Coupled-Channels Calculation of the Elastic and Inelastic Scattering of Protons by Carbon-12

The elastic scattering of protons by ${\mathrm{C}}^{12}$ exhibits the following resonances with an appreciable fraction of the single-particle width: $E_p=0.461\mathrm{}$ MeV ($s_{\frac{1}{2}}$), 1.748 MeV ($d_{\frac{5}{2}}$), and $\approx 6.7$ MeV ($d_{\frac{3}{2}}$). A narrow $d_{\frac{5}{2}}$ resonance is observed at 4.808 MeV and an anomaly in the $d_{\frac{3}{2}}$ phase shift at $\approx 5.3$ MeV. It is demonstrated by means of a coupled-channel calculation that these last two features are due to the resonance in the inelastic channel (${\mathrm{C}}^{12*}+p$) corresponding to the 0.461-MeV $s$-wave resonance in the elastic channel. The experimental $s$- and $d$-wave phase shifts and damping parameters are accurately reproduced after adjustment of the depth of square-well potentials. The coupled-channel calculation also demonstrates that a resonance may be switched from one channel to another by changing the strength of the coupling.