J. H. Malmberg

Nonlinear Interaction of a Small Cold Beam and a Plasma

Recently, a simple model was proposed for the nonlinear interaction of a low‐density monoenergetic electron beam and a relatively cold infinite homogeneous one‐dimensional plasma. The essential feature of this model is the observation that after several e‐foldings the bandwidth of the growing waves is so narrow that the electrons interact with a very nearly pure sinusoidal field. In terms of this single wave model, a properly scaled solution of the nonlinear beam‐plasma problem which depends analytically on all the basic parameters of the problem (i.e., plasma density, beam density, plasma thermal velocity, and beam drift velocity) is presented. This solution shows that the single wave grows exponentially at the linear growth rate until the beam electrons are trapped. At that time the wave amplitude stops growing and begins to oscillate about a mean value. During the trapping process the beam electrons are bunched in space and a power spectrum of the higher harmonics of the electric field is produced. Bot...

Waves and transport in the pure electron plasma

Investigations of low frequency waves and transport in a plasma consisting almost purely of electrons are presented here. This plasma is trapped in a cylindrical system with radial confinement supplied by a strong axial magnetic field and axial confinement supplied by electrostatic fields. Very long containment times are possible. Classical transport due to electron‐neutral collisions has been investigated and good agreement with the theory of Douglas and O’Neil is obtained. Externally launched diocotron waves are investigated. The modal frequencies agree well with linear theory, but the damping is governed by nonlinear effects. Experimental scaling laws for the damping rates are given. Measurements of spatial transport due to these modes are also presented. A signature of this process is that the transport is strongly localized spatially.

Reduction of radial losses in a pure electron plasma

A new pure electron‐plasma containment apparatus exhibits radial losses approximately 20 times smaller than the prior apparatus. However, the new containment times show the same (L/B)−2 scaling with plasma column length and magnetic field as obtained previously. The radial transport is apparently induced by small irregularities that break the cylindrical symmetry of the magnetic and electric containment fields. The fact that the two devices show the same (L/B)−2 scaling suggests the dominance of a single generic process, although this process has not yet been identified.

Destruction of trapping oscillations

A somewhat unconventional traveling wave tube was built to investigate the nonlinear behavior of the beam‐plasma instability beyond the first trapping oscillation of the wave amplitude. In the small cold beam limit, the equations governing the evolution of the beam‐plasma instability are mathematically identical to those describing the traveling wave tube. The traveling wave tube has the advantage that the slow wave structure will remain linear for the wave amplitudes reached in the experiments; furthermore, it does not introduce noise. Five trapped particle oscillations are observed following the saturation of a single launched wave. Two mechanisms for destroying these oscillations have been found. The first involves wave damping and can occur for decrements smaller than 0.01 k0. The second is a result of the modulation of the main wave by unstable sidebands. In addition to the experiments, the equations which describe the interaction are solved numerically. The experimental observations are in excellent...

A nonlinear diocotron mode

A perturbation procedure for the construction of a nonlinear diocotron mode of an infinite length cylindrical pure electron plasma column is presented. The plasma is modeled as a cold fluid executing E×B drift in a strong axial magnetic field. The mode has no axial variation and depends on θ (azimuthal angle) and t (time) through the combination θ−Ωt. Thus the potential and the density are stationary in the frame rotating with angular frequency Ω. In this frame, the potential satisfies the relation ∇2φ=F(φ) where F is a function of φ only. This equation is solved perturbatively to determine a nonlinear mode supported by a cylindrically symmetric equilibrium density profile. The properties of the mode depend only on the offset of the plasma column from the axis of the bounding conducting cylinder and the frequency of the mode is larger than the value predicted by linear theory by an amount roughly proportional to the square of the offset.

Nonlinear wavenumber of an electron plasma wave

The wavenumber of a large‐amplitude electron plasma wave propagating on a collisionless plasma column is measured. The wavenumber is shifted from that of a small‐amplitude wave of the same frequency. This nonlinear wavenumber shift, δkr, depends on position, frequency, and initial wave amplitude, Φ. The observed spatial oscillations of δkr agree qualitatively with recent theories. Experimentally δkr∝kiS (Φ) √Φ where ki is the linear Landau damping coefficient, S (Φ) ≡ki(Φ)/ki, and ki(Φ) is the initial damping coefficient which depends on Φ.

Wave enhancement due to a static electric field

The effect of an applied static electric field on beam electrons trapped by the wave in a traveling wave tube has been investigated experimentally. For sufficiently weak applied fields the wave power is enhanced. When the applied field is sufficiently strong the beam electrons are detrapped, and the wave power enhancement is destroyed. It is found that the beam space charge plays an important role in the detrapping process and acts to limit the wave power enhancement. In addition it is found that the wave power enhancement can be increased by increasing the rf input drive level. By launching waves near the saturation level, over 10 dB of wave power enhancement has been observed. These effects are predicted in a computer simulation, and there is good agreement between the results of the simulation and the experimental results.

Hollow electron column from an equipotential cathode

An unneutralized, magnetically confined electron column thermally emitted from an equipotential cathode is considered. An analytic, scaled solution of the radial Poisson’s equation indicates that the column is hollow and less than √8 central Debye lengths in radius.