Breaking of nonlinear cylindrical plasma oscillations

Nonlinear axisymmetric cylindrical plasma oscillations are investigated analytically and numerically. It is shown that the breaking of strongly nonlinear oscillations is attributed to a singularity in the electron density and occurs several periods after the onset of an off-axis density maximum. For weakly nonlinear conditions, an analytic dependence of the breaking time of the oscillations on their amplitude is obtained based on the effect of intersection of electron trajectories and is shown to agree with numerical results.     

Nonlinear relativistic plasma resonance: Renormalization group approach

An analytical solution to the nonlinear set of equations describing the electron dynamics and electric field structure in the vicinity of the critical density in a nonuniform plasma is constructed using the renormalization group approach with allowance for relativistic effects of electron motion. It is demonstrated that the obtained solution describes two regimes of plasma oscillations in the vicinity of the plasma resonance— stationary and nonstationary. For the stationary regime, the spatiotemporal and spectral characteristics of the resonantly enhanced electric field are investigated in detail and the effect of the relativistic nonlinearity on the spatial localization of the energy of the plasma relativistic field is considered. The applicability limits of the obtained solution, which are determined by the conditions of plasma wave breaking in the vicinity of the resonance, are established and analyzed in detail for typical laser and plasma parameters. The applicability limits of the earlier developed nonrelativistic theories are refined.     

Incoherent Scattering of Electromagnetic Waves by Langmuir Fluctuations Trapped in a Plasma Density Well

Incoherent scattering of a probing wave by Langmuir fluctuations trapped and enhanced near a local minimum of the electron density (plasma density well) in plasma with a parabolic density profile is considered. Steady-state amplitudes of fluctuations are calculated for arbitrary velocity distribution functions of plasma particles with allowance for electron collisions. It is shown that quasi-periodic oscillations with two characteristic scales can be present in the spectrum of the plasma line. The smaller scale is due to the wellknown effect of discretization of the spectrum of Langmuir fluctuations in a plasma density well. The larger scale is associated with the generation of scattered waves in two spatial regions and subsequent interference of these waves at the exit from the density well. Oscillations with this scale are more stable under unsteady plasma conditions and can be more often observed in experiments. The results of this work can be used to experimentally determine the plasma parameters, such as the electron collision frequency and the size and lifetime of the plasma density well.     

Effect of an external RF field on the beam-plasma discharge in a magnetic field

The effect of an RF field on a steady-state beam-plasma discharge with a plane electrode placed parallel to a sheetlike electron beam is studied experimentally. The plasma parameters were measured by a single probe, and the electron distribution function was determined with the use of an electrostatic analyzer. The energy and current of the electron beam were E _B=2.5 keV and J _B=0.05–1.5 A, respectively. The working pressure was p=2×10^?5–10^?3 torr. The frequency of the external RF field was 13.56 MHz. Both the steady-state regimes in which the RF field had no effect on the plasma parameters and regimes with a pronounced effect of the RF field were observed. The experiments show that the regime of the discharge depends strongly on the plasma density and the magnetic field. The parametric instability is studied theoretically in the weak-turbulence approximation. It is shown that, due to the decay nature of the spectrum of plasma oscillations, the onset of instability is accompanied by the transfer of the energy of fluctuations over the spectrum, from the pump frequency toward its harmonics.     

Nonlinear currents generated in plasma by a radiation pulse with a frequency exceeding the electron plasma frequency

It is shown that the nonlinear currents generated in plasma by a radiation pulse with a frequency exceeding the electron plasma frequency change substantially due to a reduction in the effective electron–ion collision frequency.     

On the nonlinear theory of collective Cherenkov interaction of a high-density relativistic beam with a plasma

The nonlinear dynamics of the instability of a straight high-density relativistic electron beam under the conditions of the stimulated Cherenkov effect in a plasma waveguide is studied both analytically and numerically. It is shown that, for a beam of sufficiently high density such that the stabilizing factors are nonlinear frequency shifts and for a plasma described in a linear approximation, the basic equations have soliton-like solutions and the electron beam after saturation of the instability relaxes to its initial, weakly perturbed state, provided that only one harmonic of the plasma and the beam density is taken into account. The analytical solutions obtained here for this case correlate well with the numerical ones. A more general model that accounts for the generation of higher harmonics of the plasma and the beam density does not yield soliton-like solutions for the time evolution of the amplitudes of the plasma and beam waves. In such a model, the instability will be collective again: it can be described analytically (at least, up to the time at which it saturates) by using equations with cubic nonlinearities and the method of expansion of the electron trajectories and momenta.     

