Lorentz Force Equation Research Articles

PPPL Lorentz orbit code

A code that integrates the Lorentz force equation has been developed to trace a single charged particle’s trajectory under the influence of toroidally symmetric magnetic fields found in tokamaks. This code is used primarily to design and estimate the efficiency of charged fusion product probes, which detect escaping energetic ions such as the 1 MeV tritons, 3 MeV protons, 15 MeV protons, and 3.5 MeV alphas created in TFTR. This interactive code has also been used as a teaching tool to illustrate classes of orbits such as trapped and passing, as well as subtle orbital motions, e.g., precession of banana orbits in tokamaks, or orbits in dipole magnetic field configuration. This paper describes the code as well as recent modifications which (1) include Shafranov shifts of the magnetic surfaces, (2) use more realistic current density profiles, and (3) allow modeling of the detector and limiters.

The Hall effect in copper: An undergraduate experiment

This experiment on the Hall effect uses only apparatus ordinarily found in an elementary laboratory, and the Hall effect device itself can be constructed easily from ordinary materials without special equipment. It illustrates the vector nature of the Lorentz force equation, shows that the charge carriers in copper are negatively charged, and permits determination of the charge-carrier density and other related parameters in ordinary commercial copper.

On the Lorentz particle with spin — a twistorial approach

A “square root” of Lorentz and BMT equations is obtained using twistorial methods. The formalism, with classical Lorentz force equations as input, requires the gyromagnetic ratio to be equal to two.

Relativistic dynamics of charges in external fields: the Pauli algebra approach

The Pauli algebra is used to study the relativistic motion of charged particles in external electromagnetic fields. Plane-wave solutions to Maxwell's equations, in particular standing waves with electric and magnetic fields which are everywhere parallel, are easily represented and analysed in a gauge-independent treatment. The authors derive the 4-momentum of electromagnetic fields and show how the integrals for fields of a charged particle can be evaluated with integration limits specified in the rest frame of the moving charge. The general solution of the Lorentz force equation is shown to be a Lorentz transformation with time-dependent rotation and boost parameters which, in the case of parallel fields, are trivially related to the magnetic and electric fields, respectively. With the Pauli algebra, an earlier derivation in the Dirac algebra of the 4-velocity in constant uniform fields is greatly simplified and extended to arbitrary initial motion, and solutions are found to motion in a Coulomb fields and in the presence of arbitrarily polarised (monochromatic, directed) plane waves.

Numerical Study of Gyrophase and Time-Dependent Nonadiabatic Electron Losses in Axially Symmetric Magnetic Mirror Fields

Charged-particle motion in vacuum magnetic fields with 2 degrees of freedom has been investigated by numerically integrating the relativistic Lorentz force equation. The numerical simulation demonstrated that collisionless particle losses due to the nonexistence of a globally conserved quantity other than the energy and canonical angular momentum are gyrophase dependent, thus resulting in the existence of multiple particle confinement times. This is consistent with the cited experimental results and, in addition, demonstrates the indistinct nature of the mirror loss cone for rho/L/sub M/ < 2 x 10/sup -2/.

Integral equation for the laser field in a long-pulse free-electron laser.

The free-electron laser is described by the self-consistent solution of the coupled system of the Lorentz force equation for the electrons and the Maxwell equation for the laser field. We show that, near saturation and at both low- and high-gain regime, the evolution of each longitudinal mode in the optical cavity can be written as an integral equation. We present numerical solutions of the integral equations for a free-electron-laser oscillator.

Radiation force corrections to planar wiggler FEL gain

The longitudinal electron dynamics in an FEL are governed by the coupling of the electrons transverse velocity, V ⊥, to the combined magnetic fields of the wiggler and radiation in the Lorentz force equation. The derivation of the “standard” gain expression neglects the radiation contribution to V ⊥ and assumes V ⊥ results from the wiggler field alone [1]. We show, however, that the radiation contribution to V ⊥ couples with the wiggler field to produce a force of the same order as that from the wiggler field coupling with the radiation field [2]. When this former force, the “radiation force”, is included in the electron dymanics, the “difference of Bessel function factor” in the pendulum equation differs significantly from that of the standard expression, being given by: 1 2 1+ 1 f J (f−1) 2 (fξ)− 1− 1 f J (f+1) 2 (fξ) (−1) (f−1) 2 instead of {J (f−1) 2 (fξ)−J (f+1) 2 (fξ)}(−1) (f−1) 2 The gain is then proportional to the product of these two factors, rather than the square of the latter. The corrections to the pendulum/gain expressions result in approximately a 20% increase in gain at the fundamental but a 20% decrease in gain at the third harmonic for deflection parameter K = 2. The standard expressions for helical wiggler gain and spontaneous emission cross-sections are unaffected by inclusion of the radiation force term.

