Electron Bounce Frequency Research Articles

Changing pattern of N2 dissociation in N2–Ar RF plasma during E–H mode transition

The behaviour of nitrogen plasma mixed with varying proportions of argon (10%–80%) is investigated under different RF discharge conditions. It is observed that at a relatively low RF power of 200 W (E-mode) the dissociation fraction (DF) of nitrogen increases with the growing concentration of argon, whereas the opposite happens for a higher RF power of 1000 W (H-mode), when the DF rapidly falls from a high value as the argon percentage starts to increase. This rising trend of DF closely follows the argon metastable fraction (MF) in the E-mode, and for the H-mode it is not followed until the argon percentage crosses the 20% mark. The electron density, temperature and electron energy probability function (EEPF) are obtained using a RF compensated Langmuir probe and to evaluate the vibrational and rotational temperatures, DF, MF etc, a separate optical emission spectroscopy technique is incorporated. At 5 × 10−3 mbar of working pressure and 10% argon content the EEPF profile reveals that the plasma changes from non-Maxwellian to Maxwellian as the RF power jumps from 200 W to 1000 W, and for a fixed RF power the high energy tail tends to move upwards with the gradual increment of argon. These observations are reverified theoretically by considering electron–electron collision frequency and electron bounce frequency as a function of electron temperature. Overall, all the major experimental phenomena in this study are explained in terms of EEPF profile, electron–electron collision effect, electron and gas temperature, electron density and argon metastable population.

Suppression of stimulated Raman side-scattering of a laser in a plasma in presence of co-propagating electron beam

The growth of stimulated Raman side-scattering of a laser beam propagating in the plasma may be influenced by the presence of co-propagating electron beam. The laser beam excites a pair of plasma wave and side-scattered electromagnetic wave. The side-scattered wave and pump wave couple together and exert a ponderomotive force on beam and plasma electrons which intern enhances the amplitude of plasma wave. The density perturbation in the plasma due to plasma wave couple nonlinearly with pump wave and excite a nonlinear current responsible for the growth of side-scattered electromagnetic wave. The phase matching between plasma wave and electron beam controls the growth of Raman process. Raman process saturates when bounce frequency of beam electrons exceeds the growth rate.

Electrostatic odd symmetric eigenmode in inhomogeneous Bernstein-Greene-Kruskal equilibrium

A self-consistent electrostatic odd-symmetric eigenmode (OEM) is analytically found in a solitary type Bernstein-Greene-Kruskal (BGK) equilibrium. The frequency of the OEM is order of the electron bounce frequency and it is spatially odd-symmetric with the scale comparable to that of the solitary BGK equilibrium structure. Such an OEM is consistent with the recent observation from particle-in-cell simulation of the solitary wave [Dokgo et al., Phys. Plasmas 23, 092107 (2016)]. The mode can be driven unstable by trapped electrons within the hole structure of the solitary wave. Such a low frequency, pure electron mode, which may possibly interact resonantly with the ion acoustic mode, provides a possible damping mechanism of the BGK equilibrium.

Kinetic theory of the electron bounce instability in two dimensional current sheets—Full electromagnetic treatment

