Longitudinal Plasma Waves Research Articles

磁界中のプラズマ振動のCauchy積分型分散式の誘導

The dispersion formula of Cauchy integral type for longitudinal plasma waves in a magnetic field has been for the first time exactly derived, in order to obtain a general instability criterion for magnetoplasma waves, on the basis of Vlasov's collision-free kinetic equation for arbitrary velocity distributions. It reads∫Φ(k, v)/v-ζ dv=k2/ωpwhereΦ(k, v)=φ_(k, v)=2π/NkII[kII∫A2l(k1/Ω)2l+k1∑A2l+1(v)(k1/Ω)2l+1]and ζ=ω/kII, 1mζ>0N and Ω are the total density and the gyration frequency of electrons. A2l and A2l+1 are fully determined by the 2/-th moment of the elelctron velocity distribution function integrated perpendicularly to the external magnetic field.The motion of the ions is easily taken into account by replacing Φ(k, v) by φ-(k, v)+φ+(k, v)m_/m+. The formula being important for the drift instability can be written ask2-ω_∫∞-∞g0(v)/v-ζ dv+(k1/kII)2∫∞-∞D(v.Ω/kII)/v-ζ dv=0The first two terms correspond to the dispersion relation either for the case of strong magnetic field approximation or for the case of grazing angle propagation. The third term contributes to the drift instability in the case of wave propagation at an appreciable angle to a finite magnetic field.The fourth or higher order terms give rise to the instability due to anisotropic velocity distributions.

Radiation in a Plasma. I. Čerenkov Effect

We start with the linearized plasma equations containing an isotropic pressure term, plus extra source terms ${\mathrm{J}}^{s}$ and ${\ensuremath{\rho}}^{s}$ in the Maxwell equations. The fields of (${\mathrm{J}}^{s}, {\ensuremath{\rho}}^{s}$) can be decomposed into two modes. The electromagnetic (EM) mode has all the magnetic field and no charge accumulation; it is the ordinary EM field of (${\mathrm{J}}^{s}, {\ensuremath{\rho}}^{s}$) in a dispersive medium of relative dielectric constant ${\ensuremath{\epsilon}}_{r}=1\ensuremath{-}{(\frac{{\ensuremath{\omega}}_{p}}{\ensuremath{\omega}})}^{2}$. The plasma (P) mode has all the charge accumulation and no magnetic field; at great distances from the source, it becomes a longitudinal (radial) plasma wave with the usual dispersion relation for plane plasma waves. Various potentials for the EM and P modes are given by the inhomogeneous Klein-Gordon equation. The fields of a uniformly moving charged particle are found by a Lorentz transformation. When $(\frac{u}{{v}_{0}})l1$ ($u=\mathrm{particle}\mathrm{velocity}$, ${v}_{0}=\mathrm{rms}\mathrm{thermal}\mathrm{velocity}$), the EM and P fields are exponentially screened outside oblate spheroids foreshortened in the direction of motion. When $(\frac{u}{{v}_{0}})g1$, the $P$ field exists only within the Mach (\ifmmode \check{C}\else \v{C}\fi{}erenkov) cone trailing the particle. The frequency and angular spectra of the \ifmmode \check{C}\else \v{C}\fi{}erenkov radiation are found, and the total radiated energy is found by assuming an arbitrary high-frequency cutoff due to Landau damping. The expression for total radiated energy agrees with that given by Pines and Bohm, except for the logarithmic terms.

Die Kopplungsmechanismen zwischen longitudinalen und transversalen Wellen in einem Plasma

Since electron plasma oscillations are assumed to cause nonthermal cosmical radiofrequency radiation, the problem of coupling between longitudinal and transversal plasma waves becomes important. Starting from the basic plasma equations all possible coupling mechanisms are deduced in a general way. Besides the hitherto known mechanisms as the effect of a pressure gradient, of an external magnetic field, and of a general flow or drift field, a new nonlinear “internal coupling“ effect which is able to generate electromagnetic radiation proves to be existent. Finally some remarks concerning the relativistic aspect of the problem are added, as errors are found in the literature.

Radiation by Charged Particles passing through an Electron Plasma in an External Magnetic Field

The motion of a charged particle through a low density electron plasma in an external magnetic field has been investigated. The interaction of the particle through transverse waves has only been considered, and the consequent energy losses are calculated. It is found that the coupling between the longitudinal plasma wave and the transverse electromagnetic wave modifies the nature of radiation emitted by the particle. The results show that for a velocity greater than that needed for exciting plasma waves in the medium, the particle emits non-Cerenkov type of radiation, in addition to Cerenkov radiation obtained previously by others.

The Dispersion Relation of Transyers and Longitudinal waves in Plasma in aMagnetic Field

The Dispersion Relation of Transyers and Longitudinal waves in Plasma in aMagnetic Field

Energy absorption in a longitudinal wave in a collisionless, magnetized plasma

The present paper is devoted to calculations of the energy absorption in a longitudinal plasma wave. The direction of propagation is arbitrary with respect to an external magnetic field H0.

