Cyclotron Period Research Articles

Quasineutral hybrid simulation of macroscopic plasma phenomena

A method for solving the quasineutral hybrid plasma equations in two dimensions is presented, using full ion dynamics and inertialess electrons. The method uses a predictor-corrector field solver and is extended to allow plasma-vacuum interfaces of arbitrary shape. A three-region method for treating the plasma-vacuum interfaces makes possible the simulation of slowly evolving phenomena over time scales much longer than the ion cyclotron period. The algorithm is applied to the study of rotational instabilities in theta pinch Vlasov equilibria.

Multigroup representation of fusion product orbits in a plasma column

A method is derived for describing the time-dependent behavior of α particles produced in a radially nonuniform slender plasma column as a distribution function among the possible orbits. A multigroup numerical approximation is introduced to analyze the development of the distribution function and its moments. Results are presented of calculations of the time-dependent α-particle energy spectrum and radial density, energy, and electron heating profiles in plasma columns with radii comparable to the α Larmor radius. This technique allows calculation of the α particle history at much more rapid rates than allowed by Monte Carlo techniques: The characteristic time scale is the α-electron slowing-down time rather than the cyclotron period.

Stochastic ion heating by a lower hybrid wave: II

The motion of an ion in a coherent lower hybrid wave (characterized by ‖k∥‖<<‖k⊥‖ and ω≳≳Ωi) in a tokamak plasma is studied. For ions satisfying v⊥≳ω/k⊥, the Lorentz force law for the ions is reduced to a set of difference equations which give the Larmor radius and phase of an ion on one cyclotron orbit in terms of these quantities a cyclotron period earlier. These equations exhibit stochastic behavior when the wave amplitude exceeds a threshold. The stochasticity threshold is given a simple physical interpretation. In addition, the difference equations are used to derive a diffusion equation governing the heating of the ions above the stochasticity threshold. Far above the stochasticity threshold, ion Landau damping is recovered. By including the effects of collisions, the heating rate for the bulk ions is obtained.

Confinement times in quasi-trapping orbits

Baker proposed a method of injecting charged particles into a static magnetic field such that particles on these ‘trapping orbits’ never return to the vicinity of the injector and remain permanently trapped. Their infinite confinement time requires that their energy, Eo, be specified with infinite precision. When the energy differs from this by a small amount, E, the particle escapes after a finite confinement time, of the order of In (Eo/E) cyclotron periods.

Generation mechanism of proton cyclotron echoes due to pulsed radio frequency waves in space plasma

Analyses of the proton cyclotron echoes observed on the Alouette 2 ionograms, in more than 800 examples, indicate that the echoes are produced in a range within ±25° width of the magnetic dipole latitude centered at the magnetic equator. The generation mechanism of these proton cyclotron echoes is clarified here as follows: (1) the transmitted RF pulse makes an electrostatic plasma wave that produces a fast electron stream in opposite directions with respect to the wave propagation direction during damping processes of the wave; (2) the RF pulse also produces a proton concentration surrounding the antenna due to negative excursion of the antenna potential; (3) the proton concentration produced is repeated each proton cyclotron period and makes a pseudo‐wave of the electric field; and (4) the encounter of the electron stream with the pseudo‐wave due to the proton concentration makes a plasma wave with frequency in the transmitted RF range. The place where the electron stream encounters the pseudo‐wave should be located at a position relatively close to the satellite for the observation of the RF echoes.

Magnetic insulation and microwave generation

A relativistic self-consistent equilibrium is found for the magnetic insulation of a diode for conditions where the voltage rise time is long compared to the cyclotron period. Instability of the magnetic insulation equilibrium may allow high-power pulsed microwave generation in magnetronlike configurations.

Ion Cyclotron Instability Driven by a Cross-Field Current

The electrostatic instabilities driven by a current perpendicular to an external magnetic field are studied for a homogeneous Vlasov plasma of cold electrons and hot ions. The ions are assumed magnetized. This assumption is allowed for a E × B rotating plasma in which the cross-field current flows for a much longer time than the ion cyclotron period. A new instability of reactive type is found to occur by the coupling of an electron acoustic mode and a Doppler-shifted ion cyclotron mode (in the electron frame).

Plasma ion heating by adiabatic diamagnetization

Attention is drawn to the fact that if a collisionless plasma in a static magnetic field is subjected to the sudden switch-on of an electric field perpendicular to the magnetic field, with a rise-time short compared with the ion cyclotron period, or alternatively, if the process of ionization occurs in crossed static E and B fields, then the internal energy of the ion component will comprise not only the kinetic energy of drift but also an equal kinetic energy of gyrational motion in the drifting frame of reference. For near-collisionless ions this process will give rise to ion heating. For deuterium ions nuclear fusion collisions may be expected for sufficiently high E/B values.

