Lowest energy spectrum of the $^{ 12}$C nucleus is analyzed in the 3$\alpha$ cluster model with a deep $\alpha\alpha$-potential of Buck, Friedrich and Wheatley with Pauli forbidden states in the $S$ and $D$ waves. The direct orthogonalization method is applied for the elimination of the 3$\alpha$-Pauli forbidden states. The effects of possible first order quantum phase transition are shown in the lowest $^{ 12}$C($0_1^+)$ and $^{ 12}$C($2_1^+)$ states from weakly bound phase to a deep phase. The ground and lowest $2^+$ states of the $^{ 12}$C nucleus in the deep phase are created by the critical eigen states of the Pauli projector for the $0^+$ and $2^+$ three-alpha functional spaces, respectively.
Categories: nucl-th nucl-ex