Vortices in Coulomb-Born calculations for inner-shell ionization of carbon by electron-impact

The deep minimum in the He(e,2e)He$^+$ measured TDCS [1] have been traced to vortices in the continuum wave function [2]. Using the Coulomb-Born approximation [3], we show that vortices are also present in inner-shell ionization of carbon by electron impact. We considered the energy of the incident electron to be $E_i$ = 1801.2 eV and the scattering angle to be $\theta_f = 4^\circ$. We located a vortex when the momentum of the electron ${\bf k}$ is (-0.547,0,-0.326), where the z-axis is taken along the incident momentum ${\bf K}_i$ and the x-axis is in the plane of the incident and scattered electrons. At this ${\bf k}$, both $\Re$[T-matrix] and $\Im$[T-matrix] are zero. The integral of the velocity field around a closed loop encircling the zero of the T-matrix is $2 \pi$ as is required for a vortex. We also located vortices for out-of-plane geometries ($k_y$ non-zero) for $\theta_f = 4^\circ$ and also in-plane for different scattering angles. Obtaining vortices in the Coulomb-Born approximation is important since they effect the electron momentum distribution and the approximation gives for {\it fixed scattering angles} the high-energy limit for electron-impact inner-shell ionization. [1] A.~J.~Murray and F.~H.~Read, PRA {\bf 47}, 3724 (1993). [2] J.~H.~Macek et. al, PRL {\bf 104}, 033201 (2010). [3] J.~Botero and J.~H.~Macek, PRA {\bf 45}, 154 (1992).