$\alpha$-cluster microscopic study of ${}^{12}\mathrm{C}$ $+$ ${}^{12}\mathrm{C}$ fusion toward the zero energy limit

The carbon burning process is a fundamental step of stellar evolution and governs the synthesis of chemical elements important for the formation of life. In this work, we utilize the microscopic hybrid $\alpha$-cluster $\left(\mathrm{H}\alpha \mathrm{C}\right)$ model and an analytical approach, both in the framework of the imaginary time method (ITM), to study the carbon fusion reaction towards zero energy. We obtain the values of the cross sections and astrophysical factors and we correlate our results to collective motion. We also include a calculation for the $2^{+}$ carbon fusion and discuss a possible experimental investigation. Our results confirm direct experimental and theoretical results close to the barrier, while suggesting possible $2^{+}$ mixtures in the indirect experimental data. Our study offers an accurate view of the burning process in the somewhat unexplored low energy region.

Figure 1

Evolution of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion in the $xz$ plane with $E_{\mathrm{C}.\mathrm{M}.}=3.5$ MeV. The cyan points are the densities of the $\alpha$ particles from 300 event calculations with the $\mathrm{H}\alpha \mathrm{C}$ model, while the reaction axis is defined to be the $z$ axis.

Figure 2

Evolution of the total energy (green line) of the system during fusion with $E_{\mathrm{C}.\mathrm{M}.}=3.5$ MeV, as calculated with the $\mathrm{H}\alpha \mathrm{C}$ model. The blue and red lines correspond to the ground state energies of ${}^{24}\mathrm{Mg}$ and ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$, respectively. The dashed orange line, corresponds to the final energy with the experimental $Q=13.933$ MeV value.

Figure 3

The $l=0$ channel action $A$ normalized by the same quantity in the Gamow limit $[A_G$, Eq. (12)] as function of energy (red points), as obtained via the $\mathrm{H}\alpha \mathrm{C}$ model. The dashed line corresponds to an exponential fit. We indicate the value of the effective Coulomb barrier at 5.89 MeV.

Figure 4

Top panel: The $S^{*}$ factors as a function of energy for different $l$ values (points) obtained via the $\mathrm{H}\alpha \mathrm{C}$ model and via the analytical approach with free and screened Coulomb (lines), according to the key. Bottom panel: The $S^{*}$ factors as a function of energy from several experimental [9, 11, 12, 14, 24, 25] and theoretical [7, 26] data sets. This work corresponds to the red curve for all $l$ values (g.s.) and the orange curve (4.44 MeV, $2^{+}$), as obtained with the $\mathrm{H}\alpha \mathrm{C}$ model. The pink arrow corresponds to the lowest energy direct measurement [25] and represents an upper limit.

Figure 5

The reaction rate $\langle \sigma v\rangle$ scaled by the corresponding rate ${\langle \sigma v\rangle }_0$ obtained via a constant factor $S_0={10}^{16}$ MeV b, as a function of temperature. The dashed lines denote regions of astrophysical interest [14].

Figure 6

The ratio of $S^{*}$ values calculated via the $\mathrm{H}\alpha \mathrm{C}$ model divided by the corresponding analytical values [Eqs. (10) and (18)], for different $l$ values (solid lines) as a function of $E^{*}=E_{\mathrm{C}.\mathrm{M}.}+Q$. The points represent the experimental isoscalar monopole ($E0$) spectrum of ${}^{24}\mathrm{Mg}$ [37]. We indicate the energy of isoscalar ${}^{24}\mathrm{Mg}$ quadrupole ($E2$) peak [37] with a red arrow. We also include the position of the $0^{+}$ experimental levels of ${}^{24}\mathrm{Mg}$ with red arrows [36]. The experimental points are normalized to the intersection with the $l=0$ curve at 16.4 MeV.

Figure 7

The transverse mean radius $\langle R_{xy}\rangle =\sqrt{\langle x^2+y^2\rangle }$ of the $E_{\mathrm{C}.\mathrm{M}.}=3.5$ MeV reaction, for two single events (dashed curves) and as an average of 300 events (solid curve). The vertical lines show the average turning points of the carbon-carbon system, as calculated via the $\mathrm{H}\alpha \mathrm{C}$ model.