Consistent mean-field description of the ${}^{12}\mathrm{C}\hspace{0.17em}+\hspace{0.17em}{}^{12}\mathrm{C}$ optical potential at low energies and the astrophysical $S$ factor

The nuclear mean-field potential built up by the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ interaction at energies relevant for the carbon burning process is calculated in the double-folding model (DFM) using the realistic ground-state density of ${}^{12}\mathrm{C}$ and the CDM3Y3 density dependent nucleon-nucleon (NN) interaction, with the rearrangement term properly included. To validate the use of a density dependent NN interaction in the DFM calculation in the low-energy regime, an adiabatic approximation is suggested for the nucleus-nucleus overlap density. The reliability of the nuclear mean-field potential predicted by this low-energy version of the DFM is tested in a detailed optical model analysis of the elastic ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ scattering data at energies below 10 MeV/nucleon. The folded mean-field potential is then used to study the astrophysical $S$ factor of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion in the barrier penetration model. Without any adjustment of the potential strength, our results reproduce very well the nonresonant behavior of the $S$ factor of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion over a wide range of energies.

Figure 1

The ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ overlap density given by the FDA and ADA (in ratio to the saturation density of nuclear matter ${\rho }_0\approx 0.17{\mathrm{fm}}^{-3}$) at different internuclear distances $R$. The $z$ axis is along the line connecting the centers of the two ${}^{12}\mathrm{C}$ nuclei.

Figure 2

Three versions of the double-folded ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ potential (7) obtained at $E_{\mathrm{lab}}=78.8$ MeV using either the FDA or ADA for the overlap density. DF1 is the $V_{\mathrm{HF}}$ potential based on the FDA, DF2 is the $V_{\mathrm{HF}}+V_{\mathrm{RT}}$ potential based on the FDA, and DF3 is the $V_{\mathrm{HF}}+V_{\mathrm{RT}}$ potential based on the ADA.

Figure 3

OM description of the elastic ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ scattering data at $E_{\mathrm{lab}}=78.8$ MeV given by three versions of the real double-folded ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ potential shown in Fig. 2 and the WS imaginary potential with parameters given in Table 1. $N_{\mathrm{R}}=1$ was used with the folded potentials.

Figure 4

OM descriptions of the elastic ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ scattering data at $E_{\mathrm{lab}}=16$, 18, and 20 MeV [9] given by three versions of the real OP considered in the present work. The same notation as that in Fig. 2 is used for the double-folded potential.

Figure 5

The same as Fig. 4 but for the elastic ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ scattering data at $E_{\mathrm{lab}}=35,45,$ and 74.2 MeV [29, 37].

Figure 6

The same as Fig. 4 but for the elastic ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ scattering data at $E_{\mathrm{lab}}=83.3$, 102.1, and 117.1 MeV [31].

Figure 7

Elastic ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ excitation function [29, 31, 32, 37] measured at ${\theta }_{\mathrm{c}.\mathrm{m}.}={90}^{\circ }$ in comparison with the predictions given by three versions of the real OP considered in the present work. The same notation as that in Fig. 2 is used for the double-folded potential.

Figure 8

Potential barrier $V_{\mathrm{B}}$ at $l=0$ given by the double-folded DF1, DF2, and DF3 potentials, in comparison with that given by the energy dependent WS potential (BRA) suggested by Brandan et al. [38] and the phenomenological potential (BH) used by Buck and Hopkins in the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ cluster model [70].

Figure 9

${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion cross section given by the BPM using the double-folded DF1, DF2, and DF3 potentials, BRA potential [38], and BH potential [70], in comparison with the data [5, 6, 7, 11, 12, 24] from direct measurements of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion at energies of 2–9 MeV.

Figure 10

Astrophysical $S$ factors of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion predicted by the BPM using the same potential models as those used to obtain the fusion cross sections shown in Fig. 9, in comparison with the phenomenological $S$ factor suggested by Fowler et al. [4] and the data from direct measurements of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion taken from Refs. [5, 6, 7, 11, 12, 24].

Figure 11

Reaction rates (17) of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion obtained from the $S$ factors given by the double-folded DF3, BRA, and BH potentials in comparison with that obtained from the phenomenological $S$ factor suggested by Fowler et al. [4].