Peculiarity of the $${}^{{12}}$$C(0$${{}^{+}}$$) and $${}^{{12}}$$C(2$${{}^{+}}$$) Energy Spectrum in a 3$$\alpha$$ Model

Abstract

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.