Projection operator model for heavy-ion resonance parameters below the Coulomb barrier and its application to ${}_{}{}^{12}{}_{}{}^{}$C${+}^{12}$C

A nuclear projection model is developed for heavy-ion reactions in the sub-Coulomb region. Generalized doorways are introduced as the set of resulting states obtained by coupling the shape resonances to the single and mutual excitation channels. The model is applied to the ${}_{}{}^{12}{}_{}{}^{}$C${+}^{12}$C reaction by treating the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ nucleus as deformed, and the ion-ion optical potential is taken as that of Kondo et al. with a soft repulsive core. The energies and continuum widths of the resonances below and near the Coulomb barrier are calculated and found to be consistent with observation. Intermediate structure is mainly caused by the shape resonances which are fragmented into the various generalized doorways. However, the major strength of the elastic channel remains concentrated around the energy where a potential resonance exists in the elastic channel. Levels are predicted down to 2.91 MeV c.m. and provide a first step in an impending extrapolation of ${}_{}{}^{12}{}_{}{}^{}$C${+}^{12}$C reaction rates to astrophysical energies.