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Physical Review Letters

Motion of an Electron from a Point Source in Parallel Electric and Magnetic Fields

Christian Bracher, J. B. Delos

Published Mar 15, 2006• 57 citations
Publisher

Abstract

Negative ions undergoing near-threshold photodetachment in a weak laser field provide an almost pointlike, isotropic source of low-energy electrons. External fields exert forces on the emitted coherent electron wave and direct its motion. Here, we examine the spatial distribution of photodetached electrons in uniform, parallel electric and magnetic fields. The interplay of the electric and magnetic forces leads to a surprising intricate shape of the refracted electron wave, and multiple interfering trajectories generate complex fringe patterns in the matter wave. The exact quantum solution is best understood in terms of the classical electron motion.

DOI
10.1103/physrevlett.96.100404

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Physical Review A•Nov 1, 1990•26 citations
Negative ions undergoing near-threshold photodetachment in a weak laser field provide an almost pointlike, isotropic source of low-energy electrons. External fields exert forces on the emitted coherent electron wave and direct its motion. Here, we examine the spatial distribution of photodetached electrons in uniform, parallel electric and magnetic fields. The interplay of the electric and magnetic forces leads to a surprising intricate shape of the refracted electron wave, and multiple interfering trajectories generate complex fringe patterns in the matter wave. The exact quantum solution is best understood in terms of the classical electron motion.
We study the quantal motion of electrons emitted by a pointlike monochromatic isotropic source into parallel uniform electric and magnetic fields. The two-path interference pattern in the emerging electron wave due to the electric force is modified by the magnetic lens effect which periodically focuses the beam into narrow filaments along the symmetry axis. There, four classical paths interfere. With increasing electron energy, the current distribution changes from a quantum regime governed by the uncertainty principle, to an intricate spatial pattern that yields to a semiclassical analysis.
We examine the dynamics of electrons photodetached from the H– ion in time-dependent electric and magnetic fields for the first time. The photodetachment microscopy patterns caused by a time-dependent gradient electric field and magnetic field have been analyzed in great detail based on the semiclassical theory. The interplay of the gradient electric field and magnetic field forces causes an intricate shape of the electron wave and multiple electron trajectories generated by a fixed energy point source can arrive at a given point on the microchannel-plate detector. The interference effects between these electron trajectories cause the oscillatory structures of the electron probability density and electron current distribution, and a set of concentric interference fringes are found at the detector. Our calculation results suggest that the photodetachment microscopy interference pattern on the detector can be adjusted by the electron energy, magnetic field strength, and position of the detector plane. Under certain conditions, the interference pattern in the electron current distribution might be seen on the detector plane localized at a macroscopic distance from the photodetachment source, which can be observed in an actual photodetachment microscopy experiment. Therefore, we make predictions that our work should serve as a guide for future photodetachment microscopy experiments in time-dependent electric and magnetic fields.
During photodetachment in external electric and magnetic fields, the detached electron wave propagated from the negative ion in different directions may intersect again at a large distance from the negative ion, thus creating an interference pattern in the detached electron’s current density distributions on a detector plane placed at a certain distance from the negative ion. In this work, we calculate the photodetached electron’s current density distributions in perpendicular electric and magnetic fields on the basis of the semi-classical theory. Our results suggest that the electron current density distributions on the detector plane are not only related to the electric and magnetic fields’ strength, but also related to the electron energy. Under certain conditions, the interference pattern could reach macroscopic dimensions and could be observed in a direct photodetachment microscopy experiment. Our studies may guide the future researches in the wave propagation theory and the photodetachment microscopy of negative ion in the presence of external fields.
Matter-wave interferometers, either with electrons, atoms or molecules, owe their sensitivity to the accuracy of phase-shift measurements. One may, however, raise the question whether the shift undergone by interference fringes, when external forces are applied, actually differs from the shift of classical trajectories. For the case of a magnetic perturbation, we provide experimental evidence and a vector demonstration that interference patterns only undergo a global shift. The experiment is performed with the photodetachment microscope, with a uniform magnetic field superimposed over the whole volume accessible to the interfering electron. Identity of the fringe and trajectory shifts is established for any two-wave interferometer using charged particles, submitted to a magnetic field.
The motion of an electron in a uniform electric field, as in free fall, is characterized by the existence of pairs of trajectories that rejoin on the same final point. The corresponding interference pattern was first observed in 1996, and explained by the elementary model of uniform acceleration starting from a motionless electron source. In the experiment, however, the electron source is a fast-moving negative ion, which makes direct application of the model questionable. Relativistic arguments can be used to ascertain that no phase changes are actually produced by the motion of the source. The quantum phase of the electron can also be checked all along its two interfering trajectories. This method, which produces the same reassuring result as relativity, gives us the opportunity to revisit the relativistic transformation of the Schrödinger phase. It also makes it possible to calculate the effect of an acceleration of both the free electron and the electron emitter. Only in this case is a change of the interference pattern expected, but it is quantitatively negligible in the actual experiment.
An analytic momentum-space wave function for an electron in both laser and static uniform electric fields is presented. It is used to obtain analytic multiphoton detachment cross sections in a static, uniform electric field which include effects of static-field-induced electron-photon interactions. These general results are not restricted to weak laser intensities or to weak static-field strengths and depend only on (a) the electric dipole approximation and (b) the approximation that final-state electron-atom interactions are ignored. Four specific predictions of our general formulas for the most interesting case of linearly polarized light polarized along the static-field direction are presented for weakly bound electrons initially in an s state. First, the effects of a weak static electric field on N-photon detachment cross sections near threshold are shown to be described by two modulation factors, one for odd N and one for even N, which depend only on a scaled energy. Second, for photodetachment in a static electric field, effects of static-field-induced electron-photon interactions are demonstrated. Third, the lifetime against field ionization is presented. Fourth, the cross section for electric-field-induced stimulated emission is presented. Numerical results for the latter three effects are presented for the ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ ion. The simpler case of circularly polarized light directed along the static-electric-field direction is treated briefly. In particular, we show for this case that the static- and laser-field effects are uncoupled and that the near-threshold, weak static-electric-field modulation factors for the N-photon detachment cross sections are dependent on N.