Dissipation of Energy of an AC Electric Field in a Semi-Infinite Electron Plasma with Mirror Boundary Conditions

Absorption of the electromagnetic energy in a semi-infinite electron plasma is calculated for an arbitrary degree of the electron gas degeneracy. Absorption is determined by solving the boundary-value problem on the oscillations of electron plasma in a half-space with mirror boundary conditions for electrons. The Vlasov?Boltzmann kinetic equation with the Bhatnagar–Gross–Krook collision integral for the electron distribution function and Maxwell’s equation for the electric field are employed. The electron distribution function and the electric field inside plasma are searched for in the form of expansions in the eigenfunctions of the initial set of equations. The expansion coefficients are found for the case of mirror boundary conditions. The contribution of the plasma surface to absorption is analyzed. Cases with different degrees of electron gas degeneracy are considered. It is shown that absorption of the electromagnetic energy near the surface depends substantially on the ratio between the electric field frequency and the volumetric electron collision frequency.     

Nonlinear oscillations of a semiconductor plasma with a nonrelativistic electron beam

Nonlinear oscillations of a semiconductor plasma with a low-density electron beam in the absence of an external magnetic field are studied in the hydrodynamic approximation. The beam is assumed to be nonrelativistic and monoenergetic. Cases are studied in which the Langmuir frequency of the electron oscillations in a semiconductor is much higher or much lower than the electron momentum relaxation rate. The self-similar solution obtained for the first case describes the damping of the nonlinear oscillations of the wave potential. Numerical analysis of the second case shows that the electric field distribution in the beam may correspond to that in a shock wave.     

Experimental study of magnetized plasma rotation in crossed fields

Rotation of magnetized plasma between two coaxial electrodes in crossed electric and magnetic fields was studied experimentally. Three regimes of plasma rotation were observed. In the first regime, the radial electric field is created by a beam?plasma discharge due to the charging of the inner axial electrode by electrons, the outer electrode being grounded. Plasma rotation in this case is accompanied by strong high-frequency current oscillations detected by a Mach probe. When a negative voltage was applied to the coaxial electrodes, the second regime was observed, in which weakly perturbed quasi-stationary plasma rotation occurred at a relatively low radial current. The third regime of plasma rotation was observed upon a spontaneous disruption of the second regime. It is characterized by high currents of ~1 kA, sheared plasma rotation, and excitation of high-frequency perturbations.     

Nonlinear properties of two-dimensional wake waves excited by relativistic electron bunches in plasma

The properties of a nonlinear plasma wake wave excited by an axially symmetric relativistic electron bunch are studied. It is shown that the nonlinear dependence of the wake wavelength on the transverse coordinate leads to distortion of the phase front of the wake wave and to steepening and oscillations of the transverse profile of the wakefield. The magnetic field of the wake wave is nonzero and oscillates at a frequency higher than the plasma electron frequency. Because of nonlinearity, the amplitude of the excited wake wave changes with distance from the bunch. The increase in nonlinearity leads to the development of turbulence and chaotization of the wakefield and results in the switching-on of the thermal effects and plasma heating.     

Investigation of low-frequency waves in plasma-filled microwave oscillators

The excitation of microwave oscillations by an electron beam in a hybrid plasma waveguide—a slow-wave structure (a sequence of inductively coupled resonators) with a plasma-filled transport channel—is studied both experimentally and theoretically. It is shown that the governing role in the generation of microwaves and their transmission to a feeder line is played by the spatial and temporal plasma-density variations associated with low-frequency ion plasma oscillations. The microwave pressure gives rise to low-frequency plasma oscillations with a rise time shorter than their period. This nonlinear mechanism for the excitation of low-frequency oscillations has a threshold in terms of the microwave power. The unsteady character of the spatial distribution of the plasma density results in intermittent microwave generation and shortens the duration of microwave pulses.     