In the general context of understanding the possible destabilization of a current sheet with applications to magnetospheric substorms or solar flares, a kinetic model is proposed for studying the resonant interaction between electromagnetic fluctuations and trapped bouncing electrons in a 2D current sheet. Tur et al. [A. Tur et al., Phys. Plasmas 17, 102905 (2010)] and Fruit et al. [G. Fruit et al., Phys. Plasmas 20, 022113 (2013)] already used this model to investigate the possibilities of electrostatic instabilities. Here, the model is completed for full electromagnetic perturbations. Starting with a modified Harris sheet as equilibrium state, the linearized gyrokinetic Vlasov equation is solved for electromagnetic fluctuations with period of the order of the electron bounce period. The particle motion is restricted to its first Fourier component along the magnetic field and this allows the complete time integration of the non local perturbed distribution functions. The dispersion relation for electromagnetic modes is finally obtained through the quasineutrality condition and the Ampere's law for the current density. It is found that for mildly strechted current, undamped modes oscillate at typical electron bounce frequency with wavelength of the order of the plasma sheet half thickness. As the stretching of the plasma sheet becomes more intense, the frequency of these normal modes decreases and beyond a certain threshold in ε = Bz/Blobes, the mode becomes explosive with typical growth rate of a few tens of seconds. The free energy contained in the bouncing motion of the electrons may trigger an electromagnetic instability able to disrupt the cross-tail current in a few seconds. This new instability–electromagnetic electron-bounce instability–may explain fast and global scale destabilization of current sheets as required to describe substorm phenomena.

Collisionless microinstabilities in stellarators. I. Analytical theory of trapped-particle modes

This is the first of two papers about collisionless, electrostatic micro-instabilities in stellarators, with an emphasis on trapped-particle modes. It is found that, in so-called maximum-$J$ configurations, trapped-particle instabilities are absent in large regions of parameter space. Quasi-isodynamic stellarators have this property (approximately), and the theory predicts that trapped electrons are stabilizing to all eigenmodes with frequencies below the electron bounce frequency. The physical reason is that the bounce-averaged curvature is favorable for all orbits, and that trapped electrons precess in the direction opposite to that in which drift waves propagate, thus precluding wave-particle resonance. These considerations only depend on the electrostatic energy balance, and are independent of all geometric properties of the magnetic field other than the maximum-$J$ condition. However, if the aspect ratio is large and the instability phase velocity differs greatly from the electron and ion thermal speeds, it is possible to derive a variational form for the frequency showing that stability prevails in a yet larger part of parameter space than what follows from the energy argument. Collisionless trapped-electron modes should therefore be more stable in quasi-isodynamic stellarators than in tokamaks.

Lower hybrid destabilization of trapped electron modes in tokamak and its consequences for anomalous diffusion

Parametric coupling of lower hybrid pump wave with low frequency collisionless/weakly collisional trapped electron drift wave, with frequency lower than the electron bounce frequency is studied. The coupling produces two lower hybrid sidebands. The sidebands beat with the pump to exert a low frequency ponderomotive force on electrons that causes a frequency shift in the drift wave, leading to the growth of the latter. The short wavelength modes are destabilized and they enhance the anomalous diffusion coefficient.

Nonlinear evolution of the electron two‐stream instability: Two‐dimensional particle simulations

We studied nonlinear evolution of the electron two‐stream instability in a two‐dimensional system. Electron two‐stream and bump‐on‐tail instabilities are considered to be the most probable generation mechanisms for electrostatic solitary waves and electron holes observed in various regions of the Earth's magnetosphere. We performed two‐dimensional particle‐in‐cell simulations for various sets of electron cyclotron frequencies and initial electron thermal velocities. We found that the nonlinear evolution falls into four groups. When the electron cyclotron frequency is smaller than the bounce frequency of electrons trapped by electron holes, the electron holes become unstable as reported in previous simulations. When the electron cyclotron frequency is larger than the bounce frequency, the stability of electron holes is controlled by their amplitude. In the case of the cold two‐stream instability where the potential energy of excited electrostatic waves becomes larger than the thermal energy of background electrons, electron holes decay into electrostatic whistler waves. In another case, the two‐stream instability develops to form electron holes in runs with high initial electron thermal velocities. When the electron cyclotron frequency is much larger than the electron bounce frequency, we found the formation of stable one‐dimensional electron holes through coalescence. When the electron cyclotron frequency is smaller than twice the bounce frequency, we found formation of two‐dimensional electron holes isolated in both directions parallel and perpendicular to the ambient magnetic field.

