Expectations for ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$ induced fusion cross sections at energies of astrophysical interest

The extrapolations of cross sections for fusion reactions involving ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$ nuclei down to energies relevant for explosive stellar burning have been reexamined. Based on a systematic study of fusion in heavier systems, it is expected that a suppression of the fusion process will also be present in these light heavy-ion systems at extreme sub-barrier energies due to the saturation properties of nuclear matter. Previous phenomenological extrapolations of the $S$ factor for light heavy-ion fusion based on optical model calculations may therefore have overestimated the corresponding reaction rates. A new “recipe” is proposed to extrapolate $S$ factors for light heavy-ion reactions to low energies taking the hindrance behavior into account. It is based on a fit to the logarithmic derivative of the experimental cross section which is much less sensitive to overall normalization discrepancies between different data sets than other approaches. This method, therefore, represents a significant improvement over other extrapolations. The impact on the astrophysical reaction rates is discussed.

Figure 1

Plot of $S(E)$ vs. $E-E_0$ ($E$ and $E_0$ are values of the center-of-mass energy and Coulomb barrier, respectively) for light heavy-ion fusion reactions ${}^{10}\mathrm{B}+{}^{12}\mathrm{C}$ [6], ${}^{11}\mathrm{B}+{}^{12}\mathrm{C}$ [6], ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ [7, 8, 9], ${}^{12}\mathrm{C}+{}^{13}\mathrm{C}$ [6], ${}^{12}\mathrm{C}+{}^{14}\mathrm{N}$ [6], ${}^{14}\mathrm{N}+{}^{14}\mathrm{N}$ [6], ${}^{12}\mathrm{C}+{}^{16}\mathrm{O}$ [10, 11, 12], ${}^{14}\mathrm{N}+{}^{16}\mathrm{O}$ [6], and ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ [13, 14, 15, 16]. Values of $E_0$ are 5.72, 5.63, 6.66, 6.57, 7.57, 8.61, 8.45, 9.62, and 10.76 MeV, respectively. The $Q$-values for these systems range from 32 to 15 MeV. The solid curves are optical model calculations with potential parameters $V=50$ MeV, $W=10$ MeV, $r_0=1.27$ fm and $a_0=0.40$ fm.

Figure 2

(a) and (b) The fusion reactions ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ and ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ with data taken from Ref. [19] and [28], respectively. (a) Plot of $L(E)$ vs. $E$. The solid and open circles are the logarithmic derivatives obtained from least-squares fits to three experimental data points. The dashed curve is the constant $S$ factor result for the reaction ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$. The constant $S$ factor for the reaction ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ is almost identical. The solid lines are obtained from fits with the formula $A_0+B_0/(E+Q)$. The dot-dashed curves are from conventional coupled-channels calculations. (b) Corresponding plots of $S(E)$ vs. $E$. The solid curves in (b) are extrapolations, corresponding to the solid curves in (a). (c) and (d) The fusion reaction ${}^{12}\mathrm{C}+{}^{13}\mathrm{C}$ with data taken from Ref. [6]. (c) Plot of $L(E)$ vs. $E$. The solid points are the logarithmic derivatives obtained from least-squares fits to five experimental data points, because the energy difference between the neighbouring data points is rather small. (d) Corresponding plots of $\sigma (E)$ vs. $E$ and $S(E)$ vs. $E$. The black triangles in (d) are the fusion cross sections with the scale given at the right side of the figure. The curves are identical to the ones in (a) and (b), but the fit was performed with the equation $A_0+B_0/E^{3/2}$ and the dot-dashed curves are the optical model calculations. (e) and (f) The fusion reaction ${}^{11}\mathrm{B}+{}^{12}\mathrm{C}$ with data taken from Ref. [6]. The data points and the curves have the same definitions as in (c) and (d).

Figure 3

(a) Plot of $L(E)$ vs. $E$ for the fusion of ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$. The derivatives were obtained by least-squares fits to three cross section points. The black long-dashed curve corresponds to a constant $S$ factor. (b) Plot of $S(E)$ vs. $E$ and (c) Plot of $\tilde{S}(E)$ vs. $E$ for the same reaction of ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$; the symbols and curves are the same in (a), (b), and (c). The black solid curve is from optical model calculations, and the light-blue dashed and black dot-dashed curves are the extrapolations obtained by Hulke et al. [14] and by Fowler et al. [2], respectively. The red solid line is the extrapolation obtained in the present paper. See text for details.

Figure 4

$S$ factors for the fusion reaction ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$. Data from [13, 14, 15, 16] are multiplied by different scaling factors as indicated in the plot. The dashed curves serve only to guide the eye.

Figure 5

Plots of ${\chi }^2/\mathrm{dof}$ vs. $N$ for the $L(E)$ representation (a) and the $S$ factor representation (b) for system ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$. The value $N$ is the number of experimental data points used in the calculations (starting with the lowest energy) and dof are the degrees-of-freedom. In (b), Hulke's results are not shown because they are off scale.

Figure 6

Plot of (a) $L(E)$, (b) $S(E)$ and (c) $\tilde{S}(E)$ for the fusion reaction ${}^{12}\mathrm{C}+{}^{16}\mathrm{O}$ with data taken from Refs. [10, 11, 12]. The derivatives were obtained by least-squares fits to eight, five, and five cross section points for Patterson's, Christensen's, and $\tilde{\mathrm{C}}$ujec's experiments, respectively. The black long-dashed line corresponds to a constant $S$ factor, while the black solid line is the result of an optical model calculation. The black dot-dashed curves correspond to the fit obtained by Fowler et al. [2]. The red solid lines are the results of least-squares fits to the data using Eqs. (10) and (11). See text for details.

Figure 7

Plots of ${\chi }^2/\mathrm{dof}$ vs. $N$ for the $L(E)$ representation, (a) ${}^{12}\mathrm{C}+{}^{16}\mathrm{O}$, (b) ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$. The value $N$ is the number of experimental data points used in the calculations (starting with the lowest energy). The symbol dof means degrees-of-freedom. Some data points are not shown because they are off scale.

Figure 8

$L(E),S(E)$, and $\tilde{S}(E)$ vs. $E$ plots for the system ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$. The derivatives were obtained by least-squares fits to three cross section points. The symbols and curves are similar to those of Figs. 3 and 6. The light-blue solid curves shown in (b) and (c) are the calculations for the “nonresonant part” of $\tilde{S}(E)$ given in Ref. [45] with the Krappe-Nix-Sierk (KNS) nuclear potential [46].

Figure 9

Calculated reaction rates for the fusion reaction ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$. The upper and lower dot-dashed curves are obtained from Hulke's and Fowler's extrapolations, respectively. The solid curve is obtained with the present extrapolation.

Figure 10

Calculated reaction rates for the fusion reaction ${}^{12}\mathrm{C}+{}^{16}\mathrm{O}$. The dot-dashed and the solid curves are obtained from Fowler's and the present extrapolation, respectively.

Figure 11

Calculated reaction rates for the fusion reaction ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$. The dot-dashed curves are obtained from Gasques’ and Fowler's extrapolations, respectively. These two extrapolations are very similar at low temperatures, and Gasques’ extrapolation is slightly higher than Fowler's at high temperatures. The solid curve is obtained with the present extrapolation.

Figure 12

Ratios of reaction rates for the three fusion reactions ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ (red), ${}^{12}\mathrm{C}+{}^{16}\mathrm{O}$ (green) and ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ (light-blue). These bands [made from the corresponding three $S(E)$ bands] are the ratios between the present extrapolation and that proposed by Fowler [2]. See text for details.