Origin and Consequences of ${}^{12}\mathbf{C}+{}^{12}\mathbf{C}$ Fusion Resonances at Deep Sub-barrier Energies

Previous explanations for the resonance behavior of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion at low energies were based on a nonresonant compound-nucleus background and an additional contribution from a series of resonances. This separation into “resonance” and “background” contributions of the cross section is artificial. We propose to explain this phenomenon through the impact on the cross section of the relatively large spacings and the narrow widths of ${}^{24}\mathrm{Mg}$ compound levels in the corresponding excitation-energy region.

Figure 1

Comparisons of the $S$ factors for four fusion systems. The data for systems in (a), (b), (c), (d) are taken from Refs. 3, 4, 5, 6, 7, 27, 31, 32, 33, and 16, 34, 35, respectively.

Figure 2

Total level densities for ${}^{24}\mathrm{Mg}$, ${}^{25}\mathrm{Mg}$, and ${}^{32}\mathrm{S}$ as a function of the excitation energy $U$; the curves are calculations described in the text.

Figure 3

The experimental widths of $J=0^{+}$, $2^{+}$, and $4^{+}$ levels in ${}^{24}\mathrm{Mg}$ are shown as open symbols. The solid black line and the shaded region represent the estimated average trend and range above $U>13\mathrm{MeV}$. The average level spacings, $D_J$, for levels of spin, $J^{\pi }=0^{+}$, $2^{+}$, and $4^{+}$ are given as blue, red, and green curves, respectively. The corresponding center-of-mass energy, $E_{\mathrm{c}.\mathrm{m}.}$, for ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion is given on the top. The energy $E_g$ at which $\Gamma =D_0$ is indicated by an arrow.

Figure 4

Calculations of the $S$ factor corresponding to ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion cross sections for the Woods-Saxon potential (blue curve), $\mathrm{CC}(\mathrm{M}3\mathrm{Y}+\mathrm{rep})$ IWBC (dashed black curve), and Eq. (5) (solid black curve) including the range of $\Gamma$ values indicated in Fig. 3 (shaded region) are compared with experimental values from Refs. 3, 4, 5, 6.