A vortex line for K-shell ionization of a carbon atom by electron impact

We obtained using the Coulomb-Born approximation [1] a deep minimum in the TDCS for K-shell ionization of a carbon atom by electron impact for the electron ejected in the scattering plane [2]. The minimum is obtained for the kinematics of the energy of incident electron $E_i$ = 1801.2 eV, the scattering angle $\theta_f$ = 4$^\circ$, the energy of the ejected electron $E_{\bf k}$ = 5.5 eV, and the angle for the ejected electron $\theta_k$ = 239$^\circ$. This minimum is due to a vortex in the velocity field. At the position of the vortex, the nodal lines of ${\rm Re}[T]$ and ${\rm Im}[T]$ intersect. We decomposed the CB1 T-matrix into its multipole components [1] for the kinematics of a vortex, taking the $z^\prime$-axis parallel to the direction of the momentum transfer vector. The $m = \pm 1$ dipole components are necessary to obtain a vortex. We also considered the electron to be ejected out of the scattering plane and obtained the positions of the vortex for different values of the y-component of momentum of the ejected electron, $k_y$. 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, submitted to Phys. Rev. A.