Gyrotron and peniotron modes in rotating-beam devices

Abstract Several authors have assumed the existence of the (s + 1), s and (s − 1) harmonic interactions in the electron beam immersed in an axial magnetic field (Dohler et al. 1982, Ono et al. 1983). This paper examines the interaction between a relativistically rotating electron layer and electromagnetic fields with exp (jωt − Γz − jsθ) variation. The linearized relativistic Lorentz force equation is used to evaluate the RF displacement due to harmonic fields. It can be shown from this analysis that in the non-relativistic case, the beam waves are one at the (s + 1), one at the (s − 1) and two at the s harmonic of the cyclotron frequency. These modes may be identified as the fast (s + 1) and slow (s − 1) peniotron modes. This result is consistent with existing theories. The gyrotron interaction is observed as a result of energy exchange between the forward circuit wave and two beam waves at the s harmonic; it is a three-wave interaction. Whereas the fast (slow) peniotron interaction is observed as a result of energy exchange between the forward circuit wave and the fast (slow) peniotron wave at s + 1 (s − 1) harmonics; it is a two-wave interaction. The polarization-variable approach is used to relate the RF displacement to the RF current density and the RF charge density. The dispersion relations are then derived by solving for the electromagnetic fields in cylindrical waveguide, consistent with the current and charge densities in the electron layer for each instability. The results are compared with existing theories.

Single Particle Dynamics and Gain in Free Electron Laser

Single particle dynamics in free electron laser is discussed with the help of Lorentz force equation. The treatment gives a small gain term “on resonance” where previously no gain was found by Colson.

The Vlasov equation and the description of inverse bremsstrahlung

It is shown that the Vlasov equation based on the Lorentz force equation does not describe inverse bremsstrahlung since the force by inverse bremsstrahlung is not included in the Lorentz force equation. It is also shown that quasilinear theory based on the classical Vlasov equation can adequately explain inverse bremsstrahlung.

Tachyonium, an action-at-a-distance model

A model is presented for a tachyon and a bradyon bound in circular orbits. They are taken to be charged point particles interacting through action-at-a-distance retarded potentials without radiation reaction. The superluminal Lorentz transformation used conforms to the relativity principle in that it retains the constancy of the speed of light, the forms of Maxwell’s equations and the Lorentz force equation, and requires electric charges to retain their electric character upon transformation from superluminal to subluminal frames. Interactions between a tachyon and a bradyon are restricted to regions within twointeraction cones. Backward cones are related to retarded potentials and forward cones to advanced potentials; this should be taken into account in resolving causal paradoxes. The tachyonium atom consisting of a tachyon-bradyon pair interacting through retarded potentials has discrete values of total energy and angular momentum for a given mass ratio; it is shown that this discreteness does not occur for half-retarded-half-advanced potentials. Values for conserved quantities and other properties are presented. A model that might explain many of the features of quasars is suggested.

Multimode Theory of Free-Electron Laser Oscillators

The nonlinear self-consistent theory of short-pulse free-electron laser oscillators can now be extended to include transverse diffraction within the optical resonator mode. The theory provides efficient solutions to the three-dimensional parabolic-wave equation coupled to the Lorentz force equation. The method is general enough to include arbitrary magnet designs, optical mirror arrangements, and driving currents. New mode structures are predicted which should be observed in future experiments at Stanford University.

Inclusion of space-charge effects with Maxwell's equations in the single-particle analysis of free-electron lasers

In free-electron lasers, the motion of electrons is governed by the Lorentz force equations in the presence of a radiation field. The variation of the radiation field then follows from Maxwell equations in which the modulated electron current acts as a source. A self-consistent analysis, which completely incorporates these two concepts, is presented to describe the behavior of the radiation field and the modulation of the electron beam. In this analytical study, we consider the space-charge effect as well as high interaction strength in the small-signal limit so that the result can be compared directly with the traveling-wave tube theory. It is found that the well-known three-wave solution is essentially applicable to the electron beam only. The variation of the radiation field is much more complicated. According to the analysis, there are only three controlling parameters: the pumping strength, the electron density, and the electron energy detuning. For different choices of those parameters, the field can be in a stable regime where its growth is limited or in an unstable regime where it grows exponentially. The boundary between these two regimes is defined quantitatively. The effect of the plasma resonance is observed at high electron densities as a natural result of maximum single-pass gains. The traveling-wave tube is then analyzed as a special example.

