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Physical Review A

Potential scattering of electron wave packets by large-size targets

Maxim A. Efremov, Mikhail Vladimirovich Fedorov

Published May 9, 2002• 8 citations
Publisher

Abstract

Scattering of electron wave packets by large-size targets is considered. In the ``no-spreading'' approximation the general expression for the electron distribution function after scattering is found. An alternative regime of scattering, different from both the first-order Born approximation and classical limit, is found to exist in the case of narrow (compared to a target) nonspreading wave packets. In this regime, the average angle of scattering is shown to differ from the classical one and, counterintuitively, the difference is shown to be greater, the narrower the wave packets are. Physical reasons of this effect are discussed.

DOI
10.1103/physreva.65.052725

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Scattering of electron wave packets by large-size targets is considered. In the ``no-spreading'' approximation the general expression for the electron distribution function after scattering is found. An alternative regime of scattering, different from both the first-order Born approximation and classical limit, is found to exist in the case of narrow (compared to a target) nonspreading wave packets. In this regime, the average angle of scattering is shown to differ from the classical one and, counterintuitively, the difference is shown to be greater, the narrower the wave packets are. Physical reasons of this effect are discussed.
Potential scattering of free-electron wave packets is considered in the framework of the nonstationary quantum-mechanical theory. To characterize the efficiency of this process, the average angle of scattering is calculated in the lowest (first and second) orders of perturbation theory in the potential. In the approximation of a well-localized, narrow, and nonspreading wave packet, the first-order term is shown to agree with the classical angle of scattering, linear in the potential. This term is shown to vanish at a very large size of the initial wave packet (much larger than the size of a target). An analogy between the potential scattering of wave packets and electron-light ponderomotive scattering is discussed.
Investigation of the large angle scattering behaviour of the narrow wave packet approximation is made. First an expression is derived for the scattering amplitude in the narrow wave packet approximation when the potential is arbitrary. Then the same expression is derived for the case when a potential is spherically symmetric. In many aspects the approximation behaves as Jeffrey's approximation. Since the narrow wave packet approximation breaks down in the vicinity of the singularity of a potential, the validity of the approximation for the case when the potential is short range and has a singularity near the origin is investigated.
It has recently been demonstrated experimentally that 200–300200\text{--}300200–300 keV electrons with the unusual spatial profiles can be produced and even focused to a subnanometer scale---namely, electrons carrying nonzero orbital angular momentum and also the so-called Airy beams. Since the wave functions of such electrons do not represent plane waves, the standard Born formula for scattering of them off a potential field is no longer applicable and, hence, needs modification. In the present paper, we address the generic problem of elastic scattering of a wave packet of a fast nonrelativistic particle off a potential field. We obtain simple and convenient formulas for a number of events and an effective cross section in such a scattering, which represent generalization of the Born formula for a case when finite sizes and spatial inhomogeneity of the initial packet should be taken into account. As a benchmark, we consider two simple models corresponding to scattering of a Gaussian wave packet on a Gaussian potential and on a hydrogen atom, and perform a detailed analysis of the effects brought about by the limited sizes of the incident beam and by the finite impact parameter between the potential center and the packet's axis.
We explore elastic scattering of non-relativistic electrons in the form of standard Laguerre-Gaussian (sLG) and elegant Laguerre - Gaussian (eLG) packets on atomic targets in the generalized Born approximation and compare these results to the reference with Bessel-Gaussian (BG) packets. Scattering by hydrogen-like, iron, silver, and golden targets is considered. The incident electron carries a nonzero orbital angular momentum, while sLG and eLG packets have a definite radial quantum number n as well. In scattering of sLG and eLG wave packets by a macroscopic target sensitivity of the average cross section to the orbital angular momentum is observed, which is absent for BG packets. We highlight the opportunity to employ the differences in the experimental scattering results for the revelation of the properties of incident twisted electron wave packets
We explore the elastic scattering of nonrelativistic electrons in the form of Bessel-Gaussian (BG), standard Laguerre-Gaussian (sLG), and elegant Laguerre-Gaussian (eLG) packets on atomic targets in the generalized Born approximation. Scattering by hydrogenlike, iron, silver, and golden targets is considered. The incident electron carries a nonzero orbital angular momentum, while sLG and eLG packets have a definite radial quantum number nnn as well. In the scattering of sLG and eLG wave packets by a macroscopic target, sensitivity of the average cross section to the orbital angular momentum is observed, which is absent for BG packets. We discuss the potential experimental realization of vortex-electron wave packets of these three types. The distinguishing features in the scattering results can be employed for imaging purposes and revelation of the packet properties in experiments.