Deep minimum in the Coulomb-Born TDCS for inner-shell ionization of carbon by electron impact

Recently, a minimum in the TDCS of electron impact ionization of helium has been explained in terms of a vortex [1]. We have determined the kinematics to obtain a deep minimum due to a vortex in the TDCS for K shell ionization of carbon by electron impact using the Coulomb-Born (CB1) approximation [3]. The deep minimum occurs at an angle of the ejected electron of 239$^\circ$ for an incident energy of $1801.2 eV$, a scattering angle of $4^\circ$, and energy of the ejected electron of $5.5 eV$. At the angle of the minimum, both the real and imaginary parts of the T-matrix are zero. The integral of the velocity field around a closed path encircling the vortex position is $2 \pi$ [2]. Following the treatment of Ref.~[3], we decomposed both the Born (B1) and the CB1 T-matrix into their multipole components [4]. The $\ell=1$, $m=\pm 1$ CB1 multipole components are important in determining the shape of the CB1 angular distribution.\\[4pt] [1] J.H. Macek, J.B. Sternberg, S.Y. Ovchinnikov amd J.S. Briggs, Phys. Rev. Lett. {\bf 104}, 033201 (2010). [2] S.J. Ward and J.H. Macek, http://meetings.aps.org/link/BAPS.2011.DAMOP.Q1.63. [3] J. Botero and J.H. Macek, Phys. Rev. A {\bf 45}, 154 (1992). [4] S.J.Ward and J.H. Macek, Bull. Am. Phys. Soc. {58}, no. 6, p. 61 (2013).