Signature of shallow potentials in deep sub-barrier fusion reactions

We extend a recent study that explained the steep falloff in the fusion cross section at energies far below the Coulomb barrier for the symmetric dinuclear system ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ to another symmetric system, ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$, and the asymmetric system ${}^{64}\mathrm{Ni}+{}^{100}\mathrm{Mo}$. In this scheme, the very sensitive dependence of the internal part of the nuclear potential on the nuclear equation of state determines a reduction of the classically allowed region for overlapping configurations and consequently a decrease in the fusion cross sections at bombarding energies far below the barrier. Within the coupled-channels method, including couplings to the low-lying $2^{+}$ and $3^{-}$ states in both target and projectile as well as mutual and two-phonon excitations of these states, we calculate and compare with the experimental data the fusion cross sections, $S$ factors, and logarithmic derivatives for the above-mentioned systems and find good agreement with the data even at the lowest energies. We predict, in particular, a distinct double peaking in the $S$ factor for the far sub-barrier fusion of ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$, which should be tested experimentally.

Figure 1

Coulomb plus the nuclear M3Y+repulsion, double-folding potential for ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$. Results are shown for different values of the nuclear incompressibility $K$.

Figure 2

Comparison of the double-folding potentials (with and without repulsion) and the AW potential for the system ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$.

Figure 3

Ion-ion potentials for ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$, ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$, ${}^{64}\mathrm{Ni}+{}^{74}\mathrm{Ge}$, ${}^{64}\mathrm{Ni}+{}^{100}\mathrm{Mo}$ based on the potentials employed in the present work and on the AW potential, which was used in Refs. [2, 3, 4, 5, 6]. Dashed strips show for each system the experimental boundaries of the threshold energy $E_s$ [6].

Figure 4

Experimental fusion excitation function for the system ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ [7] compared with various CC calculations based on the M3Y+repulsion potential (assuming different values of the nuclear incompressibility $K$), and on the AW potential.

Figure 5

Experimental fusion excitation functions for the systems ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ [7], ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ [4], and ${}^{64}\mathrm{Ni}+{}^{100}\mathrm{Mo}$ [5] compared with various CC calculations described in the text, and with the no-coupling (NOC) limit for the AW potential.

Figure 6

Experimental $S$ factors for the systems ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ [7], ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ [4], and ${}^{64}\mathrm{Ni}+{}^{100}\mathrm{Mo}$ [5] compared with CC calculations performed with the M3Y+repulsion and AW potentials.

Figure 7

Logarithmic derivatives of the energy-weighted cross sections for ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$, ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$, and ${}^{64}\mathrm{Ni}+{}^{100}\mathrm{Mo}$. Experimental results derived from the cross sections shown in Fig. 5 are compared with CC calculations performed with the M3Y+repulsion and AK potentials. Prediction for a constant $S$ factor is also shown.

Figure 8

Linear plot of fusion cross sections shown in Fig. 5 for the two systems ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ [4] and ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ [7].

Figure 9

Dependence of the calculated fusion cross section on the diffuseness $a_W$ of the imaginary potential. Results are compared with the data for ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ [4] and the calculation based on the IWBC without absorption. Triangular data point at the lowest energy is an upper limit.

Figure 10

Average angular momentum for the fusion of ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ obtained in CC calculations based on the M3Y+repulsion potential (with and without the effect of couplings to the two-phonon octupole states) and on the no-coupling limit using the AW potential. Data are from Ref. [56].

Figure 11

$S$ factors for the fusion of ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ obtained in the CC calculations based on the M3Y+repulsion potential with and without the effect of couplings to the two-phonon octupole states. Data are from Ref. [4]; triangle is an upper limit.

Figure 12

$V_{\mathrm{pocket}}$ (triangles) and $E_s$ (solid points) vs $Z_1{Z}_2\sqrt{\mu }$ for three reactions.