Excitation of hydrogenlike ions from ground state to arbitrarypstate by electron impact

We have formulated a generalized method for the evaluation of the cross sections for $1s$-$\mathrm{np}$ excitation of hydrogenlike ions by electron impact both in Coulomb-Born and Coulomb-Born-Oppenheimer (CBO) approximations. In this formulation we have replaced the associated Laguerre polynomial for the final-state wave function by an integral representation. The main advantage of our method is that the higher-order parametric differentiations usually adopted in order to obtain the amplitudes for the higher values of the principal quantum number $n$ are avoided by one-dimensional integration. To our knowledge, this is the first attempt to calculate the electron-ion excitation cross section in the CBO approximation for an arbitrary $p$ state without any recourse to partial-wave analysis. We have also calculated asymptotic scaled cross sections for the limit $n\to \infty$ at different energies. The direct-excitation results are in good agreement with the existing theoretical findings of Tully for different $n$ values in the intermediate- and high-energy regions.