Enhanced collisionless heating in a nonuniform plasma at the bounce resonance condition

The importance of accounting for a nonuniform density profile for modeling of collisionless electron heating in a bounded low-pressure plasma is demonstrated. A drastic enhancement of the power transfer into an inductive plasma under the condition of a bounce resonance is observed if the nonuniformity of the plasma density profile is accounted for. This enhanced plasma heating is attributed to the increase of the number of resonant electrons, for which the bounce frequency of electrons confined inside the plasma potential is equal to the rf field frequency.

Effects of a sheath boundary on electron energy distribution in Ar/He dc magnetron discharges

In this study, the effects of a sheath boundary on electron energy distribution and discharge characteristics in an unbalanced and planar-type dc magnetron sputtering system are investigated. The anode sheath potential is changed by applying dc bias voltages to the substrate. The electron energy distribution functions (EEDFs) are measured in argon and helium discharges using a single Langmuir probe in conjunction with the ac superposition method. The evolutions of the EEDFs are first observed in argon at 3 mTorr and then in helium at 30 mTorr. The results show that, as the substrate bias voltage decreases to high negative voltage, the EEDF transition from the bi-Maxwellian to the Maxwellian in the downstream region occurs at a specific bias voltage that depends on the operating gas. The major factors that affect the EEDF formation are investigated. In particular, the concept of total electron bounce frequency is introduced to represent the change of the sheath boundary condition. The observed EEDF transition is explained by comparing it with the plasma characteristic frequencies calculated from the measured EEDFs. As a result, the bi-Maxwellian distribution observed at the small substrate bias voltage is attributed to the low electron–electron collision frequency and the different loss mechanisms of two electron groups: the ambipolar diffusion loss of low-energy electron group confined by low plasma potential and the direct thermal loss of high-energy electron group, providing the electron current that compensates for the discharge current in a steady state.

Measurements of electron energy distribution functions and electron transport in the downstream region of an unbalanced dc magnetron discharge

In this study, electron energy distribution functions (EEDFs) are measured using a Langmuir probe in conjunction with the ac superposition method in the downstream region of a planar and unbalanced magnetron argon discharge and the effects of an anode sheath boundary on the discharge characteristics are investigated. The potential of the anode sheath can be controlled by applying a dc voltage to the substrate, and the nonlinear behaviour of the plasma potential with respect to the dc substrate voltage causes the distinctive evolution of the potential of the anode sheath. It is found that when the potential of the anode sheath reaches a specific value, which is related to the threshold energies of argon for the inelastic collisions, an outstanding EEDF transition from a bi-Maxwellian distribution to a single Maxwellian distribution occurs. We introduce the concept of the total electron bounce frequency as an indicator of how the electron collisions such as the electron–electron collision and the inelastic collisions affect the EEDF features as the potential of the anode sheath changes. This result provides the decisive clue to explaining the appearance of the bi-Maxwellian distribution in magnetron discharges. The results of the spatially resolved measurements of EEDF and plasma characteristics are also presented. From these results, we will discuss the electron transport in the downstream region in detail.

The stability of propagating slab electron holes in a magnetized plasma

The stability of a propagating, one-dimensional solitary, slab electron hole in a magnetized plasma is studied using the linearized Vlasov description of the small perturbations. The plasma regimes characteristic to the magnetosphere and to the Earth’s magnetotail are considered. A destabilization of the electron hole by the resonant interaction of the wave with the trapped and the marginally free particles is found, and the expressions for the corresponding growth rates are given. In the case of a cyclotron wave–particle resonance, the instability manifests predominantly in the emission of linear waves. For the low-frequency resonance, when the wave frequency is comparable to the electron bounce frequency, larger growth rates are obtained, and the possibility of the breakup of the hole into smaller structures is found.

