Ground State of Three Alpha Particles

The Faddeev equation is applied to solve for the energy of three $\alpha$ particles using a local static two-body potential obtained by fitting scattering phase shifts. A single bound state is found with a binding energy of 2.79 MeV. Although this value is not very close to the ground-state energy of ${\mathrm{C}}^{12}$, the existence of a $3-\alpha$ bound state with a binding energy of several MeV indicates that the main structure of ${\mathrm{C}}^{12}$ is like a composite of three $\alpha$ particles. The effect of inelastic processes is estimated in a rough approximation by using a two-channel two-body potential. It is shown that they can easily increase the binding energy of the $3-\alpha$ bound state by several MeV.