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arXiv:1508.00026 · Physical Review A

Scattering of wave packets on atoms in the Born approximation

Dmitry V. Karlovets, Gleb L. Kotkin, Valeriy G. Serbo

Published Nov 4, 2015• 44 citations
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Abstract

It has recently been demonstrated experimentally that $200\text{--}300$ 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such electrons do not represent plane waves, the standard Born formula for scattering of them off a potential field is no longer applicable and, hence, needs modification.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.

DOI
10.1103/physreva.92.052703

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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such electrons do not represent plane waves, the standard Born formula for scattering of them off a potential field is no longer applicable and, hence, needs modification.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.
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.
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
While a plane-wave approximation in high-energy physics works well in a majority of practical cases, it becomes inapplicable for scattering of the vortex particles carrying orbital angular momentum, of Airy beams, of the so-called Schrödinger cat states, and their generalizations. Such quantum states of photons, electrons and neutrons have been generated experimentally in recent years, opening up new perspectives in quantum optics, electron microscopy, particle physics, and so forth. Here we discuss the non-plane-wave effects in scattering brought about by the novel quantum numbers of these wave packets. For the well-focused electrons of intermediate energies, already available at electron microscopes, the corresponding contribution can surpass that of the radiative corrections. Moreover, collisions of the cat-like superpositions of such focused beams with atoms allow one to probe effects of the quantum interference, which have never played any role in particle scattering.
The projectile beam coherence effects occurring in ion-atom collisions are analyzed on the basis of the recent theory of Karlovets et al. [Phys. Rev. A 92, 052703 (2015)] developed for the elastic scattering of wave packets of particles off a potential field. This theory is generalized to estimate the loss of coherence for inelastically scattered projectiles in ionizing collisions. The results obtained by the suggested model are compared with experimental data for the ionization of hydrogen atoms and molecules by 75-keV proton impact. Significantly improved agreement is observed between the theory and experiment.
While wave-packet solutions for relativistic wave equations are oftentimes thought to be approximate (paraxial), we demonstrate, by employing a null-plane- (light-cone-) variable formalism, that there is a family of such solutions that are exact. A scalar Gaussian wave packet in the transverse plane is generalized so that it acquires a well-defined zzz component of the orbital angular momentum (OAM), while it may not acquire a typical ``doughnut'' spatial profile. Such quantum states and beams, in contrast to the Bessel states, may have an azimuthal-angle-dependent probability density and finite uncertainty of the OAM, which is determined by the packet's width. We construct a well-normalized Airy wave packet, which can be interpreted as a one-particle state for a relativistic massive boson, show that its center moves along the same quasiclassical straight path, and, which is more important, spreads with time and distance exactly as a Gaussian wave packet does, in accordance with the uncertainty principle. It is explained that this fact does not contradict the well-known ``nonspreading'' feature of the Airy beams. While the effective OAM for such states is zero, its uncertainty (or the beam's OAM bandwidth) is found to be finite, and it depends on the packet's parameters. A link between exact solutions for the Klein-Gordon equation in the null-plane-variable formalism and the approximate ones in the usual approach is indicated; generalizations of these states for a boson in the external field of a plane electromagnetic wave are also presented.