Cluster states and their potential contribution to carbon burning in massive stars

Background: A recent inelastic $\alpha$-scattering experiment [P. Adsley et al., Phys. Rev. Lett. 129, 102701 (2022)] found $0^{+}$ resonances in ${}^{24}\mathrm{Mg}$ on and above the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ break-up threshold. It has been conjectured that the states have a ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ cluster structure, and play a similar role in accelerating ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion to the manner in which the Hoyle state accelerates production of ${}^{12}\mathrm{C}$ in massive stars.

Purpose: We aim to build up a quantitative theoretical basis for the considerations of the Hoyle-state paradigm, by calculating the distribution of the $0^{+}$ states in the shell model, as well as in the relevant cluster models.

Methods: We determine the spectrum of excited $0^{+}$ states in ${}^{24}\mathrm{Mg}$ nucleus using multiconfigurational dynamical symmetry calculations leading to a unified description of the quartet (or shell), ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ and ${}^{20}\mathrm{Ne}+{}^4\mathrm{He}$ cluster configurations.

Results: The density of $0^{+}$ states in the quartet spectrum is comparable to that found in experiment; however, the density of cluster states is considerably less.

Conclusions: The recently observed $\alpha$-scattering resonances do not seem to be simple ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ cluster states, but are more plausibly interpreted as fragmented cluster states due to coupling to quartet excitations, as background states.

Figure 1

Currently known experimental energies of positive parity states as a function of $J(J+1)$ in the ${}^{24}\mathrm{Mg}$ nucleus [25]. Green dots indicate definite spin-parity, while red stars show the uncertain ones. Red lines between states indicate gamma transitions. The black lines show the assumed rotational bands, which we have labeled with $K^{\pi }$ quantum numbers. Only the ground state (GS) and $2_1^{+}$ bands are available in the experimental database [25].

Figure 2

Currently known experimental energies of negative parity states as a function of $J(J+1)$ in the ${}^{24}\mathrm{Mg}$ nucleus [25]. The notations are the same as in Fig. 1.

Figure 3

The spectrum of the semimicroscopic algebraic quartet model (upper part) in comparison with experimental data for the ${}^{24}\mathrm{Mg}$ nucleus (lower left part) and the results of Bloch-Brink $\alpha$ cluster model (lower right part). Experimental bands are labeled by $K^{\pi }$ quantum numbers, and the model states by $n(\lambda ,\mu )K^{\pi }$ labels. The width of the arrows between the states is proportional to the strength of the $E2$ transition.

Figure 4

Theoretical spectra of even-momentum states of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ clusterization compared to experimental data in different energy windows. For each angular momentum, the bottom of the energy window is $\min \left(E_{exp}\right)-0.5$ MeV and the top is $\max \left(E_{exp}\right)+0.5$ MeV. The branchings indicate states that are close to each other. Each column shows the experimental states on the right and the theoretical states on the left.

Figure 5

The landscape of the quartet and cluster band-heads in the ${}^{24}\mathrm{Mg}$ nucleus.

Figure 6

$0^{+}$ spectra in quartet and cluster model spaces in ${}^{24}\mathrm{Mg}$. The lower line represents the ${}^{20}\mathrm{Ne}+\alpha$ threshold energy and the upper one the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ threshold energy. The second from the right spectrum represents the core-plus-alpha configuration with excited ${}^{20}\mathrm{Ne}$ (see text for further explanation). The branchings denote states with multiplicities greater than 1. Dashed lines indicate the states also visible in the low-energy spectra, and red dotted lines indicate the states found in [2].