Carbon burning rates on the compound nucleus formation

. The 12 C + 12 C reaction rates based on the compound nucleus formation seem to be concordant with the standard rates. The resonant contribution in 12 C + 12 C is also discussed. To put the rates on ﬁrm ground, the resonances below E c . m . = 3 MeV will have to be studied further.


Introduction
The 12 C+ 12 C fusion reaction is one of the key reactions to understand the evolution of massive stars and various explosive scenarios. However, precise measurements of cross sections below E c.m. = 3 MeV are difficult because of the tiny amplitudes caused by the Coulomb barrier. At present, the direct measurements have been performed in E c.m. = 2.1-2.5 MeV [1]. The indirect measurements have been studied with 24 Mg(α,α ) [2] and Trojan horse method [3]. The derived rates [3] are much faster than CF88 [4], due to the resonant states at E c.m. ≈ 1.5 MeV, which may have the 12 C+ 12 C molecule-like structure. The nuclear fusion for 12,13 C+ 13 C have also been discussed experimentally to understand C+C comprehensively [5].
In this presentation, I use a barrier penetration model (BPM), and I show the calculated results of isotope dependence of fusion cross sections and reaction rates in C+C. The transmission coefficients are given by the WKB approximation, semi-classically, and the potentials used in the present work are calculated from a single-folding model [6] with [7,8]. I also discuss the contribution from the resonances in 12 C+ 12 C by comparing the result of BPM with a schematic calculation of the coupled-channels multi-level R-matrix [9].

Compound nucleus formation
Before moving on to the results, let me recall the compound nucleus (CN) formation, to understand the reaction mechanism in C+C. The 12 C+ 12 C potential obtained from the studies of elastic scattering has predicted the sequences of the rotational excitation in 24 Mg [7,10]. These resonances are the excited states with the 12 C+ 12 C molecule-like structure in 24 Mg. However, the potential resonances at E c.m. ≈ 0 are dispersed easily, because of the couplings to reaction channels. Although the inelastic channels are closed at E c.m. = 4.44 MeV, other reaction channels are open, and they work as absorption to the entrance channel. Accordingly, their fragments are distributed around the original energy positions. In fact, many fragments of J π = 2 + , 4 + resonances have been observed, in addition to 0 + [3]. Whereas most of flux are consumed by Coulomb scattering, a small amount of flux is captured into the longliving fragment levels, and exits through the proton, neutron, and α channels after forming a compound nucleus. Under the circumstance, the reactions should be described statistically,  and the emitted nuclei have to be treated as evaporation products. Therefore, I adopt BPM and R-matrix based on the CN formation in the present study.
In BPM, the energy-averaged fusion cross sections are given by σ F = (π/k 2 ) L (2L + 1) |S CN | 2 . k is the wavenumber; L is the angular momentum between nuclei. |S CN | 2 are the transmission coefficients T L , given by WKB approximation for E c.m. < E B : μ is the reduced mass. R 1 and R 2 are the inner and outer turning points of effective potentials U L . The nuclear potentials inŨ L are calculated recursively from the single-folding model [6][7][8]. The cross sections are also given by R-matrix theory, σ F = (π/k 2 ) cL (2L + 1) | S L c,0 | 2 . S L c,0 is the S -matrix deduced from the R-matrix with the resonance parameters in [3]. The reduced width of 12 C+ 12 C is statistically assumed to be a constant γ 2 iL = 0.001γ 2 W for all levels, based on the CN formation. γ 2 W is the Wigner limit. In [5], the transmission coefficients are calculated from an approximation using the unitarity relation of S -matrix. To display the cross sections, the S * factors are defined as S * ≡ σ F E c.m. exp (87.21E −1/2 c.m +0.46E c.m. ), e.g. [3].

Results & Conclusion
The calculated S * factors with BPM for 12 C+ 12 C are shown by the solid curve in Fig. 1(A), and they appear to give the trend of the energy variation in the experimental data [1,11]. The derived reaction rates are shown in ratio to CF88. (Fig. 1(B)) They seem consistent with CF88. For 12,13 C+ 13 C and 12 C+ 14,15 C, the present calculations of BPM (solid curves) reproduce the experimental data [5,12], consistently, as shown in Figs. 1(C) -1(F). Figure 2(A) illustrates the isotope dependence of S * obtained from BPM. In BPM, the 12 C+ 12 C S * factors below E c.m. = 2 MeV become the largest, so the derived reaction rates are the fastest below T 9 = 0.8 (Fig. 2(B)). In addition, the S * factors are found to be enhanced at the sub-barrier energies as the number of neutrons increases. Particularly, those of 12 C+ 15 C are enhanced larger. The barrier radius R B and barrier height energy E B in the present calculations are shown in Figs. 2(C) and 2(D), as a function of the mass number. R B (E B ) becomes large (small) as the number of neutrons increases. Especially, R B suddenly becomes large at 15 C. This is caused by the weakly-bound s-wave neutron in 15 C. Therefore, the corresponding S * and reaction rates are expected to be enhanced more by the sharp reduction of E B .
The resonant contribution in 12 C+ 12 C is shown in Fig. 3. From the result of the R-matrix calculation by the solid curve in Fig. 3(A), the values of S * are found to be much smaller than those of [3]. In the present calculation, I include the same 34 levels and four exit channels as    The experimental data are taken from [1,11]. those in [3]. If γ 2 iL = 0.05γ 2 W is used as the reduced width of 12 C+ 12 C at E c.m. ≈ 1.5 MeV, the reaction rates would increase like those in [3]. The carbon burning rates are sensitive to the reduced width of 12 C+ 12 C. In addition, the reaction rates estimated from the R-matrix extrapolation are confirmed to be reduced from the result of BPM. However, the derived rates at T 9 = 0.6 still seem to be consistent with CF88. To put the rates on firm ground, the resonances below E c.m. = 3 MeV will have to be scrutinized further.