbations due to ponderomotive force is considered as an initial trigger mechanism. Thereafter, these velocity perturbations steepen into the shocks followed by quasi-periodic rise and fall of the cool jets transporting mass in the overlying corona. This article is part of the theme issue 'Partially ionized plasma of the solar atmosphere: recent advances and future pathways'.

Altering Conformational States of Dynamic Ion Populations using Traveling Wave Structures for Lossless Ion Manipulations.

With its capacity to store and translate ions across considerable distances and times, traveling wave structures for lossless ion manipulations (TW-SLIM) provide the foundation to expand the scope of ion mobility spectrometry (IMS) experiments. While promising, the dynamic electric fields and consequential ion-neutral collisions used to realize extensive degrees of separation have a considerable impact on the empirical results and the fundamental interpretation of observed arrival time distributions. Using a custom-designed set of TW-SLIM boards (∼9 m) coupled with a time-of-flight mass spectrometer (SLIM-ToF), we detail the capacity to systematically alter the gas-phase distribution of select peptide conformers. In addition to discussing the role charge-transfer may play in TW-SLIM experiments that occur at extended time scales, the ability of the SLIM-ToF to perform tandem IMS was leveraged to confirm that both the compact and elongated conformers of bradykinin2+ undergo interconversion within the SLIM. Storage experiments in which ions are confined within SLIM using static potential wells suggest that factors aside from TW-induced ion motion contribute to interconversion. Further investigation into this matter suggests that the use of radio frequency (RF) fields to confine ions within SLIM may play a role in ion heating. Aside from interconversion, storage experiments also provide insight into charge transfer behavior over the course of extended periods. The results of the presented experiments suggest that considerations should be taken when analyzing labile species and inform strategies for the TW-SLIM design and method development.

Magnetoacoustic waves in a partially ionized astrophysical plasma with the thermal misbalance: A two-fluid approach

We consider the propagation of magnetoacoustic (MA) and acoustic waves of various frequency ranges in a partially ionized plasma at an arbitrary angle to the magnetic field, taking into account the influence of heating, radiative, and thermo-conductive cooling, as well as ion-neutral collisions. A dispersion equation that describes the evolution of nine modes was obtained in a compact mathematical form using the two-fluid model. The number and type of propagating waves (modified fast and slow MA waves, MA waves in the ion component, acoustic waves in the neutral component, as well as isothermal MA and isothermal acoustic waves) vary in different frequency ranges depending on the parameters of the medium. Analytical expressions are found for the speed and damping rates of all these propagating waves, and it is shown how dispersion and damping are formed by three processes: thermal misbalance, ion-neutral collisions, and thermal conductivity. Comparison of analytical calculations of the velocity and damping rates of MA waves with the numerical solution of the dispersion relation under conditions characteristic of the low solar atmosphere and prominences showed high accuracy of the obtained analytical expressions. The strong influence of thermal misbalance caused by gasdynamic perturbations on the speed and damping rate of modified magnetoacoustic waves in a strongly coupled region is shown as well.

Modeling the structure of the dayside Venusian ionosphere: Impacts of protonation and Coulomb interaction

The CO$_2$-dominated thick atmosphere of Venus coexists with an ionosphere that is mainly formed, on the dayside, via the ionization of atmospheric neutrals by solar extreme ultraviolet and soft X-ray photons. Despite extensive modeling efforts that have reproduced the electron distribution reasonably well, we note two main shortcomings with respect to prior studies. The effects of protonation and Coulomb interaction are crucial to unveiling the structure and composition of the Venusian ionosphere. We evaluate the role of protonated species on the structure of the dayside Venusian ionosphere for the first time. We also evaluate the role of ion-ion Coulomb collisions, which are neglected in many existing models. Focusing on the solar minimum condition for which the effect of protonation is expected to be more prominent, we constructed a detailed one-dimensional photochemical model for the dayside Venusian ionosphere, incorporating more than 50 ion and neutral species (of which 17 are protonated species), along with the most thorough chemical network to date. We included both ion-neutral and ion-ion Coulomb collisions. Photoelectron impact processes were implemented with a two-stream kinetic model. Our model reproduces the observed electron distribution reasonably well. The model indicates that protonation tends to diverge the ionization flow into more channels via a series of proton transfer reactions along the direction of low to high proton affinities for parent neutrals. In addition, the distribution of O$_2^+$ is enhanced by protonation by a factor of nearly 2 at high altitudes, where it is efficiently produced via the reaction between O and OH$^+$. We find that Coulomb collisions influence the topside Venusian ionosphere not only directly by suppressing ion diffusion, but also indirectly by modifying ion chemistry. Two ion groups can be distinguished in terms of the effects of Coulomb collisions: one group preferentially produced at high altitudes and accumulated in the topside ionosphere, which is to be compared with another group that is preferentially produced at low altitudes and, instead, depleted in the topside ionosphere. Both protonation and Coulomb collisions have appreciable impacts on the topside Venusian ionosphere, which account for many of the significant differences in the model ion distribution between this study and early calculations.

