Measurement of the ${}^{12}\mathrm{C}({}^{12}\mathrm{C},p){}^{23}\mathrm{Na}$ cross section near the Gamow energy

The fusion reaction ${}^{12}\mathrm{C}({}^{12}\mathrm{C},\mathrm{p}){}^{23}\mathrm{Na}$ has been studied from $E=2.00$ to 4.00 MeV by particle spectroscopy. The data reveal broad resonances above $E=3.00$ MeV and are compatible with previously reported resonance structure around $E=2.1$ MeV. The data were limited at low energies by low count rates as well as possible background contributions. This experiment extends the previously achieved low-energy measurement by charged particle spectroscopy to 2 MeV, which corresponds to the high-energy side of the astrophysically relevant temperature. Present knowledge of level structures and nonresonant contribution cannot explain the results of the present experiment, which may change the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ reaction rate significantly. Despite the progress decreasing the low-energy limit, any extrapolation into the astrophysical energy range remains highly uncertain based on available experimental data.

Figure 1

The total $S^{*}$ factor of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ reactions compared to some theoretical models, i.e., the hindrance model [18], a potential model calculation based on Ref. [31], and the recent model of Diaz-Torres [32], as well as the parametrization of Caughlan and Fowler [13].

Figure 2

Schematic of the experimental setup where the beam direction is from right to left.

Figure 3

A sample $\mathrm{\Delta }E-E$ matrix taken at $E=2.6$ MeV with the HOPG target. The region of interest for the protons from the ${}^{12}\mathrm{C}({}^{12}\mathrm{C},p){}^{23}\mathrm{Na}$ reaction ($p_0$ and $p_1$) is indicated and is constrained from the other proton groups by kinematics. The continuum of low-energy counts arises from the remaining deuterium contamination in the target, through the $d({}^{12}\mathrm{C},p){}^{13}\mathrm{C}$ reaction. Ejected protons corresponding to higher excited states of the ${}^{23}\mathrm{Na}$ nucleus ($p_2, p_3, p_4$, ...) are found between the region of interest of the $p_0+p_1$ group (ROI) and the deuterium protons as well as merged with the deuterium protons.

Figure 4

Thick-target yields (blue circles) of the ${}^{12}\mathrm{C}({}^{12}\mathrm{C},p){}^{23}\mathrm{Na}$ fusion reaction as a function of the center-of-mass energy $E$. The (red) solid line was obtained by a fit to the data including a nonresonant component and several broad resonances. The individual contributions to the total yield are shown as dashed and dotted lines for the nonresonant contribution and the resonances, respectively. For details, see the text.

Figure 5

The modified ${\mathrm{S}}^{*}$ factor for the sum of the $p_0$ and $p_1$ channels (solid line) is shown together with the corresponding data ($p_0$ and $p_1$ only) of Becker et al. [25] (open circles). The individual contributions for the resonances (dotted line) as well as the flat nonresonant contribution (dashed line) are also presented (for explanations, see Fig. 4). Note that a high-energy resonance at $E=4.2$ MeV as inferred from Becker et al. [25] had to be included to fit the thick-target yield data above 3.8 MeV. This resonance is not confined by the present data and thus indicated by the dashed extension of the solid line.

Figure 6

The upper panel shows the modified $S^{*}$ factor for the $p_{\text{tot}}$ (filled circles) and $p_0+p_1$ channel (open circles) from Ref. [25] above 3.2 MeV. The lower panel displays the ratio between the sum of these two components and the total $S^{*}$ factor of the proton channel. Surprisingly, this ratio seems to be described well by a linear trend and the solid line represents a linear fit to the data. This trend was used to correct the data of the present work.

Figure 7

The ratio between the proton-channel $S^{*}$ factor and the total $S^{*}$ factor, i.e., the sum of the proton and $\alpha$ channel, over a wide energy range from the measurement of Ref. [25]. Again, the ratio seems to be described reasonably well by a linear trend and the solid line represents a linear fit to the data.