Scattering of twisted electron wave packets by atoms in the Born approximation

The potential scattering of electrons carrying nonzero quanta of the orbital angular momentum (OAM) is studied in a framework of the generalized Born approximation, developed in our recent paper [D. V. Karlovets, G. L. Kotkin, and V. G. Serbo, Phys. Rev. A 92, 052703 (2015)]. We treat these so-called twisted electrons as spatially localized wave packets. The simple and convenient expressions are derived for a number of scattering events in collision of such a vortex electron with a single potential, located at a given impact parameter with respect to the wave packet's axis. The more realistic scenarios are also considered with either localized (mesoscopic) targets or infinitely wide (macroscopic) ones that consist of randomly distributed atoms. Dependence of the electron-scattering pattern on the size and on the relative position of the target is studied in detail for all three scenarios of the single-potential, mesoscopic, and macroscopic targets made of hydrogen in the ground $1s$ state. The results demonstrate that the angular distribution of the outgoing electrons can be very sensitive to the OAM and to kinematic parameters of the focused twisted beams, as well as to composition of the target. Scattering of vortex electrons by atoms can, therefore, serve as a valuable tool for diagnostics of such beams.

Figure 1

The averaged cross section (26), (73) for the scattering of twisted electrons by a macroscopic target, consisting of hydrogen atoms in their ground state. Results are presented for the incident wave packets with width ${\sigma }_{\varkappa }={\varkappa }_0/3$ (blue dashed line) and ${\sigma }_{\varkappa }\ll {\varkappa }_0$ (black solid line), and the opening angle ${\theta }_k={15}^{\circ }$ (top) and ${30}^{\circ }$ (bottom). Results of calculations are compared, moreover, with the prediction obtained for the plane-wave electrons (red dotted line). Here and in Figs. 2, 3, 4, all results are presented for the wave packets with the averaged momentum $p_i=10/a_0$ where $a_0$ is the Bohr radius (the kinetic energy is 1.4 keV).

Figure 2

The azimuthal asymmetry parameter (36) for the scattering of the superposition of two Bessel wave packets by a macroscopic hydrogenic target. The calculations have been performed for two incident electron beams with the width ${\sigma }_{\varkappa }\ll {\varkappa }_0$, difference of OAM projections $\mathrm{\Delta }m=2$, and the opening angles ${\theta }_k={10}^{\circ }$ (red dotted line), ${20}^{\circ }$ (blue dashed line), and ${30}^{\circ }$ (black solid line).

Figure 3

The (normalized) number of events $d\nu \left(\theta \right)/d\mathrm{\Omega }$ with $N_e=1$ from Eq. (71) for the scattering of the Bessel wave packet by a single hydrogen atom, placed at a distance $b=0$ (top) and $b=a_0$ (bottom). Results are presented for the incident packets with width ${\sigma }_{\varkappa }={\varkappa }_0/5$, opening angle ${\theta }_k={10}^{\circ }$, and the OAM projection $m=0$ (black solid line), $m=1$ (blue dashed line), and $m=2$ (red dotted line).

Figure 4

The azimuthal angle dependence of the normalized number of events $d\nu (\theta ,\phi )/d\nu (\theta ,\phi =0)$ for scattering of the Bessel wave packet by a single hydrogen atom, placed at a distance $b=2a_0$. The results are presented for the incident packets with the width ${\sigma }_{\varkappa }={\varkappa }_0/5$, the opening angle ${\theta }_k={10}^{\circ }$, and the OAM projection $m=-2$ (red dotted line), $m=-1$ (blue dashed line), $m=0$ (black solid line), $m=1$ (blue solid line), and $m=2$ (red solid line). We assumed, moreover, that outgoing photons are detected at the polar angles $\theta =1^{\circ }$ (top) and ${20}^{\circ }$ (bottom).

Figure 5

The relative differential cross section $R\left(b_0\right)=d{\sigma }^{\left(\mathrm{mesos}\right)}\left(b_0\right)/d{\sigma }^{\left(\mathrm{PW}\right)}$ from Eq. (49) as a function of the impact parameter $b_0$ of the target center (top) for ${\sigma }_b=1/{\varkappa }_0=10$ nm, $1/{\sigma }_{\varkappa }=50$ nm, ${\theta }_k\ll 1$, and for the following projections of the orbital angular momentum: $m=0$ (black solid line), $m=1$ (blue dashed line), $m=3$ (red dot-dashed line), and $m=5$ (black dotted line). On the bottom panel the normalized density of the incident twisted beam ${\rho }^{\left(m\right)}\left(r_{\perp }\right)/{\rho }^{\left(0\right)}\left(0\right)$ is shown for the same parameters.

Figure 6

The number of events (51) for a wide target as a function of the impact parameter $b_0$ for ${\sigma }_b=10$ nm, $1/{\sigma }_{\varkappa }=2$ nm, ${\theta }_k=\theta =1^{\circ }, \phi ={\phi }_b=0$, and for the following projections of the OAM: $m=0$ (black solid line), $m=50$ (blue dashed line), and $m=100$ (red dotted line).