Unified description of scattering and fusion phenomena in heavy-ion collisions

Abstract

An analytical recursive formula of the partial-wave scattering matrix for the total effective complex potential of nucleus-nucleus collisions is derived to conveniently analyze the data of angular variations of elastic scattering cross sections. Further, another expression of cross sections for the absorption from arbitrarily small intervals is derived. This leads to the explanation of the fusion cross section (${\ensuremath{\sigma}}_{\mathrm{fus}}$) data at various incident center-of-mass energies ${E}_{\mathrm{c}.\mathrm{m}.}$ by collecting the absorption contributions in the interior region of the effective potential. This concept is akin to that used by Udagawa et al. in the calculation of fusion cross sections in elastic channels. The interaction potential considered in the analysis is energy independent and by virtue of its weakly absorbing character it supports resonance states in different partial-wave trajectories. Consequently, occurrence of these resonances is shown to be the physical origin of the observed oscillatory structure in the variation respect to energy of the quantity $D({E}_{\mathrm{c}.\mathrm{m}.})={d}^{2}({E}_{\mathrm{c}.\mathrm{m}.}{\ensuremath{\sigma}}_{\mathrm{fus}})/{\mathit{dE}}_{\mathrm{c}.\mathrm{m}.}^{2}$, the second derivative of the product ${E}_{\mathrm{c}.\mathrm{m}.}{\ensuremath{\sigma}}_{\mathrm{fus}}$ with respect to ${E}_{\mathrm{c}.\mathrm{m}.}$. In this article, we investigate two well-known cases of heavy-ion collisions, namely $^{12}\mathrm{C}+^{208}\mathrm{Pb}$ and $^{16}\mathrm{O}+^{208}\mathrm{Pb}$, and obtain simultaneous and very successful explanations of cross sections for elastic scattering and fusion and the results of $D({E}_{\mathrm{c}.\mathrm{m}.})$. These results obtained by using a somewhat novel and convenient method demonstrate the unified description of scattering and fusion for interacting heavy-ion systems.