A free-electron laser in a uniform magnetic field

The interaction of free electrons and free electromagnetic radiation, in the presence of a uniform magnetic field, can result in stimulated emission or absorption. We analyze the dynamics of single electrons by solving the classical, relativistic Lorentz force equations of motion in these combined fields. An electron may gain energy from, or lose energy to, the radiation field, depending crucially on the phase and oscillation frequency of the electron's helical motion within the superposed, circularly polarized light wave. To first order in the radiation field strength, electrons in a monoenergetic, uniformly distributed beam become spatially bunched, but there is no net energy change. To second order, however, the beam may experience a gain or loss of energy, corresponding to attenuation or amplification of radiation. We compare the bunching of this laser process to the bunching processes involved in 1) the Stanford free-electron laser and 2) the cyclotron maser, and find significant differences in each case. Our analytic results provide a clear, simple picture of the interaction process, and can be useful in exploring light amplification in astrophysical magnetic fields, the magnetosphere, or in laboratory devices.

One-body electron dynamics in a free electron laser

The free electron laser is analyzed by solving the one-body classical Lorentz force equation in the presence of periodic magnetic field and a plane electromagnetic wave. Phase space paths for electrons are related to those of a simple pendulum and describe laser gain, saturation, and coherent electron beam modulation.

Elimination of Retarded Functions from Two-Body Classical Relativistic Electrodynamics

Interaction problems in classical relativistic electro-dynamics are frustratingly difficult to solve, since the differential equation of motion involves functions of both the present and retarded times. In this paper the author eliminates the retarded functions from the equation of motion for the two-body problem in which two equal masses with opposite charges that are equal in magnitude are released from rest and allowed to fall into one another. In the final form of the equation of motion only the present time, the present position, and derivatives of the present position with respect to the present time appear—in other words, it is an equation with one unknown and the present time only. To accomplish this simplification of the equation of motion, the author employs the Lorentz condition, ∇·A+δΦ/cδt=0, and the expression E=−∇Φ−δA/cδt to obtain a partial differential equation in the retarded separation distance R that can be solved. When the resulting relationships are substituted into the Lorentz force equation it contains only functions of the present time.

Theory of electron cyclotron resonance heating. I. Short time and adiabatic effects

The theory of single particle electron cyclotron resonance heating in a magnetic mirror is treated analytically and numerically, using the techniques of (a) integration of the Lorentz force equation and (b) transformation to a Hamiltonian approximation, to study both short time scale and adiabatic effects. The force equation is analytically integrated in the vicinity of the resonance plane to obtain the energy dependence of the effective time spent in resonance per bounce te. For electrons passing through the resonant zone at constant parallel velocity vzR, te varies as vzR-12/. For electrons which turn in or near the resonant zone, te varies as vperpendicular to R, P=2/3, where vperpendicular to R is the transverse velocity at resonance. These results agree with the exact numerical integration of the force equation, for which P approximately=0.5-0.7.

Relativistic effects in the traveling-wave amplifier

A relativistically correct large-signal theory is developed for the analysis of high-power, axially symmetric traveling-wave amplifiers in order to investigate the physical phenomena involved in the interaction process. The nonlinear integro-differential system equations are developed from the Lorentz force equation, the one-dimensional equivalent circuit equation, the wave equation, and the continuity of charge relation. These equations are applied to two electron stream models: a ring model which permits the effects of nonlaminar flow and space-charge forces to be evaluated, and a disk-electron model in which these effects are ignored. The ring model space-charge fields are obtained from the appropriate Green's function for Poisson's equation in a moving frame of reference. Numerical solutions are presented and discussed with major emphasis on the disk-model solutions. The principal results are that the gain per unit length decreases with increasing beam velocity, the circuit phase velocity for optimum power output approaches the dc beam velocity u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> , as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u_{0}/c</tex> approaches unity, and the conversion efficiency is almost independent of u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> for the synchronous case. The linearized one-dimensional theory of the traveling-wave tube is also discussed. Several of the large-signal results are predicted from the small-signal theory.

Motion of a charged particle in electric and magnetic fields

The motion of a charged particle in time-varying uniform electric and magnetic fields has been determined exactly by writing the Lorentz force equation in a matrix form. The general solution is obtained by solving the ordinary first-order linear differential equation.

A mathematical technique for an exact small-signal field analysis of multiple-stream interaction in a finite longitudinal magnetic field

A mathematical technique for solving Maxwell's equations and the Lorentz force equation with no approximations except the small-signal approximation is presented. A finite dc magnetic field parallel to the dc velocity of the charges is included. Polarization variables are used, and the boundary conditions include ac surface charge density and surface current density. The advantages of the method are that both fast waves and slow waves are included without a quasi-static approximation, and only the determinantal equation requires computer solution. The partial differential equations are solved directly and need not be solved by computer.