Mass-Energy Compensation Effect of 3\(\alpha\) Hamiltonian

The \(3\alpha\) phenomenological model describes the structure of the carbon-12 nucleus as a cluster of three alpha particles. This model includes a pairwise \(\alpha{-}\alpha\) interaction and a three-body force. To fit the three-body potential, the \({}^{12}\) C data are used, while ensuring that the pair potential reproduces the \(\alpha{-}\alpha\) scattering data. Alternatively, the mass–energy compensation (MEC) effect can be used to simulate the effect of the three-body potential by adjusting the mass of the \(\alpha\) particle within the effective-mass approach. We demonstrate the MEC effect for the \(3\alpha\) ground state by numerically solving the differential Faddeev equation, in which the \(\alpha{-}\alpha\) interaction is described by the Ali–Bodmer potential. The effective masses of \(\alpha\) particles are evaluated for the ground and excited \(0^{+}\) and bound \(2^{+}\) states. We demonstrate a coupling between the ground and first excited \(0^{+}\) states, indicated by an anti-crossing of these energy levels in the energy–mass coordinates. A correspondence between the effective mass and a three-body potential is demonstrated. We discuss the results of the \(0^{+}_{2}\) calculations for various models of the \(\alpha{-}\alpha\) interaction.