$\mathrm{Triple}-\alpha$ continuum structure and Hoyle resonance of ${}^{12}\mathrm{C}$ using the hyperspherical slow variable discretization

We develop a method for calculating the bound and continuum energy spectrum of three particles interacting through both short-range and Coulomb potentials. Our method combines hyperspherical coordinates with the slow variable discretization approach. A complex absorbing potential is employed to describe accurately the continuum wave functions. The method is well known in atomic and molecular physics. It is extended here to nuclear physics, with a special emphasis on the long-range Coulomb interaction. The method is applied to compute the energy spectrum of ${}^{12}\mathrm{C}$ in a $3\alpha$-particle model, focusing on an accurate calculation of the Hoyle resonance width of the narrow near-threshold $J_n^{\pi }=0_2^{+}$ state, which plays an important role in stellar nucleosynthesis. We employ an effective $\alpha \text{−}\alpha$ interaction potential which reproduces both the energy and width of ${}^8\mathrm{Be}$, while a three-body force is added in order to fix the ${}^{12}\mathrm{C}$ energy levels at the experimental values. We also analyze the structure of the bound and resonance states by calculating the wave functions and one-dimensional distribution functions.

Figure 1

Two-dimensional contour plot of the potential energy surface without three-body term, ${\sum }_{i<j}^{3}v\left(r_{ij}\right)$, at a fixed hyper-radius $R=10$ fm as a function of the hyperangles $\theta$ and $\phi$ for the three-$\alpha$ system. The solid lines show the lowest contour line shown here ($-4$ MeV); the dashed lines show the second lowest contour line ($-2$ MeV). The other lines correspond to 0 MeV and 2 MeV. The closed circles indicate the two-body coalescent points.

Figure 2

The 10 lowest adiabatic hyperspherical potential curves $U_{\nu }\left(R\right)$ of the three-$\alpha$ system for (a) $J^{\pi }=0^{+}$ and (b) $J^{\pi }=2^{+}$. In both panels (a) and (b), the inset shows the crossings between the potential curves corresponding to the ${}^8\mathrm{Be}$ resonance.

Figure 3

Contributions of the centrifugal potential $\langle \langle {\Phi }_{\nu }|{\Lambda }^2/\left(2\mu R^2\right)|{\Phi }_{\nu }\rangle \rangle$, the nuclear potentials $\langle \langle {\Phi }_{\nu }|{\sum }_{ij}{v}_{\alpha \alpha }\left(r_{ij}\right)|{\Phi }_{\nu }\rangle \rangle$, and the Coulomb potentials $\langle \langle {\Phi }_{\nu }|{\sum }_{ij}{v}_C\left(r_{ij}\right)|{\Phi }_{\nu }\rangle \rangle$ to the two $0^{+}$ lowest potential curves (a) $U_1\left(R\right)$ ($\nu =1$) and (b) $U_2\left(R\right)$ ($\nu =2$), as functions of the hyper-radius $R$.

Figure 4

The squared hyper-radial wave functions $|F_{\nu }{\left(R\right)|}^2$ for the bound and resonance states of the three-$\alpha$ system. (a) $J_n^{\pi }=0_1^{+}$ state: the first $\nu =1$ hyperradial component in the expansion in Eq. (7) is shown; (b) $J_n^{\pi }=0_2^{+}$ state: the first $\nu =1$ and second $\nu =2$ hyper-radial components in the expansion in Eq. (7) are shown respectively as solid and dashed lines; and (c) the same as for panel (b) but for $J_n^{\pi }=2_1^{+}$ state. The $\nu =2$ components drawn here are magnified 10 times.

Figure 5

Contour plots of the 2D probability density functions for the $J_n^{\pi }=0_1^{+}$ and $0_2^{+}$ states of the three-$\alpha$ system (a) in $R\theta$ space, (b) in $\theta \phi$ space, and (c) in $R\phi$ space. The $0_1^{+}$ state is indicated by solid lines and the $0_2^{+}$ state by dashed lines.

Figure 6

(a) Pair distribution $P\left(r\right)$ and (b) angle cosine distribution $P(cos\vartheta )$ for the $J_n^{\pi }=0_1^{+}$, $0_2^{+}$, and $2_1^{+}$ states.