Further evidence for a dynamically generated secondary bow in ${}^{13}\mathrm{C}+{}^{12}\mathrm{C}$ rainbow scattering

The existence of a secondary bow is confirmed for ${}^{13}\mathrm{C}+{}^{12}\mathrm{C}$ nuclear rainbow scattering in addition to the ${}^{16}\mathrm{O}+{}^{12}\mathrm{C}$ system. This is found by studying the experimental angular distribution of ${}^{13}\mathrm{C}+{}^{12}\mathrm{C}$ scattering at the incident ${}^{13}\mathrm{C}$ energy $E_L=250\mathrm{MeV}$ with an extended double-folding (EDF) model that describes all the diagonal and off-diagonal coupling potentials derived from the microscopic wave functions for ${}^{12}\mathrm{C}$ using a density-dependent nucleon-nucleon force. The Airy minimum at $\theta \approx {70}^{\circ }$, which is not reproduced by a conventional folding potential, is revealed to be a secondary bow generated dynamically by a coupling to the excited state $2^{+}$ (4.44 MeV) of ${}^{12}\mathrm{C}$. The essential importance of the quadruple $Y2$ term (reorientation term) of potential of the excited state $2^{+}$ of ${}^{12}\mathrm{C}$ for the emergence of a secondary bow is found. The mechanism of the secondary bow is intuitively explained by showing how the trajectories are refracted dynamically into the classically forbidden angular region beyond the rainbow angle of the primary rainbow.

Figure 1

Angular distributions in (a) ${}^{13}\mathrm{C}+{}^{12}\mathrm{C}$ scattering at $E_L=250\mathrm{MeV}$ and (b) ${}^{16}\mathrm{O}+{}^{12}\mathrm{C}$ scattering at $E_L=330\mathrm{MeV}$ obtained using the CC method (blue [gray] solid lines) and single-channel (dotted lines) calculations are displayed in comparison with the experimental data (points) from Refs. [30, 41]. The far-side and near-side components of the CC calculations are shown by the green (gray) dashed lines and the brown (dark gray) dash-dotted lines, respectively. The CC results are obtained in panel (a) with coupling to the $2^{+}$ and $3^{-}$ states of ${}^{12}\mathrm{C}$ and in panel (b) with coupling to the $2^{+}$ and $3^{-}$ states of ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$. The potential parameters used in panel (b) are taken from Ref. [15].

Figure 2

Angular distributions in ${}^{13}\mathrm{C}+{}^{12}\mathrm{C}$ scattering at $E_L=250\mathrm{MeV}$ calculated using the CC method and a single channel (black dotted lines) are displayed in comparison with the experimental data (points) [30]. The blue (gray) solid, brown (dark gray) dash-dotted, and green (gray) long-dashed lines represent the CC results with coupling to both the $2^{+}$ and $3^{-}$ states of ${}^{12}\mathrm{C}$, those with coupling to the $2^{+}$ state only and those with coupling to the $3^{-}$ state only, respectively. The upper figures $\left(W=0\right)$ are calculated by switching off the imaginary potential. The purple (dark gray) short-dashed line represents the CC calculations with coupling to the $2^{+}$ state without the quadruple $Y2$ term (reorientation term) for ${}^{12}\mathrm{C}(2^{+})$.

Figure 3

The illustrative figures of refractive trajectories by the attractive potential in nuclear rainbow scattering. (a) Refraction in the case without coupling. The refracted angle ${\theta }_R$ is a rainbow angle for the primary nuclear rainbow caused by the optical potential (Luneburg lens) of the nucleus (indicated by a circle). The angular region $\theta \le {\theta }_R$ is the bright side of the primary nuclear rainbow. $\theta >{\theta }_R$ is not allowed to refract classically and is the dark side. (b) Refraction in the case with coupling to the collective excited $2^{+}$ state of ${}^{12}\mathrm{C}$. The trajectories of a secondary bow (red [gray] line) strongly refracted (refracted angle ${\theta }_S$) beyond ${\theta }_R$ by the additional attractive potential in the inner region induced dynamically by the coupling to the excited state of the nucleus.

Figure 4

The energy evolution of the Airy structure in the angular distributions of the cross sections in ${}^{13}\mathrm{C}+{}^{12}\mathrm{C}$ scattering calculated with coupling to the $2^{+}$ and $3^{-}$ states of ${}^{12}\mathrm{C}$ (blue [gray] solid lines) and the far-side components (green [gray] dashed lines) are displayed at $E_L=200$, 260, and 330 MeV in comparison with the experimental data (points) from Ref. [40]. The dotted lines represent the single-channel calculation.