Cyclotron Frequency Ωc Research Articles

Weakly relativistic nonlinear orbit dynamics for intense ordinary-mode propagation near electron cyclotron resonance

The nonlinear orbit equations are investigated analytically and numerically for a constant-amplitude electromagnetic wave (ωs, ks) with ordinary-mode polarization propagating perpendicular to a uniform magnetic field B0 êz. Relativisitic electron dynamics are essential in determining the maximum orbit excursion when the incident wave frequency ωs is near the nth harmonic of the electron cyclotron frequency Ωc. Coupled nonlinear equations are obtained for the slow evolution of the amplitude A(т) and phase ø(t) of the perpendicular orbit when ωs ≈ nΩc. The dynamical equations are reduced to quadrature and simplified analytically. At exact resonance (ωs = nΩc), if relativistic effects are (incorrectly) neglected, the maximum orbit excursion Amax approaches the unacceptably large value determined from the first (non-zero) solution to Jn(ns ΩsAmax) = 0, which totally invalidates the non-relativistic approximation. (Here ns Ωs = cks/Ωc.) As a contrasting illustrative example, for n = 1, ωs = Ωc and moderate pump strength, weak relativistic effects limit the maximum orbit excursion to the approximate value , where єs = ns ωs (Vz/c) Vq/c ≪ 1, and Vq is the axial quiver velocity in the applied oscillatory field. Physically, relativistic effects produce a nonlinear frequency shift that dynamically ‘detunes’ the particle motion from exact cyclotron resonance, thereby limiting amplitude growth.

Electromagnetic radiation and nonlinear energy flow in an electron beam-plasma system

The mechanism by which unstable electron plasma waves are converted into electromagnetic waves in a uniform plasma is investigated. Electromagnetic radiation is generated upon injection of an electron beam (500 eV) into a collisionless quiescent magnetoplasma (ne ≲1012 cm−3, kTe ≊2 eV). The emission (ω0) is observed to peak near the plasma frequency (ωp) which is well above the cyclotron frequency (ωc ≪ωp≲ω0). It is shown that electromagnetic waves (ω0, k0) are produced by the scattering of electrostatic plasma waves (ωe, ke) off self-consistently produced ion-acoustic waves (ωi, ki). At low beam intensities the frequency and wave vector matching conditions are experimentally verified (ωe=ωi+ω0, ke=ki+k0≂ki). The emission is found to be polarized, negligible in intensity at ω0=2ωp, and its source is localized. The space-time evolution of the three-wave interaction is presented. Besides these nonlinear wave–wave interactions the wave–particle interactions are investigated. It is found that the strong Langmuir turbulence exhibits a three-dimensional character. Cross-correlation surfaces are measured, and their characteristic scale lengths are found to decrease with increasing wave intensity toward the Debye length (λD ≂20 μm) although the resolution is probe limited (Lmin ≂1 mm). The beam electrons are diagnosed using a novel directional velocity analyzer which is capable of resolving the true three-dimensional distribution function. After interacting with the intense Langmuir waves, the beam electrons are scattered in velocity space, both parallel and perpendicular to their injection velocity. The background electron distribution is observed to develop an energetic anisotropic tail (E≲50 eV, kTe ≂1 eV). The energization of the background electrons coincides with the strong damping of the Langmuir waves. The nonlinear damping is seen as an anomalous ac resistivity produced by large amplitude ion-density fluctuations (δni/ni ≂5%) in the electron beam region.

Non-linear analysis of a high-harmonic rectangular gyrotron

Abstract Non-linear analysis of a gyrotron using an axially grooved rectangular waveguide operating at the high harmonic l of the cyclotron frequency Ωc is given. We use 120 electrons in the simulation of the electron beam. The growth rate, electric field, relativistic factor γ and phase factor are calculated as functions of the slip parameter χ = lΩ0ωγ0 for the chosen optimized operating parameters of the gyrotron. Efficiency of ∼20% in the electron-beam frame is possible for the sixth harmonic of the cylotron frequency which increases to 58% in the second harmonic.

Observation of beam-generated VLF hiss in a large laboratory plasma

The paper describes a lab experiment on a beam-plasma instability under conditions for which the theory of VLF hiss is applicable. An energetic electron beam is injected into a cold, dense, almost collisionless magnetoplasma. Beam and plasma dimensions are so large that oblique whistler wave propagation and growth are readily observable. The observations of frequency and wave number spectra, growth rate, and instability level are consistent with theory and satellite observation in regions of electron precipitation.

Theory of allowed resonance Raman scattering of first and second order in a magnetic field

AbstractThe effect of a magnetic field on first‐ and second‐order resonance Raman scattering from nonpolar optical phonons is investigated theoretically. Both the Landau and spin quantizations are considered within a simple spheric and parabolic two‐band model. Magnetic field effects on the dispersion of the scattering amplitude can be observed which are similar to those of the dielectric susceptibility in magneto‐optics. New effects appear which have no counterpart in magnetooptics if the ratio of phonon frequency ω0 and cyclotron frequency ωc is an integer. For ωc = ωo e.g., in first‐order scattering all Landau resonances of the scattering amplitude, except one, compensate each other. The magnetic field dependence of the scattering amplitude is similar to its dependence on the incident frequency, so that resonance Raman scattering can be investigated experimentally by changing the magnetic field strength instead of the laser frequency. The two processes of second‐order Raman scattering show essential different dispersions in the presence of a magnetic field. The magnetic field induced circular anisotropy of the scattering cross section is considered and the polarization of the scattering light is investigated.

