Fresh study of simultaneous electron-photon excitation of a Hydrogen atom based on Bethe-Born approximation

The advent of powerful laser sources has made it possible to observe a relatively large cross section of the excited state of Hydrogen atom. This is due to the effect of joint collisions of a linearly polarized $N$-photon and high-energy electron. For such a process, we evaluate the excitation cross section for geometries, in which the laser field is perpendicular or parallel to the initial momentum of the electron. The second-order, time-dependent perturbation theory together with Bethe-Born approximation suitable for an electron with a large incident energy is employed to obtain the transition amplitude. The amplitudes are calculated for the S-S and S-D transitions of the Hydrogen atom in the Sturmian representation of the non-relativistic Green's function. In particular, we investigate the excitation cross sections for transitions, which have an initial state $1S$ and final state nS with $n\in \lbrace 2,3,4,5\rbrace$. The characteristic dependence of the excitation cross section on the momentum of the projectile is shown and discussed. Our investigation indicates that the Bethe-Born approximation yields reasonable results for the excitation cross section of the simultaneous electron photon excitation process when a high energy projectile is treated.