Rutherford scattering of electron vortices

By considering a cylindrically symmetric generalization of a plane wave, the first-order Born approximation of screened Coulomb scattering unfolds two new dimensions in the scattering problem: transverse momentum and orbital angular momentum of the incoming beam. In this paper, the elastic Coulomb scattering amplitude is calculated analytically for incoming Bessel beams. This reveals novel features occurring for wide-angle scattering and quantitative insights for small-angle vortex scattering. The result successfully generalizes the well-known Rutherford formula, incorporating transverse and orbital angular momentum into the formalism.

Figure 1

Schematic of the convergent beam scattering experiment. The relation of the transverse momenta and the aperture dimensions is shown, and an on-axis pinhole detector is shown. See Sec. 5 for the details of such an experiment.

Figure 2

Transverse wave functions for the Bessel [Eq. (11)], aperture far field, and Laguerre–Gaussian [Eq. (15)] beams, using the HSV color map as shown in the legend at the bottom. They all contain an $\ell =1$ vortex. The Laguerre–Gaussian has two intensity lobes ($n=2$), clearly showing the strongest localization of the three in the transverse plane.

Figure 3

Poles and contours of the integrand in Eq. (20). The red (top) contour is used for $\ell >0$. The blue (bottom) contour is used for $\ell <0$. In both cases, there is an $\ell$th-order pole in the other half of the complex plane due to the first factor of Eq. (20). Only one pole of those due to the second factor in Eq. (20) lies in a contour for every value for $\ell$.

Figure 4

The elastic-scattering cross section for a screened Coulomb potential and Bessel beam for various values of $\ell$, $\kappa =25a_0^{-1}$, $Z=26$, and $k_z=169a_0^{-1}$ (corresponding to a 300 kV acceleration voltage). See Sec. 4c for details on the screening potential model used for an atom with atomic number $Z$. The scattering amplitude becomes smaller with larger $\ell$. Panel (a) shows only the ${(\cdots )}^{\left|\ell \right|}$ factor. Panel (b) shows the full equation (24).

Figure 5

The elastic scattering cross section for a screened Coulomb potential and Bessel beam for various values of $\kappa$ (in $a_0^{-1}$), $\ell =0,\pm 1$, $Z=26$, and $k_z=169a_0^{-1}$ (corresponding to a 300 kV acceleration voltage). See Sec. 4c for details on the screening potential model used for an atom with atomic number $Z$. The scattering amplitude becomes smaller with larger $\ell$, due to the larger spatial spread of higher-$\ell$ Bessel functions.