Extending the Hoyle-State Paradigm to ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ Fusion

Carbon burning is a key step in the evolution of massive stars, Type 1a supernovae and superbursts in x-ray binary systems. Determining the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion cross section at relevant energies by extrapolation of direct measurements is challenging due to resonances at and below the Coulomb barrier. A study of the ${}^{24}\mathrm{Mg}(\alpha ,{\alpha }^{\prime }){}^{24}\mathrm{Mg}$ reaction has identified several $0^{+}$ states in ${}^{24}\mathrm{Mg}$, close to the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ threshold, which predominantly decay to ${}^{20}\mathrm{Ne}(\text{ground state})+\alpha$. These states were not observed in ${}^{20}\mathrm{Ne}(\alpha ,{\alpha }_0){}^{20}\mathrm{Ne}$ resonance scattering suggesting that they may have a dominant ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ cluster structure. Given the very low angular momentum associated with sub-barrier fusion, these states may play a decisive role in ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion in analogy to the Hoyle state in helium burning. We present estimates of updated ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction rates.

Figure 1

Analogy between the Hoyle state in ${}^{12}\mathrm{C}$ as the driver of $3\alpha$ fusion and candidate cluster states in ${}^{24}\mathrm{Mg}$ as drivers of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion. Left: The Hoyle state in ${}^{12}\mathrm{C}$ with its associated E0 decay to the ground state, suggesting a large radial difference between the states. The illustration to the left of the experimental state shows the Hoyle state as a bent arm of three $\alpha$ particles close to the $3\alpha$ threshold. To the right, the Gamow window is shown (in orange) for helium burning around $T_9=0.1$. Right: A subset of the excited ${}^{24}\mathrm{Mg}$ $0^{+}$ states that are strongly populated by ${}^{24}\mathrm{Mg}(\alpha ,{\alpha }^{\prime }){}^{24}\mathrm{Mg}$. The size of the bars is proportional to the fraction of the energy-weighted sum rule (EWSR). On the left are schematic diagrams of the predicted ${}^{24}\mathrm{Mg}$ cluster structures and their breakup thresholds. (For simplicity of presentation, the ${}^{16}\mathrm{O}+{}^8\mathrm{Be}$ configuration is omitted.) The 6.432-MeV $0^{+}$ state suggested to be a highly deformed counterpart to the ${}^{24}\mathrm{Mg}$ ground state is indicated along with its associated E0 decay [8]. On the right are the Gamow windows corresponding to superbursts (pink; $T_9=0.4$), massive stars (green; $T_9=0.6$), and Type 1a supernovae (blue; $T_9=1.8$).

Figure 2

Diagram of the K600 and the CAKE in 0-degree mode. In this experiment only four of the possible five CAKE detectors were used. The configuration shown is that used in the present experiment.

Figure 3

Excitation-energy spectra for states excited in the ${}^{24}\mathrm{Mg}(\alpha ,{\alpha }^{\prime }){}^{24}\mathrm{Mg}$ reaction at 0°. The inclusive data, shown in black, are the same in each panel. Superimposed are the coincidence data for different breakup channels. From top to bottom: ${\alpha }_0$, ${\alpha }_1$, $p_0$, and $p_1$. The center-of-mass energy for the colliding ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system is given at the top of top panel. The locations of the four $J^{\pi }=0^{+}$ states of particular importance in the current Letter are marked with black diamonds.

Figure 4

The ratio of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion rate calculated in the present Letter (see text for details) to the reference rates of Caughlan and Fowler [46]: the solid black curve shows the impact on the ${}^{12}\mathrm{C}({}^{12}\mathrm{C},\alpha ){}^{20}\mathrm{Ne}$ reaction rate and the broken red curve that for (${}^{12}\mathrm{C}({}^{12}\mathrm{C},p){}^{23}\mathrm{Na}$). The parameters used in the calculation were $\mathrm{\Gamma }=10\mathrm{keV}$ for both states, with $R=7.2\mathrm{fm}$ (see the text). The reduced width was taken to be ${\theta }^2=0.1$ and the carbon width for the states was computed using this value along with the penetrability corresponding to the resonance energies in Table 1.