Nonlinear dust-ion acoustic solitary waves for collisional electronegative dusty plasma in the presence of trapped electron distribution

We have investigated the characteristics of nonlinear propagation of dust-ion acoustic solitary waves in collisional electronegative unmagnetized dusty plasma, which consists of trapped electrons, Boltzmann negative ions, mobile positive ions, mobile negative dust particulates, and a uniform background of neutral particles. In account of ion-neutral collisions, the modified Korteweg–de Vries relation has been derived by employing the standard reductive perturbation method. Analytical and numerical solutions of the damped Korteweg–de Vries equation has been presented in which finite difference method is used for numerical solution. On the other hand, the dust charging equation has been solved by using Newton’s Raphson method. It is found that the temperature ratio of free to trapped electrons, ion-neutral collision, concentration of negative ions, dust number density, and dust density perturbation modify the basic properties of the dust-ion acoustic solitary waves. The temporal evolution of dust-ion acoustic solitary waves is crucial as it affects the amplitude and width of wave structure. In addition, the analytical and numerical solutions are compared, and their deviation is graphically illustrated.

Damped Kadomtsev Petviashvilli equation for magnetosonic waves in a dissipative OH plasma in the ionospheric F-Layer

In this paper we study the linear and nonlinear dynamics of magnetosonic waves in a dissipative Oxygen-Hydrogen (OH) plasma. It is shown that such waves can propagate nonlinearly as solitary structures for both super and sub, acoustic and Alfvenic regimes. We use the hyperbolic tangent method to solve the collisional Kadomtsev-Petviashvili (KP) equation and obtain a damped solitary wave solution. Both compressive and rarefactive damped solitary structures are obtained and are significantly affected by the oxygen ion-neutral collision frequency, temperature, propagation angle, and ambient magnetic field present in the F-layer of Earth’s ionosphere.

Impulsively generated waves in two-fluid plasma in the solar chromosphere: Heating and generation of plasma outflows

Context. We present new insights into impulsively generated Alfvén and magneto-acoustic waves in the partially ionized two-fluid plasma of the solar atmosphere and their contribution to chromospheric heating and plasma outflows. Aims. Our study attempts to elucidate the mechanisms responsible for chromospheric heating and excitation of plasma outflows that may contribute to the generation of the solar wind in the upper atmospheric layers. The main aim of this work is to investigate the impulsively generated waves by taking into account two-fluid effects. These effects may alter the wave propagation leading to attenuation and collisional plasma heating. Methods. The two-fluid equations were solved by the JOint ANalytical Numerical Approach (JOANNA) code in a 2.5-dimensional (2.5D) framework to simulate the dynamics of the solar atmosphere. Here, electrons + ions (protons) and neutrals (hydrogen atoms) are treated as separate fluids, which are coupled via ion-neutral collisions. The latter acts as a dissipation mechanism for the energy carried by the waves in two-fluid plasma and may ultimately lead to the frictional heating of the partially ionized plasma. The waves in two-fluid plasma, which are launched from the top of the photosphere, are excited by perturbations induced by localized Gaussian pulses in the horizontal components of the ion and neutral velocities. Results. In the middle and upper chromosphere, a substantial fraction of the energy carried by large amplitude waves in the two-fluid plasma is dissipated in ion-neutral collisions, resulting in the thermalization of wave energy and generation of plasma outflows. We find that coupled Alfvén and magneto-acoustic waves are more effective in heating the chromosphere than magneto-acoustic waves. Conclusions. Large-amplitude waves in the two-fluid plasma may be responsible for heating the chromosphere. The net flow of ions is directed outward, leading to plasma outflows in the lower solar corona, which may contribute to the solar wind at higher altitudes The primary source of wave energy dissipation in the current paradigm comes from collisions between ions and neutrals.

Experimental quantification of ion flux reduction at the sheath edge due to ion–neutral collisions in low temperature plasmas

Reduction of the ion flux at the sheath edge due to ion–neutral collisions in low temperature DC plasmas is experimentally quantified for low to intermediate neutral gas pressures (<102 mTorr). The reduction factor is defined as a ratio of the ion flux at the sheath edge in a collisional plasma to that in a collisionless limit in this work. Its quantification as a function of the collisionality with a Langmuir probe has been hindered since the measured data contain two intermingled effects, namely, the flux reduction and the sheath expansion, which are difficult to isolate one from the other. The sheath expansion effect with and without the flux reduction effect are analyzed, and by comparing the two, the reduction factor as a function of the collisionality has been estimated with Langmuir probe data from approximately 1000 systematic scans of the plasma conditions. Neutral gas pressures ranging from 0.2–30.0 mTorr for Ar and 1.0–65.0 mTorr for He discharges are generated in a multidipole chamber with hot filaments. The estimated reduction factors are found to agree with the results from the particle-in-cell simulations for He discharges [Beving et al., Plasma Sources Sci. Technol. 31, 084009 (2022)]. The estimated reduction factors for Ar discharges are larger than those for He discharges, and the dependence of the reduction factor on species is discussed. Reduction of the ion flux at the sheath edge at intermediate gas pressures highlights the importance of taking into account ion–neutral collisions in many plasma applications.