Simple Approximation for Multistep Amplitudes

A surface approximation to the multistep amplitude for direct reactions is presented. Within this approximation, the distorted-wave Born-approximation series is summed in closed form, and the result is tested by comparing with exact coupled-reaction-channels calculations for ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}(p, d, p){}_{}{}^{17}{}_{}{}^{}\mathrm{O}$ and ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}(d, p, d){}_{}{}^{16}{}_{}{}^{}\mathrm{O}$. The approximation gives a good representation of the exact results, especially for the ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}(d, p, d){}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ process which is dominated by strong absorption of the deuteron.