Characterizing the astrophysical $S$ factor for ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion with wave-packet dynamics

A quantitative study of the astrophysically important subbarrier fusion of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ is presented. Low-energy collisions are described in the body-fixed reference frame using wave-packet dynamics within a nuclear molecular picture. A collective Hamiltonian drives the time propagation of the wave packet through the collective potential-energy landscape. The fusion imaginary potential for specific dinuclear configurations is crucial for understanding the appearance of resonances in the fusion cross section. The theoretical subbarrier fusion cross sections explain some observed resonant structures in the astrophysical $S$ factor. These cross sections monotonically decline towards stellar energies. The structures in the data that are not explained are possibly due to cluster effects in the nuclear molecule, which need to be included in the present approach.

Figure 1

Specific cuts in the collective potential-energy landscape of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system as a function of the internuclear distance and three alignments: Equator-Equator (EE), Equator-Pole (EP), Pole-Pole (PP). The EE alignment (dashed line) facilitates the access by tunneling to the potential pockets. All the alignments coexist and compete with each other, the kinetic energy operator driving the system towards either reseparation or fusion in the potential pocket of the PP alignment (solid line) [25].

Figure 2

Transmission-coefficient excitation function for ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ central collisions through the Coulomb barrier of the Broglia-Winther potential [31], calculated with the two methods indicated. The barrier height is $\sim 10$ MeV.

Figure 3

The same as in Fig. 2, but for ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ central collisions. The calculations are performed with the present method using the fusion imaginary potential centered at two different radii as indicated. The resonant structure disappears when this absorption operates around the potential pockets of dinuclear configurations other than the PP configuration in Fig. 1 (comparing the dotted and solid lines).

Figure 4

(a) Energy-resolved fusion cross sections for different values of the mean energy $E_0$ of the initial wave function. The energy components far from $E_0$ have small amplitudes, and the associated cross sections are very inaccurate. Only the overlapping parts of these excitation curves determine the physical, converged fusion excitation function. (b) Angular momentum decomposition of the converged fusion excitation function that shows some bumps originating from specific partial waves.

Figure 5

The astrophysical $S$-factor excitation function for ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$. Measurements [2, 3, 4, 5, 6, 7] (symbols) are compared to model calculations (dashed and solid lines), indicating that molecular structure and fusion are interconnected.

Figure 6

The effective real potentials for nonaxial symmetric molecular configurations and the scattering phase-shift analysis (plot inserted) for specific partial waves. The occupation of these potential resonances (circles in plot inserted) causes the structures in the $S$-factor excitation function in Fig. 5.

Figure 7

The same as in Fig. 5, but the present model calculations are shown for (i) two global factors that multiply the collective potential-energy landscape in Fig. 1, and (ii) a reduction by $15\%$ of the curvature of the potential pockets. This last greatly improves the location of the predicted resonant structures (thin solid line).