Generalized Coulomb-projected Born-approximation cross sections for atomic excitation by charged-particle impact

An expression is given for the ($n_i$, $l_i=0\mathrm{}\to n_f$, $l_f=0\mathrm{}$) excitation cross section in terms of a single one-dimensional integral with the aid of a transformation to prolate spheroidal coordinates. Results are given for the $1s-2s$ excitation of arbitrary-$Z$ hydrogenic targets. For our choice of wave functions the cross sections are found to scale as $Z^{-4\mathrm{}}$ at fixed $\frac{E_{\mathrm{inc}}}{Z^2}$. The principal difference between the Coulomb-projected Born-approximation (CPBA) and Born results is in the large-angle portion of the differential cross section. This, however, leads to CPBA and Born momentum-transfer cross sections which differ by less than 20% at high energies for $1s-2s$ transitions.