Equations Of Motion For Electrons Research Articles

Characterization of a radio-frequency line-plasma source in longitudinal line cusps created by permanent magnets

Characterization of a compact, high density, and uniform rf line-plasma source has been investigated. Plasma production is conducted in a rectangular discharge chamber (200×100×20 mm3) with a pair of permanent magnets placed on top and bottom of the chamber. A rf current is applied to an internal antenna covered with a quartz tube, and the plasma is produced by an inductive rf discharge. A magnetic field B0 structure of longitudinal line cusps is produced around the edge of the magnets. Ion-saturation current-density Jis profile is controlled by varying the width of the magnets and/or the distance between the antenna and the magnets, because the electrons created in the low-B0 region are reflected in the high-B0 region. The measured Jis profiles are explained by solving the equation of motion for electrons under the cusped magnetic field. A 140 mm wide plasma [plasma density ≃(1.8–2.5)×1012 cm−3 for electron temperature=4–8 eV] of a uniformity variation within 90% is produced using a 140 mm long antenna for an Ar pressure of 20 mTorr and a rf power of 3 kW.

Coherent switching with decay of mixing: an improved treatment of electronic coherence for non-Born-Oppenheimer trajectories.

The self-consistent decay-of-mixing (SCDM) semiclassical trajectory method for electronically nonadiabatic dynamics is improved by modifying the switching probability that determines the instantaneous electronic state toward which the system decoheres. This method is called coherent switching with decay of mixing (CSDM), and it differs from the previously presented SCDM method in that the electronic amplitudes controlling the switching of the decoherent state are treated fully coherently in the electronic equations of motion for each complete passage through a strong interaction region. It is tested against accurate quantum mechanical calculations for 12 atom-diatom scattering test cases. Also tested are the SCDM method and the trajectory surface hopping method of Parlant and Gislason that requires coherent passages through each strong interaction region, and which we call the "exact complete passage" trajectory surface hopping (ECP-TSH) method. The results are compared with previously presented results for the fewest switches with time uncertainty and Tully's fewest switches (TFS) surface hopping methods and the semiclassical Ehrenfest method. We find that the CSDM method is the most accurate of the semiclassical trajectory methods tested. Including coherent passages improves the accuracy of the SCDM method (i.e., the CSDM method is more accurate than the SCDM method) but not of the trajectory surface hopping method (i.e., the ECP-TSH method is not more accurate on average than the TFS method).

Radio frequency line-plasma source using permanent magnets

A high-density and uniform line-plasma source is developed by an inductive rf discharge using a rectangular discharge chamber (200×100×20mm) with a pair of permanent magnets placed on top and bottom of the chamber. Ion-saturation current-density Jis profile is controlled by varying the width of the magnets and the distance between the antenna and the magnets. A 140-mm-wide plasma [plasma density ≃(1.8-2.5)×1012cm−3 for electron temperature =4–8eV] of a uniformity variation within 90% is produced using a 140-mm-long antenna for an Ar pressure of 20mTorr and a rf power of 3kW. The measured Jis profiles are explained by solving the equation of motion for electrons under a magnetic field structure of longitudinal line cusps.

Double-gate field emitters with planar carbon-nanoparticle cathodes: Simulation studies

Simulation studies for the field electron emission from double-gate emitters with planar cathodes were carried out using the finite-element method, the Fowler–Nordheim field-emission equation, and the equation of motion for electrons. We systematically investigated radial position dependence of electric field, radial distribution of emission current, and trajectories of emitted electrons for various double-gate geometries and bias configurations. In particular, we studied the simplest operation mode of double-gate emitters, grounding both the cathode and the second gate adjacent to it, and the dependence of emission-electron focusing on the vertical position and the negative biasing of the second gate in detail. The flexibility in double-gate-emitter operation due to the separate control of electron emission and focusing was also discussed.

Simulation of microwave devices with external cavities using magy

A self-consistent large-signal beam-field interaction model for vacuum electronic microwave sources with external cavities is described. The model includes a self-consistent solution of the three-dimensional equations of electron motion and the time-dependent field equations. The RF fields are decomposed into the fields inside the beam region and the fields inside outer resonators. The RF fields inside the beam region are represented as a superposition of local waveguide modes. The RF fields inside resonators are represented as a sum over resonator modes. The various modes are coupled together due to gaps connecting cavities with each other and with the beam region. The numerical implementation of the model requires additional analytical steps to obtain an effective, convergent, and stable numerical solution. The modified version of the code MAGY has been tested by a comparison with known results and also with measured data.

Nonlinear ineraction of an annular electron beam with azimuthal surface waves

A nonlinear theory is developed that describes the interaction between an annular electron beam and an electromagnetic surface wave propagating strictly transverse to a constant external axial magnetic field in a cylindrical metal waveguide partially filled with a cold plasma. It is shown theoretically that surface waves with positive azimuthal mode numbers can be efficiently excited by an electron beam moving in the gap between the plasma column and the metal waveguide wall. Numerical simulations prove that, by applying a constant external electric field oriented along the waveguide radius, it is possible to increase the amplitude at which the surface waves saturate during the beam instability. The full set of equations consisting of the waveenvelope equation, the equation for the wave phase, and the equations of motion for the beam electrons is solved numerically in order to construct the phase diagrams of the beam electrons in momentum space and to determine their positions in coordinate space (in the radial variable-azimuthal angle plane).

