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 ${}_{12}C+{}_{12}C$ break-up threshold. It has been conjectured that the states have a ${}_{12}C+{}_{12}C$ cluster structure, and play a similar role in accelerating ${}_{12}C+{}_{12}C$ fusion to the manner in which the Hoyle state accelerates production of ${}_{12}C$ 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), ${}_{12}C+{}_{12}C$ 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 ${}_{12}C+{}_{12}C$ cluster states, but are more plausibly interpreted as fragmented cluster states due to coupling to quartet excitations, as background states.