Vortices for K-shell ionization of carbon by electron impact

Using the Coulomb-Born approximation [1], we obtained a deep minimum in the TDCS for K-shell ionization of carbon by electron impact [2,3]. The minimum is due to a vortex in the velocity field [2,3]. We considered the electron to be ejected in the scattering plane, which we took to be the $xz$-plane. The minimum was obtained for the kinematics of an incident energy $E_i = 1801.2\,eV$, scattering angle $\theta_f = 4^\circ$, energy of ejected electron $E_k = 5.5\,eV$, and angle of the ejected electron $\theta_k = 239^\circ$. We analyzed the importance of various multipole components in an expansion of the Coulomb-Born T-matrix [1,3]. We also considered the electron ejected out of the scattering plane for $E_i = 1801.2\,eV$ and $\theta_f = 4^\circ$ and located the positions of vortices for small but nonzero values of $k_y$, the $y-$component of the momentum of the ejected electron [2]. We constructed the vortex line for the kinematics of $E_i = 1801.2\,eV$ and $\theta_f = 4^\circ$.\\[4pt] [1] J. Botero and J. H. Macek, Phys. Rev. A {\bf 45}, 154 (1992).\\[0pt] [2] S. J. Ward and J. H. Macek, http://meetings.aps.org/link/BAPS.2011.DAMOP.Q1.63.\\[0pt] [3] S. J. Ward and J. H. Macek, Bull. Am. Phys. Soc. {\bf 58}, no. 6, p. 61 (2013), http://meetings.aps.org/link/BAPS.2013.GEC.HW1.19.