Nuclear rainbow of the symmetric nucleus-nucleus system: Interchange of the nearside and farside scattering

Extensive elastic scattering data measured at energies around 10 to 20 MeV/nucleon for some light identical systems, like ${}_{12}C+{}_{12}C$ and ${}_{16}O+{}_{16}O$, were shown to exhibit the nuclear rainbow pattern of broad Airy oscillations of the elastic scattering cross section at medium and large angles. Because of the identity of the scattered projectile and recoiled target, the smooth rainbow pattern at angles around and beyond ${\theta }_{c.m.}\approx {90}^{\circ }$ is strongly deteriorated by the boson exchange in the ${}_{12}C+{}_{12}C$ and ${}_{16}O+{}_{16}O$ systems at low energies. The exchange symmetry of two identical nuclei implies the Mott interference of the direct and exchange scattering amplitudes, which destroys the nuclear rainbow pattern. The nuclear rainbow features in the elastic scattering of two identical nuclei have been discussed so far based on the nearside-farside (NF) decomposition of the scattering amplitude given by an optical model calculation neglecting the projectile-target exchange symmetry. Moreover, the NF decomposition method was developed in the 1970s by Fuller for nonidentical dinuclear systems only, and the details of how the exchange symmetry of an identical system affects the evolution of nuclear rainbow remain unexplored. For this purpose, the Fuller method is generalized in the present work for the elastic scattering of two identical (spin-zero) nuclei, with the projectile-target exchange symmetry taken explicitly into account. The results obtained for elastic ${}_{12}C+{}_{12}C$ and ${}_{16}O+{}_{16}O$ scattering at low energies show that the exchange symmetry results in a symmetric interchange of the nearside and farside scattering patterns at angles passing through ${\theta }_{c.m.}={90}^{\circ }$, which requires a more subtle interpretation of nuclear rainbow. We found further that a similar NF interchange also occurs in a nonidentical nucleus-nucleus system with the core-core symmetry at low energies, where the elastic cross section at backward angles is due mainly to the elastic transfer of cluster or nucleon between two identical cores. This interesting effect is illustrated in the elastic ${}_{16}O+{}_{12}C$ scattering at low energies where the elastic $\alpha$ transfer between two ${}_{12}C$ cores has been proven to enhance the elastic cross section at backward angles.