A New Electromagnetic Mode in Cu

For a thin slab of Cu with a static magnetic field applied normal to the slab and approximately along the [110], we have studied the circularly polarized transmission of electromagnetic waves in the 1.0 MHz range. We find that the previously reported oscillations propagating in the Doppler shifted cyclotron resonance (DSCR) absorption region have a polarization sense opposite to that of the helicon and we interpret these as a mode associated with the DSCR. We have computed the dispersion using an approximate copper Fermi surface and find agreement with the experimental dispersion and attenuation. The mode is similar in origin to that predicted for a model Fermi surface with electrons which drift along the field with a constant displacement per cyclotron period.

Temporal linear Bernstein echoes

The appearance of a magnetic field perpendicular to the direction of propagation discretizes the continuum spectrum of electrostatic oscillations. For weak magnetic field B this discretization gives linear temporal echoes of shape independent of B and separated by a cyclotron period.

Oscillatory magnetoconductivity in a linear chain of orbits coupled by magnetic breakdown

A method is given for calculating the magnetoconductivity of a metal in the presence of magnetic breakdown. The method is based on a Kubo-Greenwood formula and the propagators are treated using a Pippard network model which is equivalent to a semi-classical approximation. Impurity scattering is represented simply by making the propagators attenuate. The method is applied to a linear chain network model, and it is found that when the relaxation time is very long compared with the cyclotron period the result is the same as suggested by Pippard using a 'quasi particle' picture. It is argued that this picture and a 'quantum interferometer' picture suggested by Stark are complementary. A brief discussion is given on 'zone oscillations' although no quantitative calculations have been attempted.

On the theory of electromagnetic waves in anisotropic metals

Doppler-shifted cyclotron resonance is studied for a simple model Fermi surface which includes a small pocket of paraboloidal form. This model is simple enough for the field dependence of all the wave characteristics to be found analytically. The results depend on the relative distances moved by the different groups of charge carriers, in the direction of the applied magnetic field, during one cyclotron period. When this displacement is larger for the carriers on the paraboloid than for the main carriers, the helicon wave is coupled with the DSCR mode. It is found that the properties of the coupled waves depend significantly on the sign of the resonant term in the conductivity with respect to the Hall conductivity due to the main carriers. If the signs coincide, the coupling results in extrema in the wave spectrum and anomalies in the impedance of the metal. If the signs are different the helicon wave transforms smoothly into the DSCR mode at the resonance. When the displacement of the carriers on the paraboloid is smaller than for the majority of carriers a DSCR propagating mode appears below the helicon absorption edge.

Resistance oscillations in external electric fields

The oscillations of the transverse magnetoresistance in the electric field reported in reference 1 follow from the interference condition that the Hall drift of carriers during one cyclotron period must equal a lattice vector. The period of the resistance oscillations found experimentally in reference 1 agrees exactly with that predicted theoretically in reference 2.

Perpendicularly Propagating Plasma Cyclotron Instabilities Simulated with a One-Dimensional Computer Model

Perpendicular propagating (k‖ = 0) cyclotron instabilities of velocity distributions f(υ⊥) = δ(υ⊥ − υ0)/2πυ⊥ and similar loss cone distributions with velocity spreads are simulated with a one-dimensional computer model. Essentially noise-free starting conditions are created by loading phase space in a uniform way. Growth starting from machine roundoff levels can be observed at precisely the linear Vlasov theory predictions, up to the saturated level, many orders of magnitude above the starting point. When the growth is dominated by a single wavenumber, the post-saturation time behavior of the electric field is dominated by recurring peaks separated by the cyclotron period. This causes the major frequency components of the nonlinear signal to be at or near harmonics of the cyclotron frequency rather than the linear real frequency, which is nonharmonic. The amplitude of succeeding bursts decays slowly, suggesting the existence of quasistable nonlinear states. A nonlinear instability has also been observed. Furthermore, we have examined another k‖ = 0 instability, a double hump mode arising from the combination of a stable loss cone plus a small amount of cold ions.

Wave packet approach to lattice electrons in external galvanomagnetic fields

The probability for Bragg reflections of wave packets in a lattice and an external electric fieldF is no longer invariant against inversion: (a) Reflections anti-parallel toF are favored compared to reflections parallel toF owing to the energy gain involved. (b) Reflection amplitudes caused in successive cyclotron periods cancel by interference, except when the respective energy gain equals an integer number of magnetic quanta. The effective current parallel toF is calculated up to terms of second order in the lattice potential. It is found proportional to 1/B 2 for scattering times shorter (low field limit) and proportional to 1/B for scattering times longer (high field limit) than one cyclotron period. The effect of magnetic breakdown interpolates between the results of a band model for closed and open electron orbits. From the reflection condition (b) one can deduct oscillations of the effective current inF/B 2 in the 1/B-region.

