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arXiv:2211.16534 · Physical Review A

Elastic scattering of Airy electron packets on atoms

D. V. Grosman, Nadezhda Sheremet, I. Pavlov, Dmitry V. Karlovets

Published Jun 21, 2023• 4 citations
arXivHTMLPDF

Abstract

The problem of elastic scattering of a nonrelativistic electron Airy beam on a hydrogen atom in the ground $1s$ state is considered in the Born approximation. It is demonstrated that the angular dependence of the scattering probability density is, in general, azimuthally asymmetric. When the position of the atom happens to coincide with a minimum of the probability density of the Airy beam, the asymmetric pattern is represented by four separated peaks. We show that this behavior is very sensitive to the precision with which the relative position of the atom and the minima is defined and study how uncertainty in the position can affect observation of the azimuthal asymmetry. Finally, we consider a spatially localized target and discuss the difficulties of observing the asymmetry for targets with sizes exceeding a critical value determined by the beam parameters and by the position of the target center.

DOI
10.1103/physreva.107.062819

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The problem of elastic scattering of a nonrelativistic electron Airy beam on a hydrogen atom in the ground 1s1s1s state is considered in the Born approximation. It is demonstrated that the angular dependence of the scattering probability density is, in general, azimuthally asymmetric. When the position of the atom happens to coincide with a minimum of the probability density of the Airy beam, the asymmetric pattern is represented by four separated peaks. We show that this behavior is very sensitive to the precision with which the relative position of the atom and the minima is defined and study how uncertainty in the position can affect observation of the azimuthal asymmetry. Finally, we consider a spatially localized target and discuss the difficulties of observing the asymmetry for targets with sizes exceeding a critical value determined by the beam parameters and by the position of the target center.
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 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.
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
Within a plane-wave approximation in scattering, an incoming wave packet's Wigner function stays positive everywhere, which obscures such purely quantum phenomena as nonlocality and entanglement. With the advent of the electron microscopes with subnanometer-sized beams, one can enter a genuinely quantum regime where the latter effects become only moderately attenuated. Here we show how to probe negative values of the Wigner function in scattering of a coherent superposition of two Gaussian packets with a nonvanishing impact parameter between them (a Schrödinger's cat state) by atomic targets. For hydrogen in the ground 1s state, a small parameter of the problem, a ratio a/σ_{⊥} of the Bohr radius a to the beam width σ_{⊥}, is no longer vanishing. We predict an azimuthal asymmetry of the scattered electrons, which is found to be up to 10%, and argue that it can be reliably detected. The production of beams with the not-everywhere-positive Wigner functions and the probing of such quantum effects can open new perspectives for noninvasive electron microscopy, quantum tomography, particle physics, and so forth.
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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.