Plasma transport during substorm growth phase and relation to breakup

For ω<ωbi,ωbe, the electron and ion bounce frequencies, the response of a plasma to an externally applied electromagnetic perturbation is nonlocal. This implies, via the quasi-neutrality equation, the development of an electrostatic potential which is constant for a given magnetic field line. In the near equatorial region the corresponding potential electric field is shown to oppose the effect of the induced electric field associated with the externally applied perturbation. Thus the effect of the induced electric field is partially shielded; the total azimuthal electric field (i.e. induced plus potential) tends to be small, which explains why the radial flow velocity is slow during quasi-steady conditions prevailing during the growth phase and after the active phase. The nonlocal response of the plasma also leads to the development of a parallel current that may generate current driven Alfven (CDA) waves, which mode convert into shear Alfven (SA) waves. CDA/SA waves are systematically observed at early breakup; they grow very fast and produce a parallel diffusion of electrons. As soon as the diffusion time is shorter than the bounce time (τd<τb), the nonlocal response vanishes. Thus the shielding disappears, and an enhanced transport is restored at the rate fixed by the induced electric field alone. We show that fast flows effectively occur when CDA waves have enough power to diffuse electrons (over τd<τb). Electron parallel diffusion also leads to an interruption of the parallel current and therefore to a disruption of the perpendicular current.

Nonlocal electron kinetics in a planar inductive helium discharge

A measurement of the electron energy distribution function (EEDF) using the ac superposition method is done over a helium pressure range of 10-100 mTorr in a planar inductive plasma, and the electron energy diffusion coefficient which describes the electron heating is calculated based on the same discharge conditions using a two-dimensional simulation. It is found that the measured EEDF shows a bi-Maxwellian distribution with a low-energy electron group at low pressures below 20 mTorr even in the inductive discharge using helium of the non-Ramsauer gas. The major factors which can affect the EEDF formation are investigated. In particular, the concept of the total electron bounce frequency, i.e., the electron residence time, is introduced as an indicator of how the electron-electron collision affects the EEDF shape. As a result, it is shown that the observed bi-Maxwellian distribution at low pressures is attributed to the combined effects of the formation of low-energy electrons through the cooling mechanism of energetic electrons enhanced by the capacitive field, the low heating rate of the low-energy electrons, the confinement of low-energy electrons by the ambipolar space potential, and the low electron-electron collision frequency which can be estimated from the total electron bounce frequency presented in this paper.

Electron kinetic effects in standing shear Alfvén waves in the dipolar magnetosphere

The electron kinetic response to an electric current driven by a standing shear Alfvén wave (SAW) is considered for the case of a dipolar geometry. The parallel electric field is found from the electron gyrokinetic equation along with the SAW dispersion coefficient. Electron trapping in the dipolar magnetic field significantly reduces the parallel electric conductivity and in this way increases the amplitude of the parallel electric field and SAW dispersion. It is demonstrated that the two-fluid hydrodynamic equations used by many authors significantly underestimate the electron response and, consequently, the magnitude and location of the parallel electric field under conditions where the electron bounce frequency is larger than the SAW frequency. This is especially important in a plasma with density depressions near the Earth’s polar magnetosphere.

Resonant Particle Heating of an Electron Plasma by Oscillating Sheaths

The first unambiguous measurements of a bounce-resonance effect in a non-neutral plasma are presented. Two striking signatures are observed when a magnetized electron column is heated by oscillating an end sheath. First, the heating rate increases by a factor of ${10}^{4}$ as the oscillation frequency is increased by a factor of 10 near the thermal electron bounce frequency. Second, the heating is enhanced when the sheaths at both ends are oscillated out of phase, but is suppressed when they are in phase. The measured rates are in quantitative agreement with a resonant particle heating theory.

