Existence of a fourth Airy elephant in the nuclear rainbows for $^{12}$C+$^{12}$C scattering

Abstract: The number of gross structures in the 90$^\circ$ excitation function for $^{12}$C+$^{12}$C elastic scattering, often called Airy elephants, has been of great interest. These structures are caused by refractive scattering and are separated by Airy minima. Their importance stems from their close relationship to the interaction potential between two $^{12}$C nuclei, which also describes the molecular resonances of the compound system at lower energies. Although a unique deep potential was usually determined from rainbow scattering at higher energies, a puzzling discrepancy persisted: the energy at which the Airy minimum $A1$ crosses 90$^\circ$ was $E_{c.m.}\approx$67 MeV for $^{12}$C+$^{12}$C. This is remarkably low compared to approximately 100 MeV for both the $^{16}$O+$^{12}$C and $^{16}$O+$^{16}$O systems. This question remained unanswered until the discovery of the secondary rainbow in the $^{12}$C+$^{12}$C system. We report for the first time that the highest energy at which the dynamically generated Airy minimum of the secondary rainbow crosses 90$^\circ$ is about 100 MeV. This demonstrates that the fourth Airy elephant exists between the Airy minimum $A1$ of the primary nuclear rainbow and that, $A1^{(S)}$, of the secondary rainbow. The long-standing problem concerning the Airy minima and Airy elephants has finally been resolved after decades of concern by recognizing the existence of a dynamically generated secondary rainbow in $^{12}$C+$^{12}$C scattering.