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

• URL: http://arxiv.org/abs/2210.11763v2
• Date: Tue, 25 Oct 2022 09:40:37 GMT
• Title: Spatial Structure of the $^{12}$C Nucleus in a 3$\alpha$ Model with Deep Potentials Containing Forbidden States
• Authors: E. M. Tursunov, M. Z. Saidov and M. M. Begijonov
• Abstract summary: Pauli forbidden states in the $12$C nucleus are studied within the 3$alpha$ model. The estimated probability density functions for the $12$C(0$_1+$) ground and 2$+$ excited bound states show mostly a triangular structure. In the Hoyle resonance the last $alpha$ particle moves far from the doublet at the distance between $R=3.0$ fm and $R=5.0$ fm.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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