Longitudinal Ion Oscillations in a Hot Plasma

Linearized, longitudinal waves in a hot plasma include, besides the familiar electron plasma oscillations, in which the frequency ω is of order ωp = (4πne2/m)½, also ion plasma oscillations with ω ≈ ωp(m/M)½. The properties of the latter are explored using a Vlasov equation description of the plasma. For equal ion and electron temperatures, Te = Ti, there exists a discrete sequence of ion oscillations, but all are strongly damped, i.e., have -Im ω/Re ω ⪞ 0.5, and hence are not likely to be observable. The ratio Im ω/Re ω can be made to approach zero (facilitating detection of the waves) by either increasing Te/Ti or by producing a current flow in the plasma. In the latter case, Im ω can even be made positive (corresponding to growing waves), the current required for this being smaller the larger the value of Te/Ti. This growing wave is just the familiar two-stream instability which is thus seen to be an unstable ion oscillation. It is also noteworthy that the ion oscillations, which for small k have the properties usually associated with an acoustic wave (longitudinal polarization, ω ∝ k), are obtained using a formalism which is sometimes designated as ``collisionless.''

Scattering of Electromagnetic Waves by Longitudinal Plasma Waves

It is shown that Bragg reflection of electromagnetic waves from plasma oscillations is possible. The wave equation for the electromagnetic field passing through a medium of plasma waves is derived, and it is found to be similar to that of Brillouin for a wave in a medium of variable dielectric constant.

Propagation of Plasma Waves across a Density Discontinuity

The linearized plasma equations and the boundary conditions to be satisfied across a surface of density discontinuity are considered. It is shown that an incident longitudinal plasma wave generates reflected and transmitted transverse plasma waves, as well as reflected and transmitted longitudinal waves. The inverse process also occurs. The transverse waves are generated only when the longitudinal wave is almost normally incident; in the inverse process, the generated longitudinal waves are propagated normal to the boundary.

Wave theory of plasmas

The fluctuating electric microfileld in a plasma in thermal equilibrium can be divided into two components. One has the character of rare sharp peaks, due to the occasional close approach of single ions or electrons to the observation point, while the other, the background, results from the simultaneous action of a large number of more distant charges. This longrange component may be imagined as arising from the random superposition of intercrossing longitudinal plasma waves, with all, wave-lengths down to a small multiple of the Debye length. Their spectral energy distribution is obtained by counting the modes of oscillation by Jeans’s method, combining this with the dispersion law given by Bohm & Gross, and giving each rnode an energy kT . The cut-off near the Debye length is estimated by various considerations, which lead, to the conclusion that the spectrum extends approximately over one octave above the critical or ‘plasma frequency’. The argument is supported by showing that there is a thermodynamical upper limit to the energy of the plasma waves. The scattering of electrons in a plasma by long-range effects is calculated from their momentum and energy exchange with the random plasma, waves. It is found to be much smaller than the scattering due to close encounters, which proves conclusively that the phenomena first observed in low-pressure arc discharges by Langmuir, and which appear to suggest a very strong electron-electron interaction, cannot be accounted for by the dynamics of equilibrium or steady-state plasmas. This suggests very forcibly that these arc plasmas are the seat of strong, partially organized, self excited oscillations, to which the considerations of the present paper do not apply.

Plasma Oscillations in a Static Magnetic Field

A theory of the small-amplitude oscillations of an ionized gas in a static magnetic field is developed, including the effects of temperature motions. The Boltzmann equation is solved for this problem, and exact expressions are obtained for the distribution function and dispersion relation. A general feature of the dispersion relation is the existence of gaps in the spectrum at frequencies which are approximately multiples of ${\ensuremath{\omega}}_{c}=\frac{\mathrm{eH}}{\mathrm{mc}}$. The magnitude of the gap depends on the temperature of the gas, being proportional to it for long wavelengths. This leads to the prediction of selective reflection of waves impinging on a plasma with frequency in the forbidden range.For $\mathrm{ck}\ensuremath{\gg}{\ensuremath{\omega}}_{p}$, ${\ensuremath{\omega}}_{c}$ the waves split into approximately longitudinal plasma waves and transverse waves. Detailed analysis is made of the plasma waves for ${\ensuremath{\omega}}_{c}$ small and ${\ensuremath{\omega}}_{c}$ large. At long wavelengths the frequency is ${\ensuremath{\omega}}^{2}\ensuremath{\simeq}\ensuremath{\omega}_{p}^{}{}_{}{}^{2}+\ensuremath{\omega}_{c}^{}{}_{}{}^{2}+\ensuremath{\beta}(\frac{\ensuremath{\kappa}T}{m}){k}^{2}$, where $\ensuremath{\beta}$ depends on ${\ensuremath{\omega}}_{p}$ and ${\ensuremath{\omega}}_{c}$. For waves near the Debye length the waves are heavily damped.Two simplified treatments of plasma oscillations based on transport equations are compared with the above treatment. Expressions of the form ${\ensuremath{\omega}}^{2}\ensuremath{\simeq}\ensuremath{\omega}_{p}^{}{}_{}{}^{2}+\ensuremath{\omega}_{c}^{}{}_{}{}^{2}+\ensuremath{\beta}(\frac{\ensuremath{\kappa}T}{m}){k}^{2}$ are obtained where the factor $\ensuremath{\beta}$ is independent of ${\ensuremath{\omega}}_{c}$ and ${\ensuremath{\omega}}_{p}$. In addition, the transport treatments fail to predict the heavy damping near the Debye length and the existence of gaps in the frequency spectrum.