Dynamical $\alpha$-cluster model of ${}^{16}\mathrm{O}$

We calculate the low-lying spectrum of the ${}^{16}\mathrm{O}$ nucleus using an $\alpha$-cluster model which includes the important tetrahedral and square configurations. Our approach is motivated by the dynamics of $\alpha$-particle scattering in the Skyrme model. We are able to replicate the large energy splitting that is observed between states of identical spin but opposite parities. We also provide a novel interpretation of the first excited state of ${}^{16}\mathrm{O}$ and make predictions for the energies of $6^{-}$ states that have yet to be observed experimentally.

Figure 1

Scattering mode of four $\alpha$ particles in the Skyrme model. Each time step shows a surface of constant energy density which is colored according to the field value as in [16].

Figure 2

$\alpha$ particles are restricted to lie on a surface with six punctures. Regions with the same coloring are related by $D_2$ symmetry. The scattering mode in Fig. 1 is represented by the thick black line.

Figure 3

Relation between $\mathcal{M}$ (left) and $\mathcal{F}$ (right). Tetrahedral configurations are at the points where three colored regions meet, while the square configurations are at points where four colored regions meet. The scattering mode in Fig. 1 is represented by the thick black lines.

Figure 4

Vibrational wave functions which lie in the trivial representation. From left to right: ground state, first excited state, and lowest-lying state with negative parity.

Figure 5

Vibrational wave functions which lie in the sign representation. From left to right: lowest-energy state, first excited state, and lowest-lying state with positive parity.

Figure 6

Lowest-energy vibrational wave functions which lie in the standard representation. Clockwise from top left: ${\varphi }_1$ with positive parity, ${\varphi }_2$ with positive parity, ${\varphi }_2$ with negative parity, and ${\varphi }_1$ with negative parity.

Figure 7

Energy spectrum of our model up to 12.7 MeV compared with all the low-lying experimentally observed states [19]. States are colored according to the type of vibrational wave function from which they arise. The partially dotted lines represent the trivial irrep (black), sign irrep (green), and standard irrep (blue) of $S_3$. Full vertical lines represent states identified with the $F$ vibration (red) and $A$ vibration (yellow). The $1^{-}$ state at 9.6 MeV is unexplained in our model.