Interaction of a test particle with a magnetized plasma

In the presence of a sufficiently strong magnetic field, the shielding of a test particle in a two-dimensional Vlasov plasma disappears. An intermediate regime is investigated, for both a stationary and a moving test particle, in which the plasma frequency ωp and the cyclotron frequency ωc are of the same order of magnitude. In the case of a stationary test particle, this intermediate regime is approximately in the range ½ ≤ a ≤ 2, where a = ωp/ωc. For a < ½, only partial shielding occurs, the potential retaining its vacuum logarithmic dependence. For a < 2, the potential approximates to the field-free case, and falls off exponentially. For a test particle moving at electron thermal speed, the typical bow wave and wake phenomenon, associated with an unmagnetized plasma, is no longer present. We find, instead, that the potential possesses upstream—downstream symmetry.

Further Note on the Quasi‐Static Theory of a Cylindrical Impedance Probe for Magnetoplasma

In a previous paper (Wait, 1966), an expression was derived for the fields of two parallel line charges in an anisotropic medium. A quasi‐static theory was used, and to facilitate the analysis, the line sources were encased in a cylindrical dielectric rod. An important conclusion was that the resonant frequencies of the driving point impedance did not depend on the geometry of the device; at least, this was true to within a first order. This is in contrast to many other plasma diagnostic devices (Heald and Wharton,. 1965).It is the purpose of the present note to generalize the above‐mentioned derivation to include the effect of a sheath at the boundary of the plasma next to the probe. The influence of a cylindrical glass container for the dielectric rod is also considered. The situation is illustrated in figure 1 where the common z axis of the Cartesian (x, y, z) and the cylindrical coordinate system is normal to the paper. Two infinitesimally thin parallel wires are located at (ρ0, φ0) and (ρ0, φ0 + π), as indicated. For purposes of the quasi‐static analysis, these are assumed to have charges of q and −q coulombs per unit length, respectively. The thin wires are encased in a cylindrical rod of dielectric constant ε1 and radius c. The dielectric rod is enclosed by a cylindrical glass container of dieiectric constant ε2 and outer radius b. The sheath separating the plasma from the device is then regarded to be a cylindrical layer of vacuum of dielectric constant ε0 which has an outer radius a. The external medium is a homogeneous electron plasma and it is characterized by a (angular) plasma frequency ω0, collision frequency v, and (angular) cyclotron frequency ωc. If the d‐c magnetic field lies in the (x, z) plane and subtends an angle θ with the z axis, the relevant components of the dielectric tensor of the plasma are (Ratcliffe, 1959)

Cyclotron-resonance amplification of VLF and ULF whistlers

Electromagnetic signals that propagate through the magnetosphere exchange energy with nonthermal charged particles by the cyclotron-resonance mechanism. The interaction is due to a Doppler shift of the propagation frequency ω to the cyclotron frequency ωc of the charged particle that allows the particle to resonate with the electromagnetic fields. The dominant exchange is between electrons around 1 kev and VLF whistlers that propagate in the right-hand mode and between protons around 1 kev and ULF whistlers (Pc 1 geomagnetic micropulsations) that propagate in the left-hand mode. The complex wave number k(ω) = kr + iki for propagation in a hot uniform plasma parallel to a static magnetic field is assumed to describe whistler properties locally in the magnetosphere. The power transfer function A(ω), which relates the input and output power spectrums, is determined by the path integral of ki along a geomagnetic flux tube. Spectrums of A for both VLF and ULF whistlers have been evaluated for a variety of energy and pitch-angle distributions of the form E−n sinm α. In general, A has an amplification maximum just below a sharp absorption cutoff, which is consistent with observations. The detailed input-output spectrums of whistlers that are necessary to test the theory are not available yet. When these spectrums are measured, this interaction may provide a possible method for mapping the phase space distribution of nonthermal particles in the magnetosphere.

Plasma Resonance Probe Characteristics in a Magnetic Field

The operation of the plasma resonance probe in the absence of a magnetic field is now well understood, and the resonant frequency can be interpreted to give the electron density in the plasma. In many potential applications of this diagnostic technique to laboratory and space plasmas, a static magnetic field is present. This paper considers the resulting modifications to the resonance. The analysis is based on the simplified model of the probe behavior that has been validated for the zero magnetic field case by the work of Harp and Crawford. It is concluded that the technique can be applied to obtain the electron plasma frequency ωp and the electron density for ωc2≪ωp2; and the electron cyclotron frequency ωc, and hence the local magnetic field when ωc2≫ωp2. An approximate correction formula is derived for the intermediate range ωc2≈ωp2.