Numerical simulation of electron orbits in a magnetized plasma excited by a surface wave

The objective of this study is to model a surface-wave plasma (SWP) source. An additional objective is to use the simulation results to design a high–density and low–pressure operation plasma source. One of the solutions for low-pressure SWP is done in this study by applying a magnetic field. Moreover, a large dielectric disk is used for excitation and control of the surface wave. A precise and fast solution to the equation of electron motion for computer calculation is presented to predict the effects of magnetic and evanescent electric fields on plasma. A standing surface wave discharge is studied assuming that the electrical orbits in the SWP are affected by magnetic confinement and ponderomotive force. The net result is that the electrons in the static magnetic field are trapped firmly and the effects on the electron orbits by the electric field are relatively small.

Nonlocal nature of the electron energy spectrum in a glow discharge in pure O2: I. Nonlocal character of the electron distribution function

A study is made of the nonlocal nature of the electron energy distribution function in the positive column of a glow discharge in a tube filled with pure oxygen. The distribution function and the axial (Ez) and radial (Er) electric fields as functions of radius are measured using an array of mobile probes. The experimentally obtained spatial profiles of the distribution function are used to test the applicability of the two-term approximation to the distribution function of the electrons with a nonlocal energy spectrum. The distribution function in a specified electric field E=Ez +Er (where Ez ⊥ Er) is calculated by solving the coordinate-dependent Boltzmann equation in the two-term approximation and by directly integrating the equations of electron motion using the Monte Carlo method. A comparison between the experimental data and the results of simulations carried out for a broad parameter range shows that, in the case of a highly nonlocal electron energy spectrum, the two-term approximation makes it possible to calculate the electron distribution function with a fairly good accuracy, in which case, however, in imposing the boundary conditions, the electron losses at the plasma surface should be treated in the kinetic approximation. It is shown that using the reflection coefficient of the plasma surface for electrons instead of the loss cone in space makes it possible to accurately calculate the electron energy distribution function over the entire parameter range under consideration, including the transient region in which the electron-energy relaxation length is comparable to the characteristic plasma dimension.

Acceleration of electrons in thin metal films by strong ultra-short laser pulses

The possibility of effective electron acceleration in a thin solid film exposed to irradiation by two power ultra-short laser pulses is discussed. The acceleration effect is caused by resonant action of the radiation on electrons oscillating across the film with frequencies close to the laser frequency. The relativistic equations of electron motion are numerically analyzed.

Electrical Current Resonance and Current Echoes: A Proposed new Experimental Technique in Solid State Physics

The similarity between the electron equation of motion in electric and magnetic fields and the Bloch equations for nuclear magnetic moments in a magnetic field leads to a novel resonance phenomenon involving the electrical current. This suggests a new experimental technique for the investigation of transport properties in conducting materials. In this work we propose a new set of Bloch Equations for the electrical current with different electron-electron and electron-lattice scattering rates and discuss the consequences of their solutions to the study of the electronic properties of conducting materials. We outline the experimental conditions for detecting electrical current resonance and estimate some relevant experimental parameters for the observation of current echoes and free current decays.

Classical electrodynamics and the quantum nature of light

A review of old inconsistencies of classical electrodynamics (CED) and some new ideas that solve them is presented. Problems with causality violating solutions of the wave equation, the electron equation of motion, and problems with the non-integrable singularity of its self-field energy tensor are well known. The correct interpretation of the two (advanced and retarded) Lienard - Wiechert solutions are in terms of the creation and annihilation of particles in classical physics. They are both retarded solutions. Previous work on the short-distance limit of CED of a spinless point electron are based on a faulty assumption which causes the well known inconsistencies of the theory: a diverging self-energy (the non-integrable singularity of its self-field energy tensor) and a causality-violating third-order equation of motion (the Lorentz - Dirac equation). The correct assumption fixes these problems without any change in the Maxwell's equations and let exposed, in the zero-distance limit, the discrete nature of light: the flux of energy from a point charge is discrete in time. CED cannot have a true equation of motion, only an effective one, as a consequence of the intrinsic meaning of the Faraday - Maxwell concept of field that does not correspond to the classical description of photon exchange, but only to the smearing of its effects in the space around the charge. This, in varied degrees, is transferred to QED and other field theories that are based on the same concept of fields as space-smeared interactions.

Scattering of relativistic electrons by high-power laser light tightly focused

We obtain an approximate solution for the drift and oscillatory components of the motion of relativistic electrons in the field of temporally extended high-power laser light under strong focusing of the light (the size of the focal region is of the order of the light wavelength). This makes it possible to start numerically integrating the equations of electron motion near the focal region. We estimate the impact parameters of the electrons when they are still efficiently accelerated in the focal region.