Onset of collisionless magnetic reconnection in sheared field reversals

An extended analysis of the kinetic stability theory of a current sheet in a curved and sheared magnetic field reversal with respect to a tearing mode perturbation is presented. The coupling between quasi‐parallel and quasi‐perpendicular mode components is newly taken into account. We obtain that an unstable branch of tearing mode instability with a growth rate less than the electron bounce frequency can be excited. The dependencies of the growth rate of this instability on the shear (Bs) and normal (Bn) field components as far as on the wave length of the mode are illustrated. The resulting tearing mode destabilization takes place favorably for an inclination of the magnetic field near the quasi‐neutral plane at about 45° with respect to the neutral plane. Our theory reveals that for proper geometry of magnetic field lines in field reversal region a collisionless ion tearing mode can be triggered and magnetic flux ropes are formed. The growth of the unstable tearing mode can be accompanied by the generation of magnetic field aligned electric fields and ion acceleration.

Stimulated Cherenkov emission in dielectric-lined waveguides.

Particle-in-cell simulations of a dielectric Cherenkov maser experiment are used to analyze the emissive characteristics of an annular relativistic beam in a dielectric-lined waveguide. Good agreement with recent experimental results is obtained. The large gain in the device follows linear theory until saturation, when particle bunching and trapping is observed. The beam density becomes highly modulated with pulse length \ensuremath{\sim}\ensuremath{\lambda}/2, where \ensuremath{\lambda} is the guide wavelength. The performance of the device in the high-current regime is highly nonlinear. A simple nonlinear theory, confirmed by numerical results, shows that the field saturates at a value corresponding to ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}}$=\ensuremath{\Delta}\ensuremath{\omega}/2, where ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}}$ is the electron bounce frequency in the wave and \ensuremath{\Delta}\ensuremath{\omega} is the detuning parameter.

Low-pressure RF discharge in the free-flight regime

The self-consistent equations system for low-pressure RF discharge in the free-flight regime is formulated. The expressions for the electron energy diffusion coefficient due to electron-neutral collisions and to the electron collisions with the plasma-space charge moving boundary (stochastic heating) are derived. If the electron-neutral elastic collisions frequency exceeds the inelastic one, the conventional two-term approximation for the electron distribution function (EDF) can be generalized, and the space-time-averaged electron kinetic equation can be reduced to the one-dimensional energy diffusion one. The fast electrons attached to the electrode surface can also be accounted for in this equation. It is shown that in the cases of (a) spatially uniform ion profile, (b) for frequencies that are small compared with the electron bounce frequency, and (c) for frequencies exceeding the electron plasma one in the sheath, the stochastic heating vanishes. >

Radial Transport Induced by Rotating RF Fields andBreakdown of Intrinsic Ambipolarity in a Magnetic Mirror

Radial transport induced by rotating RF fields in a magnetic mirror is studied. The RF frequency used is in the ion cyclotron range of frequency and is comparable to the electron bounce frequency. The radial particle fluxes for ions and electrons are obtained by use of the gyrokinetic equation. With a positive azimuthal mode number of the RF field, the radial particle flux is outward for Maxwellian ions and inward for Maxwellian electrons. This shows that the intrinsic ambipolarity is broken in this RF-induced transport. The present analyses can be applied to the thermal-barrier pumping of a tandem mirror.

Electron bounce resonance heating by r.f. waves in a bumpy cylinder

In bumpy cylinder geometry, electrons are classified into trapped and passing particles. The interaction between a wave near the electron bounce frequency and the electrons is studied both numerically and analytically. It is shown that coupling of the waves to the electron bounce motion parallel to the magnetic field can lead to heating of those electrons near the passing/trapped boundary in velocity space. The stochastic threshold condition is eE0k0/m omega b2 approximately=0.1, which is an order lower than the quadratic magnetic well results. For this mechanism, the wave energy density required to induce stochastic heating in the ELMO Bumpy Torus (EBT) by radio frequency (in the frequency range of ion cyclotron resonance heating) is about an order of magnitude more than that estimated on the basis of cold plasma wave theory. However, it is suggested that the bounce resonance can enhance the electron cyclotron resonance heating efficiency in an EBT-like heating scheme.