Calculation of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ sub-barrier fusion cross section in an imaginary-time-dependent mean field theory

The ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ sub-barrier fusion cross section is calculated within the framework of a time-dependent Hartree-Fock-based classical model using the Feynman path-integral method. The modified astrophysical $S^{*}$ factor is compared to direct and indirect experimental results. A good agreement with the direct data is found. In the lower-energy region where recent analyses of experimental data obtained with the Trojan horse method (THM) lead to contrasting results, the model predicts a nonresonant $S^{*}$ factor half-way between those results. Low-energy resonances revealed in the THM data are added to the calculation, and the relative reaction rate in the Gamow region is calculated. In particular, including $0^{+}$ resonances result in some agreement with the THM data. The role of different resonances is discussed in detail, and their influence on the reaction rate at temperatures relevant to stellar evolution is investigated.

Figure 1

Astrophysical $S^{*}$ factor as a function of the center-of-mass energy in ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ collisions. The calculated smooth black curve increasing from ${10}^{16}\mathrm{MeV}\mathrm{b}$ to $\sim {10}^{17}\mathrm{MeV}\mathrm{b}$ is obtained when no resonances are included. The analytical result, Eqs. (1) and (5), is given by the cyan curve [36]. The result including resonances with their natural widths is represented by small squares (green). Experimentally derived data are orange diamonds [7], blue triangles [8], orange circles [11], and red squares [12]. The latter two represent conflicting THM analyses of the same THM data [11, 12]. Note that the experimental resolution of 30 KeV, not included in the calculations, broadens the peaks. The double narrow peaks below 1 MeV are included in the calculations assuming they are both $J^{\pi }=0^{+}$ [11]. The resonance at 2.56 MeV [11] is not included since it is most probably $J^{\pi }>3^{-}$ [13]. For more recent direct data see Refs. [33, 34].

Figure 2

Ratio of reaction rate divided by assumed constant $S_0=1\times {10}^{16}\mathrm{MeV}\mathrm{b}$ vs $T^9$. Green circles: ratio without resonances; red squares: ratio with all included resonances. Some typical regions for carbon burning are indicated in the figure, compare to Ref. [11].