Sequential amplitudes in heavy-ion induced two-proton transfer reactions

Angular distributions have been measured for the reactions ${}_{}{}^{88}{}_{}{}^{}\mathrm{Sr}$(${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{14}{}_{}{}^{}\mathrm{C}$)${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}$ and ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}$(${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{14}{}_{}{}^{}\mathrm{C}$)${}_{}{}^{92}{}_{}{}^{}\mathrm{Mo}$ at 80 MeV, and ${}_{}{}^{88}{}_{}{}^{}\mathrm{Sr}$(${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{10}{}_{}{}^{}\mathrm{Be}$)${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}$ and ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}$(${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{10}{}_{}{}^{}\mathrm{Be}$)${}_{}{}^{92}{}_{}{}^{}\mathrm{Mo}$ at 60 MeV. The data were analyzed with full recoil, coupled-channel Born approximation calculations, in which both direct (one-step) and sequential (two-step) reaction routes were considered. Detailed shell model wave functions were used to construct all form factors. The calculated sequential contributions to the cross sections were found to be significantly larger than those associated with direct transfer, and inclusion of the two-step routes substantially improved agreement with experiment. The calculated angular distribution shapes were found to depend on both the intermediate $Q$ value of the sequential process, and also on the microscopic configurations involved in the transfer. In addition, a previously unreported configuration dependence of the recoil corrections was noted in the calculations.