Breaking of a wake wave excited by a narrow laser pulse in a low-density plasma

The spatial structure of a wake wave excited in a low-density plasma by a laser pulse with a small focal spot radius is studied both analytically and numerically. Numerical study shows that, in a small-amplitude laser field, a wake wave breaks after the formation of an off-axis density maximum, which grows in height away from the pulse to become infinitely high after several periods. Analytical and numerical calculations show that the singularity in the density arises from the intersection of the trajectories of neighboring particles. Numerical simulations demonstrate that, as the laser field amplitude increases, the breaking point of the wake wave rapidly approaches the pulse trailing edge. For weakly nonlinear conditions, an analytic dependence of the coordinate of the breaking point on the amplitude and transverse size of the laser pulse is obtained.     

Excitation of RF Oscillations in a Discharge with Negative Differential Conductivity

The excitation of oscillations in a discharge with negative differential conductivity is studied experimentally. The possibility is demonstrated of amplifying oscillations in the cathode dark space at frequencies close to the electron plasma frequency of the positive-column plasma. The phase velocities of waves at these frequencies are determined. When the waves pass from the cathode dark space to the discharge positive column, their phase velocities decrease; the closer the frequency is to the electron plasma frequency, the more pronounced the decrease in the phase velocity. As the intensity of oscillations increases, the discharge becomes non-steady-state. This is confirmed by the time evolution of the current-voltage characteristic. The shape of the current-voltage characteristic, its splitting, and the rate at which it varies depend on the input RF power. The decrease in the cathode dark space indicates that the ionization processes in the discharge are strongly influenced by electron plasma oscillations excited due to the collective interaction of the electron beam formed at the cathode with the discharge plasma. It is these processes that determine the maximum values of both the frequency of the excited oscillations and the power that can be withdrawn from the discharge.     

Microwave measurements of the time evolution of electron density in the T-11M tokamak

Unambiguous diagnostics intended for measuring the time behavior of the electron density and monitoring the line-averaged plasma density in the T-11M tokamak are described. The time behavior of the plasma density in the T-11M tokamak is measured by a multichannel phase-jump-free microwave polarization interferometer based on the Cotton-Mouton effect. After increasing the number of simultaneously operating interferometer channels and enhancing the sensitivity of measurements, it became possible to measure the time evolution of the plasma density profile in the T-11M tokamak. The first results from such measurements in various operating regimes of the T-11M tokamak are presented. The measurement and data processing techniques are described, the measurement errors are analyzed, and the results obtained are discussed. We propose using a pulsed time-of-flight refractometer to monitor the average plasma density in the T-11M tokamak. The refractometer emits nanosecond microwave probing pulses with a carrier frequency that is higher than the plasma frequency and, thus, operates in the transmission mode. A version of the instrument has been developed with a carrier frequency of 140 GHz, which allows one to measure the average density in regimes with a nominal T-11M plasma density of (3–5)×10¹³ cm^?3. Results are presented from the first measurements of the average density in the T-11M tokamak with the help of a pulsed time-of-flight refractometer by probing the plasma in the equatorial plane in a regime with the reflection of the probing radiation from the inner wall of the vacuum chamber.     

Stability and variations of plasma parameters in the L-2M stellarator during excitation of the induction current in the regime of ECR plasma heating

Results are presented from experimental studies of variations in the plasma parameters during the excitation of a multiaxis magnetic configuration by the induction current (up to 17 kA) in the basic magnetic configuration of the L-2M stellarator in the regime of ECR heating at a microwave power of ～200 kW (～1 MW m^?3) and an average plasma density of (1–2) × 10¹⁹ m^?3. The current direction was chosen to reduce the net rotational transform (the so-called “negative“ current). The current was high enough for the rotational transform to change its sign inside the plasma column. Computer simulations of the L-2M magnetic structure showed that the surface with a zero rotational transform is topologically unstable and gives rise to magnetic islands, i.e., to a multiaxis magnetic configuration. Magnetic measurements showed that, at negative currents above 10 kA, intense bursts of MHD oscillations with a clearly defined toroidal mode number n = 0 were observed in the frequency range of several kilohertz. Unfortunately, the experimental data are insufficient to draw the final conclusion on the transverse structure of these oscillations. The radial temperature profiles along the stellarator major radius in the equatorial plane were studied. It is found that the electron temperature decreases by a factor of 1.3 in the plasma core (r/a ≤ 0.6) and that the temperature jump is retained near the boundary. A change in turbulent fluctuations of the plasma density during the excitation of a negative current was studied using wave scattering diagnostics. It is found that the probability density function of the increments of fluctuations in the plasma core differs from a Gaussian distribution. The measured distribution is heavy-tailed and broadens in the presence of the current. It is found that the spectrum of turbulent fluctuations and their Doppler shift near the plasma boundary are nonuniform in the radial direction. This may be attributed to the shear of the poloidal velocity. The experimental results indicate that the formation of regions with a zero rotational transform in the plasma core somewhat intensifies plasma transport.     

