The $0^{ +}$ states of $^{ 12}\mathrm{ C}$ are considered within the framework of the microscopic three-$\alpha$-cluster model. The main attention is paid to accurate calculation of the width of the extremely narrow near-threshold $0^+_2$ state which plays a key role in stellar nucleosynthesis. It is shown that the $0^{ +}_2$-state decays by means of the sequential mechanism ${ ^{ 12}\mathrm{ C}} \to \alpha+{ ^8\mathrm{ Be}} \to 3\alpha$. Calculations are performed for a number of effective $\alpha - \alpha$ potentials which are chosen to reproduce both energy and width of $^8\mathrm{ Be}$. The parameters of the additional three-body potential are chosen to fix both the ground and excited state energies at the experimental values. The dependence of the width on the parameters of the effective $\alpha - \alpha$ potential is studied in order to impose restrictions on the potentials.
Categories: nucl-th