Absence of a maximum in the $S$ factor at deep sub-barrier energies in the fusion reaction ${}^{40}$Ca + ${}^{40}$Ca

The fusion reaction ${}^{40}$Ca + ${}^{40}$Ca was very recently reinvestigated by the Legnaro group and cross sections down to $\mu$b were measured. In the present work this reaction is analyzed within the coupled-channel formalism including $2^{+}$, $3^{-}$, and $5^{-}$ phonon states. The heavy-ion entrance channel potential is calculated within the double-folding prescription and comprises a direct local term and a local equivalent of the nonlocal knock-on exchange component. The single-particle densities entering the folding integrals are prescribed according to the density matrix expansion. Coupling potentials are calculated from the $G$-matrix Michigan three-range Yukawa-type semirealistic nucleon-nucleon interaction (M3Y) as well as one modern parametrization of the density-dependent Gogny effective interaction. Realistic deformation parameters are used. The best fit to the data is not pointing to a possible maximum in the astrophysical $S$-factor as in a previous study on sub-barrier fusion involving ${}^{48}$Ca projectiles.

Figure 1

Entrance channel potential for the system ${}^{40}$Ca + ${}^{40}$Ca using the M3Y-Reid and Gogny-D1N forces. The dashed line corresponds to the lowest energy attained in experiment.

Figure 2

Fusion cross section for the system ${}^{40}$Ca + ${}^{40}$Ca compared to the experimental data of Ref. [3] (circles). In (a) and (c) the cross sections are represented in logarithmic scale, whereas in (b) and (d) the representation is made in linear scale.

Figure 3

Experimental to theoretical cross-sections ratios for ${}^{40}$Ca + ${}^{40}$Ca. The thin dashed region is centered around a ratio equal to 1.

Figure 4

Calculated $S$ factor for the system ${}^{40}$Ca + ${}^{40}$Ca is compared to the experimental data of Ref. [3].

Figure 5

Calculated logarithmic derivatives $d\ln \left({\sigma }_{f}E\right)/dE$ for the system ${}^{40}$Ca + ${}^{40}$Ca is compared to the experimental data of Ref. [3]. Numerical derivatives are calculated by a five-points formula.