Fusion hindrance for the positive $Q$-value system ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$

Background: The fusion reaction ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$ is a link between heavier cases studied in recent years, and the light heavy-ion systems, e.g., ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}, {}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ that have a prominent role in the dynamics of stellar evolution. ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$ fusion itself is not a relevant process for astrophysics, but it is important to establish its behavior below the barrier, where couplings to low-lying collective modes and the hindrance phenomenon may determine the cross sections. The excitation function is presently completely unknown below the barrier for the ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$ reaction, thus no reliable extrapolation into the astrophysical regime for the C+C and O+O cases can be performed.

Purpose: Our aim was to carry out a complete measurement of the fusion excitation function of ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$ from well below to above the Coulomb barrier, so as to clear up the consequence of couplings to low-lying states of ${}^{30}\mathrm{Si}$, and whether the hindrance effect appears in this relatively light system which has a positive $Q$ value for fusion. This would have consequences for the extrapolated behavior to even lighter systems.

Methods: The inverse kinematics was used by sending ${}^{30}\mathrm{Si}$ beams delivered from the XTU Tandem accelerator of INFN-Laboratori Nazionali di Legnaro onto thin ${}^{12}\mathrm{C}$ ($50\mu \mathrm{g}/{\mathrm{cm}}^2$) targets enriched to $99.9\%$ in mass 12. The fusion evaporation residues (ER) were detected at very forward angles, following beam separation by means of an electrostatic deflector. Angular distributions of ER were measured at $E_{\mathrm{beam}}=45$, 59, and 80 MeV, and they were angle integrated to derive total fusion cross sections.

Results: The fusion excitation function of ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$ was measured with high statistical accuracy, covering more than five orders of magnitude down to a lowest cross section $\simeq 3\mu \mathrm{b}$. The logarithmic slope and the $S$ factor have been extracted and we have convincing phenomenological evidence of the hindrance effect. These results have been compared with the calculations performed within the model that considers a damping of the coupling strength well inside the Coulomb barrier.

Conclusions: The experimental data are consistent with the coupled-channels calculations. A better fit is obtained by using the Yukawa-plus-exponential potential and a damping of the coupling strengths inside the barrier. The degree of hindrance is much smaller than the one in heavier systems. Also a phenomenological estimate reproduces quite closely the hindrance threshold for ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$, so that an extrapolation to the C+C and O+O cases can be reliably performed.

Figure 1

Measured fusion excitation function for ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$ covering more than five orders of magnitude. The reported uncertainties are purely statistical, and do not exceed the symbol size except for the lowest energies. The inset shows the ER angular distribution obtained at $E_{\mathrm{lab}}=59\mathrm{MeV}$, compared with a pace4 calculation [26, 27] (line).

Figure 2

The excitation function measured in this work was compared to the results of various CC calculations employing the WS (a) and (c), and the YPE (b) and (d) potentials, with and without damping of the coupling strengths. The no-coupling limit is also reported in both cases. (c) and (d) Expanded views of the low-energy range. See text for details.

Figure 3

Logarithmic slope of the excitation function. It appears that the slope overcomes the $L_{\mathrm{CS}}$ value at the lowest energies. The results of theoretical calculations are also shown and within the experimental uncertainties all of them give a reasonable fit of the data. The full black line in the upper panel is a phenomenological extrapolation of the slope based on Ref. [14], that allows a clear identification of the crossing point with the value $L_{\mathrm{CS}}$.

Figure 4

Astrophysical $S$ factor for ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$ in comparison with the CC calculations. The experimental evidence is that a maximum of $S$ factor vs energy tends to develop around 10.5 MeV. The presence of this maximum is not reproduced by the present calculations which, however, fit the overall trend of the data. In particular, it appears that the low-energy damping of the coupling strengths is needed. The blue curve in the upper panel is a phenomenological extrapolation of $S$ based on Ref. [14].

Figure 5

Systematics of $E_s$ in several light- and medium-light mass systems [14]. The location for ${}^{12}\mathrm{C}+{}^{30}\mathrm{Si}$ is very close to the astrophysically relevant ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ cases. For these systems several data sets exist, but they are sometimes contradictory to each other and the errors are large. This results in large uncertainties in the expected values for their hindrance threshold. The corresponding points (open symbols) have therefore been obtained only from extrapolations.