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 $(\lambda ,\ell )=(4,4)$. As was found earlier, these bound states are created by the critical eigenstates of the three-body Pauli projector in the 0${}^{+}$ and 2${}^{+}$ functional spaces, respectively. These special eigenstates of the Pauli projector are responsible for the quantum phase transitions from a weakly bound "gas-like" phase to a deep "quantum liquid" phase. 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 ${}^8$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 distance up to $R=30.0$ fm.

arXiv comment12 pages, 4 figures, 2 tables

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