Volume source of hydrogen negative ions based on reflexive discharge: Theory and experiment (abstract)a)

The results of theoretical and experimental studies of the stationary volume source of hydrogen negative ions based on reflexive discharge are presented. Measurements of plasma parameters and emission characteristics are accomplished, and the optimization of ion source geometry is provided. The best parameters of the ion beam extracted from the source across the magnetic field are as follows: current 40 mA, current density—80 mA/cm2, emittance along the magnetic field—3×10−5 cm rad, emittance across a magnetic field—1×10−5 cm rad. The calculation of basic beam properties is performed by numerical solution of the Boltzmann equation for electrons, balance equations for more than 300 elementary processes in gas discharge plasma, and equations of ion and electron motion. The calculated dependencies of hydrogen negative ions’ current density in the plane of emission slit on the gas pressure and the discharge current are found to be in a good agreement with the experimental data.

Volume source of hydrogen negative ions based on reflexive discharge: Theory and experimenta)

The results of theoretical and experimental studies of the stationary volume source of hydrogen negative ions based on reflexive discharge are presented. Measurements of plasma parameters and emission characteristics are accomplished, and the optimization of ion source geometry is provided. The best parameters of the ion beam extracted from the source across the magnetic field are as follows: current 40 mA, current density—80 mA/cm2, emittance along the magnetic field—3×10−5 cm rad, emittance across a magnetic field—1×10−5 cm rad. The calculation of basic beam properties is performed by numerical solution of the Boltzmann equation for electrons, balance equations for more than 300 elementary processes in gas discharge plasma, and equations of ion and electron motion. The calculated dependencies of hydrogen negative ions’ current density in the plane of emission slit on the gas pressure and the discharge current are found to be in a good agreement with the experimental data.

Chaotic characteristics of electron motion in the static magnetic fields and self-fields of the electron beam

In this paper We study the properties of electron motion when a nonneutral electron beam goes through a static magnetic field. The self-electronic and magnetic fields also affects on the test electron. The Hamiltonian equations of electron motion are deduced and the Poincare surface of section and the Lyapunov exponent are used to prove that the electron motion becomes chaotic when the fields are strong enough.

Nonlinear low frequency electromagnetic waves in a relativistic electron-positron plasma

Using an exact solution of Maxwell's equations and the relativistic equations of motion for electrons and positrons we demonstrate the existence of an electromagnetic wave with a frequency considerably lower that the electron cyclotron frequency. For such a wave the cutoff frequency is determined by the Langmuir frequency and by the wave amplitude. For realistic pulsar parameters a remarkable reduction in the cutoff frequency is then controlled by nonlinear relativistic effects. Modulation instability of this wave is also investigated both analytically and numerically.

Deflection and Re-focusing of Relativistic Electrons in a Cylindrical Condenser

The relativistic equation of motion for electrons in a cylindrical condenser is presented precisely. The analytical expression for the trajectory of electrons in the vicinity of the circular trajectory is given approximately, which has an error of about 1% in comparison with numerical calculations. The properties of the focuses are discussed with the theoretical results of this paper.

Focusing properties of an undulator with anharmonic magnetic fields

The focusing properties of an undulator are considered for the common case that the undulator magnetic fields are not pure sinusoidal (anharmonic). The exact 3D solution of the Laplace equations for the magnetic field near the axis is found. Using equations of electron motion in the fields it is shown that an admixture of higher harmonics in the undulator fields can significantly increase the focusing strength of the undulator. It is demonstrated that this higher harmonic transverse focusing of the electron beam does not violate the phase synchronism condition between the electron and the field, what is similar to the natural undulator focusing of the basic harmonic field.

‘‘Phase mixing’’ of bunches and decrease of negative-mass instability increments in cyclotron resonance masers

Negative-mass (space-charge) instability in magnetized electron beams is considered for a flat, thin (having a thickness about the cyclotron diameter) beam of weakly relativistic electrons oscillators. Pitch angle and energy spreads of particles, as well as magnetic tapering are investigated on the base of averaged differential-integral equations of electron motion. For the energy spread, the phenomenon of ‘‘phase mixing’’ of particles rotating with different frequencies limits the number of cyclotron harmonics taking part in the instability and significantly decreases the increments. For the magnetic tapering, this effect leads to linear saturation at the length comparable with the characteristic scale of the tapering.

Anharmonic oscillator modeling of nonlinear susceptibilities and its application to conjugated polymers

Molecular optical susceptibilities are calculated by deriving equations of motion for the single electron reduced density matrix, and solving them using the time dependent Hartree–Fock (TDHF) approximation. The present approach focuses directly on the dynamics of the charges in real space and completely avoids the tedious summations over molecular eigenstates. It further maps the system onto a set of coupled harmonic oscillators. The density matrix clearly shows the electronic structures induced by the external field, and how they contribute to the optical response. The method is applied to calculating the frequency-dispersed optical susceptibility χ(3) of conjugated linear polyenes, starting with the Pariser–Parr–Pople (PPP) model. Charge density wave (CDW) like fluctuations and soliton pair like local bond-order fluctuations are shown to play important roles in the optical response of these systems.