Existence of inelastic supernumerary nuclear rainbow in ${}^{16}\mathrm{O}+{}^{12}\mathrm{C}$ scattering

The existence of a supernumerary nuclear rainbow in inelastic scattering is reported. This is done by studying inelastic ${}^{16}\mathrm{O}$ scattering from ${}^{12}\mathrm{C}$, exciting the $2^{+}$ (4.44 MeV) state of ${}^{12}\mathrm{C}$ and elastic scattering at the incident energies in the range 124–200 MeV, using the coupled channels method. An extended double folding potential is used. This is derived from realistic wave functions for ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$ calculated with a microscopic $\alpha$ cluster model and a finite-range density-dependent nucleon-nucleon force. Excitations to the $2^{+}$ (4.44 MeV), $3^{-}$ (9.64 MeV), and $4^{+}$ (14.08 MeV) states of ${}^{12}\mathrm{C}$, and the $3^{-}$ (6.13 MeV) and $2^{+}$ (6.92 MeV) states of ${}^{16}\mathrm{O}$ are included in the coupled channels calculations. The emergence of the supernumerary bow is understood by the properties of both the Luneburg-lens-like potential in the internal region and diffuse attraction in the outer region. The existence of a supernumerary rainbow for inelastic scattering in addition to the existence of a dynamically created secondary rainbow and a dynamically refracted primary rainbow for elastic scattering, which are not observed in meteorological rainbows, further deepens the understanding of nuclear rainbows.

Figure 1

Angular distributions in ${}^{16}\mathrm{O}+{}^{12}\mathrm{C}$ scattering at $E_L=124$, 170, 181 (180), and 200 MeV calculated with the potentials in Table 1 using the coupled channels method (solid lines) are compared with the experimental data (filled circles with vertical error bars) taken from Refs. [16, 28, 40], (a) elastic scattering and (b) inelastic scattering to the $2^{+}$ state of ${}^{12}\mathrm{C}$. The dotted lines are calculated by switching off the imaginary potentials in Table 1 in (a) and only the imaginary potentials in the $2^{+}$ channel are switched off in (b). In (a) far-side components calculated by switching off the imaginary potentials are displayed by with dashed lines for $E_L=124$ and 170 MeV.

Figure 2

Energy evolution of the angular distributions in ${}^{16}\mathrm{O}+{}^{12}\mathrm{C}$ scattering calculated by switching off the imaginary potential for $E_L=170–124$ MeV at intervals of 10 MeV using the coupled channels method are displayed for (a) elastic (ratio to Rutherford scattering) and (b) inelastic scattering to the $2^{+}$ state of ${}^{12}\mathrm{C}$. For the inelastic scattering in (b) only the imaginary potential in the $2^{+}$ channel is switched off. In (a) the solid lines represent the far-side components. The labels $A1,A2$, and $A3$ indicate the position of the Airy minimum of the order one, two, and three, respectively.

Figure 3

The far-side contribution of the angular distribution (ratio to Rutherford scattering) in ${}^{16}\mathrm{O}+{}^{12}\mathrm{C}$ elastic scattering at $E_L=124$ MeV calculated using the coupled channels method and the experimental data (filled circles) [40]. The dashed line, solid line and the dotted line represent the coupled channels calculation with $W=16$ (Table 1), $W=4$, and $W=0$, respectively.

Figure 4

Comparison of the ${}^{16}\mathrm{O}+{}^{12}\mathrm{C}$ folding potential for the inelastic channel to the ${}^{12}\mathrm{C}$($2^{+}$) state (open circles) at 124 MeV with that for elastic channel (blue solid line) and a Luneburg lens potential (red filled circles) with $V_0=233$ MeV and $R_0=4.26$ fm. The Luneburg lens potential is given by [46] $V(R\le R_0)=V_0\left(R^2/R_0^2-1\right)$ and $V(R>R_0)=0$ with $R_0$ being the size of the lens.

Figure 5

The positions of the Airy minimum observed in inelastic scattering to the $2^{+}$ state of ${}^{12}\mathrm{C}$ (filled squares) and elastic scattering (filled circles) of ${}^{16}\mathrm{O}+{}^{12}\mathrm{C}$ are displayed as a function of the inverse center-of-mass (c.m.) energies. The labels $A1$ and $A2$ indicate the order of the Airy minima. The lines are to guide the eye.