Fusion cross section of 12 C + 13 C at sub-barrier energies

In the recent work at Notre Dame, correlations between three carbon isotope fusion systems have been studied and it is found that the fusion cross sections of 12C+13C and 13C+13C provide an upper limit on the fusion cross section of the astrophysically important 12C+12C reaction. The aim of this work is to continue such research by measuring the fusion cross section of the 12C+13C reaction to lower energies. In this experiment, the off-line activity measurement was performed in the ultra-low background laboratory and the fusion cross section for 12C+13C has been determined in the energy range of Ec.m.=2.5-6.8 MeV. Comparison between this work and several models is also presented.


Introduction
Heavy-ion fusion reactions between light nuclei such as carbon and oxygen isotopes have been intensively studied because of their importance in a wide variety of stellar burning scenarios.Among them, carbon burning driven by the 12 C+ 12 C fusion is a crucial process for the formation of white dwarfs, nucleosynthesis in massive stars, and ignition in type Ia supernovae and superbursts [1,2].The temperatures for the hydrostatic carbon burning process range from 0.8 to 1.2 GK, corresponding to E c.m. =1-3 MeV.Unfortunately, because of the very low cross sections, this important energy range is only partially measured at energies above E c.m. =2.1 MeV.For the unmeasured energy ranges, one has to rely on extrapolation methods.Moreover, the situation is further complicated by the existence of the strong, relatively narrow resonances in 12 C+ 12 C reactions.The large resonance reported at energies around E c.m. = 2.1 MeV has not been confirmed by following experiment [3].
In an attempt to learn about the resonance structures of the low-energy 12 C+ 12 C reaction, the carbon isotope fusion reactions were systematically studied at the University of Notre Dame (UND) [4].It was found that the cross sections of the 12 C+ 12 C fusion reaction at resonant energies match with the cross sections in the 12 C+ 13 C and 13 C+ 13 C systems within their quoted uncertainties (see Fig. 1).The observed correlation is explained by the level density differences among the three carbon isotope systems [4,5].As a result, the 12 C+ 13 C and 13 C+ 13 C systems provide an upper limit for 12 C+ 12 C in a wide range from E c.m. =2.6 MeV up to more than 20 MeV.Since the two carbon fusion cross sections are much easier to be modeled due to their smooth behaviors, such an upper limit could be predicted a e-mail: zhangningtao@impcas.ac.cn within the astrophysical energy range.The coupled-channel calculation with the M3Y+Rep potential was used to fit the 12 C+ 13 C and 13 C+ 13 C data and constrain the effective nuclear potential, which was then used for the prediction of the 12 C+ 12 C fusion cross sections [4,6].It was found that the coupledchannel calculation using the constrained M3Y+Rep potential provides an excellent upper limit for almost all the data except for the strong resonance at 2.14 MeV, which has not been confirmed [4].Measurement of 12 C+ 13 C and 13 C+ 13 C at deep sub-barrier energies gives us not only an opportunity to model the resonance strengths in 12 C+ 12 C but also a test of the predictive powers of various theoretical models for the carbon fusion cross sections at deep sub-barrier energies.Lacking of experimental data within the energies of astrophysical interest, large discrepancies exist among different nuclear reaction models.Therefore, it is important to push the measurements of the fusion cross sections of 12 C+ 13 C and 13 C+ 13 C down towards lower energies.

1C+ 13 C experiment at IFIN-HH
We report an experiment to measure the cross section of 12 C+ 13 C reaction by detecting the residual nucleus 24 Na which β-decays with a half-life of 15.0 h.Similar measurements have been performed by Notani and Dayras [4,8].In the present experiment, the 13 C beam was produced by a cesium sputter ion source and injected into a HVEE Tandetron 3 MV electrostatic accelerator of IFIN-HH [10].The 13 C beam impinges a natural graphite target with thickness of 1 mm.The reaction has been studied by varying the beam energies between 5.2 and 6.8 MeV in steps of 0.2 MeV.The 13 C beam current used in this experiment varies in the range of 2 to 8 pμA.
After each irradiation, the target sample would be quickly transported to an underground counting station (μBq) in the Unirea salt mine for offline γ-ray measurement [11].This salt mine is located in the vicinity of Slanic-Prahova city, about one hundred kilometers away from the Bucharest.In this salt mine, the μBq underground laboratory is situated at a depth of 208 m below surface (estimated to 560 m water equivalent).The total gamma background spectrum between 40 keV and 3 MeV was 100 times smaller at laboratory level with respect to the same spectrum recorded at surface in open field.In the μBq, a well shielded HPGe detector was used to detect two cascading γ rays (1369-and 2754-keV) emitted from the γ decay of 24 Na.One typical gamma spectrum is displayed in Fig. 2. In some cases, the measurement was performed in the Low Background Gamma-Ray Spectrometry Laboratory (GamaSpec) in a basement of IFIN-HH [12].In this lab, limited by the background γ rays, only target samples irradiated at higher beam energies (>5.8 MeV) could be measured.Furthermore, this measurement was used to cross check the experimental setup in the two laboratories and validate our results.
The thick-target yield (Y) for 12 C( 13 C,p) 24 Na reaction was obtained by normalizing the observed yield to the total incident 13 C beam flux.From the thick-target yield excitation function, the differential yield dY/dE are determined and then the corresponding cross sections are calculated using the equation σ(E)=dY/dE*dE/d(ρX)/N v , where N v is the number of atoms per unit of volume and dE/d(ρX) is the stopping power in the target material, given by the SRIM code.Finally, the total fusion cross sections of 12 C+ 13 C are deduced from the proton emission channel using the theoretical branching ratio given by Hauser-Feshbach model [8].

Preliminary results and summary
The preliminary results are shown in Fig. 3.In this work, the lowest cross section for 12 C( 13 C,p) 24 Na reaction has been measured down to 3 nb as shown in Fig. 3(a), representing the lowest energy reached for this reaction.This is the great advantage of the ultra-low background underground laboratory.Figure 3(b) shows the modified S factor (S*) deduced from the total fusion cross section.The result agrees with that of the two previous measurements in the energy region from 2.6-3.3MeV.Limited by the beam time, only one new data point (E c.m. =2.5 MeV) is added in our first experiment.It has been observed that the optical model with Woods-Saxon type potential reproduces the experimental data only at energies above 4 MeV.At deep sub-barrier energies, it significantly overestimates the cross section, which is quoted as hindrance effect.The equivalent square-well (ESW) model and coupledchannels (CC) with M3Y+Rep potential can predict the experimental data very well.The hindrance model prediction obtained by fitting the Dayras data also shows a reasonable agreement to the experimental data above 2.7 MeV, but predicts a much sharper decrease at astrophysical energies [13].In order to test the predictive power of the extrapolation models, we will continue our measurement towards lower energies.

Figure 2 .
Figure 2. A typical gamma spectrum measured in the underground μBq lab within 46 hours.Beam energy for this spectrum is 5.2 MeV (E c.m. =2.5 MeV), which is the lowest energy point in the experiment.The statistical error for 1369-keV γ peak is 11%, much lower than that in the Notani measurement [4].

Figure 3 .
Figure 3.The preliminary fusion cross section of 12 C( 13 C,p) 24 Na reaction obtained from the present work (a) and the deduced S* factor for the 12 C+ 13 C reaction system (b).The results from the previous experiments are also shown.