A Nonadiabatic Theory for Ultrafast Catalytic Electron Transfer: A Model for the Photosynthetic Reaction Center

A non-adiabatic theory of Electron Transfer (ET), which improves the standard theory near the inversion point and becomes equivalent to it far from the inversion point, is presented. The complex amplitudes of the electronic wavefunctions at different sites are used as Kramers variables for describing the quantum tunneling of the electron in the deformable potential generated by its environment (nonadiabaticity) which is modeled as a harmonic classical thermal bath. After exact elimination of the bath, the effective electron dynamics is described by a discrete nonlinear Schrödinger equation with norm preserving dissipative terms and a Langevin random force, with a frequency cut-off, due to the thermalized phonons. This theory reveals the existence of a specially interesting marginal case when the linear and nonlinear coefficients of a two electronic states system are appropriately tuned for forming a Coherent Electron-Phonon Oscillator (CEPO). An electron injected on one of the electronic states of a CEPO generates large amplitude charge oscillations (even at zero temperature) associated with coherent phonon oscillations and electronic level oscillations. This fluctuating electronic level may resonate with a third site which captures the electron so that Ultrafast Electron Transfer (UFET) becomes possible. Numerical results are shown where two weakly interacting sites, a donor and a catalyst, form a CEPO that triggers an UFET to an acceptor. Without a catalytic site, a very large energy barrier prevents any direct ET. This UFET is shown to have many qualitative features similar to those observed in the primary charge separation in photosynthetic reaction centers. We suggest that more generally, CEPO could be a paradigm for understanding many selective chemical reactions involving electron transfer in biosystems.     

Microwave amplification in a coaxial slow-wave plasma transmission line

Microwave generation by an electron beam in a coaxial transmission line in which the inner and outer conductors are both corrugated is studied theoretically. An annular electron beam propagates in a transport channel filled entirely with plasma. The eigenmodes of the plasma-filled coaxial line are studied, as well as how they are affected by the plasma density. It is shown that, in the presence of a plasma, the microwaves are amplified to a significantly greater extent and the spectrum of the generated microwaves is broader. The nonlinear amplification regime is analyzed. The maximum possible amplitude of the longitudinal electric field and the interaction efficiency are determined as functions of the plasma density. A comparison between the results obtained and the analogous parameters of a vacuum structure shows that plasma-filled hybrid structures are more promising than vacuum sources.     

Saturation of two-stream instability of an electron beam in plasma

The development and nonlinear saturation of two-stream instability of a warm nonrelativistic electron beam in a cold plasma are investigated numerically in the framework of a one-dimensional model. It is shown that, for a sufficiently large velocity spread of the electron beam, instability develops and saturates according to a universal law, the wave phase velocity remains the same in the saturation stage, and the maximum field is somewhat lower than that predicted by classical estimates and depends in a different way on the growth rate. The damping of plasma oscillations not only changes the instability growth rate, but also substantially decreases the maximum wave field.     

Nonlinear dynamics and chaotization of oscillations of a virtual cathode in an annular electron beam in a uniform external magnetic field

Results are presented from a numerical study of the effect of an external magnetic field on the conditions and mechanisms for the formation of a virtual cathode in a relativistic electron beam. Characteristic features of the nonlinear dynamics of an electron beam with a virtual cathode are considered when the external magnetic field is varied. Various mechanisms are investigated by which the virtual cathode oscillations become chaotic and their spectrum becomes a multifrequency spectrum, thereby complicating the dynamics of the vircator system. A general mechanism for chaotization of the oscillations of a virtual cathode in a vircator system is revealed: the electron structures that form in an electron beam interact by means of a common space charge field to give rise to additional internal feedback. That the oscillations of a virtual cathode change from the chaotic to the periodic regime is due to the suppression of the mechanism for forming secondary electron structures.     

Evolution of the electron Langmuir oscillations in an inhomogeneous magnetized plasma

The conditions under which the energy of the electron Langmuir oscillations can escape from the plasma into vacuum are determined in the simplest model of a plane slab of an inhomogeneous cold magnetized plasma in a uniform magnetic field.