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Re-examination of fusion hindrance in astrophysical $^{12}$C+$^{12}$C and $^{12}$C+$^{13}$C reactions

Kotaro Uzawa, Kouichi Hagino

Abstract

To determine the energy dependence of fusion cross sections at extremely low energies is crucial for various astrophysical processes. In the previous study by Jiang et al. [Phys. Rev. C75, 015803 (2007)], it was concluded that fusion cross sections for the $^{12}$C+$^{12}$C system rapidly drop off as the energy decreases. We here re-examine this hindrance phenomenon. While the previous study fitted the logarithmic slope $L(E)$ of fusion cross sections with a function of $L(E)=A+B/E^n$ and searched the optimum value of $A$ and $B$ with $n=1.5$, we refit the data with the same function for $L(E)$ but by releasing the restriction on $n$. We find that the optimum values of $n$ significantly deviates from $n=1.5$, resulting in the absence of hindrance of fusion cross sections both in the $^{12}$C+$^{12}$C and the $^{12}$C+$^{13}$C systems.

Re-examination of fusion hindrance in astrophysical $^{12}$C+$^{12}$C and $^{12}$C+$^{13}$C reactions

Abstract

To determine the energy dependence of fusion cross sections at extremely low energies is crucial for various astrophysical processes. In the previous study by Jiang et al. [Phys. Rev. C75, 015803 (2007)], it was concluded that fusion cross sections for the C+C system rapidly drop off as the energy decreases. We here re-examine this hindrance phenomenon. While the previous study fitted the logarithmic slope of fusion cross sections with a function of and searched the optimum value of and with , we refit the data with the same function for but by releasing the restriction on . We find that the optimum values of significantly deviates from , resulting in the absence of hindrance of fusion cross sections both in the C+C and the C+C systems.

Paper Structure

This paper contains 13 equations, 5 figures, 2 tables.

Figures (5)

  • Figure 1: (a) The chi-square values $\chi^2_w$ for the WLS fitting to the $S$ factor for the $^{12}$C+$^{12}$C fusion reaction. Here, the value of $n$ is fixed, and the other parameters $A, B$ and $\sigma'$ in Eq. (\ref{['S_int']}) are optimized for each $n$. The fitting excludes $n=0$. The star shows the minimum of the chi-square. (b) Same as (a), but for the OLS fitting.
  • Figure 2: (a) The astrophysical $S$ factors for the $^{12}$C+$^{12}$C system calculated with Eqs. (\ref{['S_int']}) and (\ref{['S_int2']}) obtained with the parameters the WLS (the red dashed line) and the OLS (the blue dot-dashed line) fittings, respectively. The $S$ factors from the hindrance model of Ref. Jiang2007a are also shown for comparison by the green solid line. The experimental data are taken from Refs. Patterson1969Mazarakis1973High1977Aguilera2006Jiang2018b. (b) The corresponding logarithmic slopes, $L(E)$. The logarithmic slope with a constant $S$ factor, $L_{cs}$, is also shown by the dotted line.
  • Figure 3: Same as Fig. \ref{['fig1']}, but for the $^{12}$C+$^{13}$C fusion reaction.
  • Figure 4: Same as Fig. \ref{['fig2']}, but for the $^{12}$C+$^{13}$C fusion reaction.
  • Figure 5: Same as Fig. \ref{['fig4']}(a) but with the expanded scale for the vertical axis.