Eikonal calculation of electron-capture cross sections from an arbitrary $\mathrm{nlm}$ shell of a hydrogenic target into an arbitrary $n^{\prime }{l}^{\prime }{m}^{\prime }$ shell of a fast bare projectile

Using techniques similar to those previously employed, we apply the eikonal approximation to the evaluation of the cross section for electron capture from an arbitrary $\mathrm{nlm}$ shell of a hydrogenic target atom into an arbitrary $n^{\prime }{l}^{\prime }{m}^{\prime }$ of a fast hydrogenic projectile. The results are obtained in exact analytical closed form. Numerical results are presented for the case ${\mathrm{H}}^{+}+\mathrm{H}\mathrm{}\left(1s\right)\mathrm{}\to \mathrm{H}\left(n^{\prime }{l}^{\prime }{m}^{\prime }\right)+{\mathrm{H}}^{+}$ when $n^{\prime }=2\mathrm{} \mathrm{and} 3$. Comparison is made with the corresponding Oppenheimer-Brinkman-Kramers (OBK) results.