$3\alpha$-cluster structure of the $0^{+}$ states in ${}^{12}\mathrm{C}$ and the effective $\alpha \text{‐}\alpha$ interactions

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 $\alpha \text{‐}\alpha$ 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 $\alpha \text{‐}\alpha$ potential is studied in order to impose restrictions on the potentials.

Figure 1

Three lowest effective potentials $U_n\left(\rho \right)$. Shown are also the asymptotic two-cluster dependence $U_1\left(\rho \right)\approx E_{2\alpha }+\tilde{q}∕\rho$ (thin dashed line) and the energy of the $0_2^{+}$ state $E=0.3795\mathrm{MeV}$ (thin horizontal line).

Figure 2

The first eigenfunction ${\Phi }_1(\xi ,\phi ,\rho )$ at different $\rho$.