Adiabaticity of Charged-Particle Trajectories in a Cusped Magnetic Field

The region of adiabatic charged-particle motion for a spindle-cusp field is determined on the basis of confinement times both by numerical integration of the particle orbits and by an experiment utilizing electrons. If the critical flux line dividing the adiabatic and nonadiabatic regions is expressed in Grad's form, ψc = (aλ0)3, where λ0 is the cyclotron radius of the particle corresponding to its total velocity, the numerical analysis yields a value for the constant a of 3.5–4.2 and the experiment yields a = 4.36. It is further shown that the transition zone between the regions of adiabatic and nonadiabatic motion is about two-thirds of an electron cyclotron radius in width, and that the relative field variation encountered during one cyclotron period by an electron whose guiding center is located in this zone is between 11 and 19%. A random-walk model of particle motion, based on the near noncorrelation between fluctuations of the magnetic moment on successive traversals of the field, is used to calculate the effect on particle lifetime of the location of the first reflection point, and thereby explain the apparently excessive random scatter of the data.

Invariants in the Motion of a Charged Particle in a Spatially Modulated Magnetic Field

In this paper we study the effect of a spatial modulation in the magnetic field on invariants such as the magnetic moment μ=V⊥2/B. In particular, we investigate whether an invariant still exists when the wavelength of the modulation is comparable to the gyro radius of the particle. In an axially symmetric magnetic field, with a square-wave modulation, the orbit equations reduce to algebraic relations convenient for numerical study. We find from such studies that orbits are of two types: (a) regular orbits which generate an invariant; (b) orbits which are quasi-ergodic. We have also calculated an invariant by perturbation theory with the depth of modulation of the field as a small parameter. For this we developed a modified form of perturbation theory which overcomes the difficulty of infinities arising at resonance between the perturbation and the cyclotron period. This difficulty in fact corresponds to a change in topology of the invariant curves. The invariant calculated from this theory shows very good agreement with the numerically computed orbits of type (a). The transition to quasi-ergodic behavior cannot be predicted analytically, but some indication of it may exist in the complex topology of the invariant curves in the ergodic regions.

Radio‐Frequency Size Effect on Electrons in Boundary Cross‐Sections Having Extremum Displacement during a Cyclotron Period

physica status solidi (b)Volume 29, Issue 1 p. K23-K26 Short Note Radio-Frequency Size Effect on Electrons in Boundary Cross-Sections Having Extremum Displacement during a Cyclotron Period V. P. Naberezhnykh, V. P. Naberezhnykh Physico-Technical Institute, Ukrainian Academy of Sciences, DonetskSearch for more papers by this author V. P. Naberezhnykh, V. P. Naberezhnykh Physico-Technical Institute, Ukrainian Academy of Sciences, DonetskSearch for more papers by this author First published: 1968 https://doi.org/10.1002/pssb.19680290152Citations: 1AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume29, Issue11968Pages K23-K26 RelatedInformation

Ion Cyclotron Electrostatic Instabilities

An investigation of the instabilities of longitudinal electrostatic oscillations in an infinite magnetized plasma at or near the ion cyclotron frequency has been made. This work extends that initiated by Harris. An approximate mechanical analysis of the coupling of the motion to the electrostatic field oscillations has been developed, which provides some degree of physical intuition and guides the more abstract dispersion-equation approach used in the suceeding analysis. The possible instabilities arising from the ion motion have been classified into two types. Type-A instabilities are characterized by (1) the circumstance that the electrostatic field propagates nearly transversely to the magnetic field, and (2) as the analysis shows, by the requirement that the transverse energy distribution must be peaked at other than zero energy. A monotonically decreasing transverse energy distribution is necessarily stable in this instance. Type-B instabilities arise when the oscillating electric field has a significant component along the magnetic field. Because of the effect of ion motion along the magnetic field, it is now possible for instabilities to arise when the transverse energy distribution decreases monotonically, unlike Type A. Both types of instabilities have been examined as to their dependence on the type of coupling to the plasma electrons, and the various instability criteria are stated. In these electron-ion instabilities a central feature is the coupling of the transverse ion motion to the motion of the electrons along field lines. The most rapidly growing modes are those in which electrons move one axial wavelength in one ion cyclotron period. For actual experimental plasmas, this axial wavelength may be too long to be supported in the experimental device. This can put severe restrictions on the occurrence of these particular instabilities. Another class of instabilities is of those which arise from a coupling of the ion distribution with itself, the electrons playing a passive role only. Of the several kinds of instabilities of this class discussed, the most important are those which arise from a double distribution of ions; e.g., a cold plasma in the presence of a hot ion plasma. The important feature of these instabilities is that they can occur for very low densities. The relevance of this analysis to several current experimental programs is discussed.

KINETIC EQUATION FOR A HOMOGENEOUS PLASMA IN A UNIFORM MAGNETIC FIELD

The theory of Prigogine and Balescu has been applied to a homogeneous single-species plasma in a static uniform magnetic field. A kinetic equation has been obtained for the one-particle velocity distribution, which is assumed initially isotropic in directions perpendicular to the field. The only stationary solutions of the kinetic equation are the Maxwellian equilibrium distributions, and an H theorem has been established. The neglect of "collisions" of the order of duration of one cyclotron period or less (a strong magnetic-field approximation) modifies the kinetic equation so that it no longer predicts any relaxation of the velocity components parallel to the magnetic field. The assumptions and approximations of the theory are stated, and are discussed in some detail.