Fusion of ${}^{12}\mathrm{C}+{}^{24}\mathrm{Mg}$ far below the barrier: Evidence for the hindrance effect

Background: The phenomenon of fusion hindrance may have important consequences on the nuclear processes occurring in astrophysical scenarios, if it is a general behavior of heavy-ion fusion at extreme subbarrier energies, including reactions involving lighter systems, e.g., reactions in the carbon and oxygen burning stages of heavy stars. The hindrance is generally identified by the observation of a maximum of the $S$ factor vs energy. Whether there is an $S$-factor maximum at very low energies for systems with a positive fusion $Q$ value is an experimentally challenging question.

Purpose: Our aim has been to search for evidence of fusion hindrance in ${}^{12}\mathrm{C}+{}^{24}\mathrm{Mg}$ which is a medium-light system with positive $Q$ value for fusion, besides the heavier cases where hindrance is recognized to be a general phenomenon. ${}^{12}\mathrm{C}+{}^{24}\mathrm{Mg}$ is very close to the ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ and ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ systems that are important for the late evolution of heavy stars.

Methods: The experiment has been performed in inverse kinematics using the ${}^{24}\mathrm{Mg}$ beam from the XTU Tandem accelerator of LNL in the energy range 26–52 MeV with an intensity of 4–8 pnA. The targets were ${}^{12}\mathrm{C}$ evaporations $50\mu \mathrm{g}/{\mathrm{cm}}^2$ thick, isotopically enriched to $99.9\%$. The fusion-evaporation residues were detected at small angles by a $E-\mathrm{\Delta }E$-ToF detector telescope following an electrostatic beam deflector.

Results: Previous measurements of fusion cross section for ${}^{12}\mathrm{C}+{}^{24}\mathrm{Mg}$ were limited to above-barrier energies. In the present experiment the excitation function has been extended down to $\simeq 15\mu \mathrm{b}$ and it appears that the $S$ factor develops a clear maximum vs energy, indicating the presence of hindrance. This is the first convincing evidence of an $S$ factor maximum in a medium-light system with a positive fusion $Q$ value. These results have been fitted following a recently suggested method and a detailed analysis within the coupled-channels model that has been performed using a Woods-Saxon potential and including the ground state rotational band of ${}^{24}\mathrm{Mg}$. The coupled-channels calculations give a good account of the data near and above the barrier but overpredict the cross sections at very low energies.

Conclusions: The hindrance phenomenon is clearly observed in ${}^{12}\mathrm{C}+{}^{24}\mathrm{Mg}$, and its energy threshold is in reasonable agreement with the systematics observed for several medium-light systems. The fusion cross sections at the hindrance threshold show that the highest value (${\sigma }_s=1.6\mathrm{mb}$) is indeed found for this system. Therefore it may even be possible to extend the measurements further down in energy to better establish the position of the $S$-factor maximum.

Figure 1

Comparison for cross sections and $S$ factors for the three fusion reactions ${}^{16}\mathrm{O}+{}^{18}\mathrm{O}$ (c), ${}^{12}\mathrm{C}+{}^{24}\mathrm{Mg}$ (b), and ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$ (a). The curves for ${}^{16}\mathrm{O}+{}^{18}\mathrm{O}$ and ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$ (SG fit) are the results of three-parameter fits that have been interpolated to obtain the single-Gaussian predictions for ${}^{12}\mathrm{C}+{}^{24}\mathrm{Mg}$ (see Ref. [19] for more details).

Figure 2

Top panel: Excitation function and $S$ factor for the system ${}^{12}\mathrm{C}+{}^{24}\mathrm{Mg}$, compared with standard CC calculations (see text). Bottom panel: Logarithmic derivative of the excitation function compared with the $L_{CS}$ value and with the CC calculations. The two blue arrows mark the threshold energy of the hindrance. Only statistical errors are reported in both panels.

Figure 3

Systematic of energy threshold of hindrance $E_S$ vs the $\zeta$ parameter, for several medium-light systems. The points for the astrophysically relevant system (open symbols) have only been obtained from extrapolations. The threshold for ${}^{12}\mathrm{C}+{}^{24}\mathrm{Mg}$ is marked by a green full dot and its uncertainty is within the symbol size, as for ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$.