Spatial Structure of the $${}^{{12}}$$C Nucleus in a 3$$\alpha$$ Model with Deep Potentials Containing Forbidden States

Abstract

The spatial structure of the lowest 0 $${}_{1}^{+}$$ , 0 $${}_{2}^{+}$$ , 2 $${}_{1}^{+}$$ and 2 $${}_{2}^{+}$$ states of the $${}^{12}$$ C nucleus is studied within the 3 $$\alpha$$ model with the Buck, Friedrich and Wheatley $$\alpha\alpha$$ potential with Pauli forbidden states in the $$S$$ and $$D$$ waves. The Pauli forbidden states in the three-body system are treated by the exact orthogonalization method. The largest contributions to the ground and excited 2 $${}_{1}^{+}$$ bound states energies come from the partial waves $$(\lambda,\ell)=(2,2)$$ and $$(\lambda,\ell)=(4,4)$$ . In contrast to the bound states, for the Hoyle resonance 0 $${}_{2}^{+}$$ and its analog state 2 $${}_{2}^{+}$$ , dominant contributions come from the $$(\lambda,\ell)=(0,0)$$ and $$(\lambda,\ell)=(2,2)$$ configurations, respectively. The estimated probability density functions for the $${}^{12}$$ C(0 $${}_{1}^{+}$$ ) ground and 2 $${}_{1}^{+}$$ excited bound states show mostly a triangular structure, where the $$\alpha$$ particles move at a distance of about 2.5 fm from each other. However, the spatial structure of the Hoyle resonance and its analog state have a strongly different structure, like $${}^{8}\textrm{Be}+\alpha$$ . In the Hoyle state the last $$\alpha$$ particle moves far from the doublet at the distance between $$R=3.0$$ fm and $$R=5.0$$ fm. In the Hoyle analog 2 $${}_{2}^{+}$$ state the two alpha particles move at a distance of about 15 fm, but the last $$\alpha$$ particle can move far from the doublet at the distanse up to $$R=30.0$$ fm.