Nuclear incompressibility and its enduring impact on fusion cross sections

The fusion mechanism of reactions involving even-even ${}^{112–124}\mathrm{Sn}$, doubly magic ${}^{132}\mathrm{Sn}, {}^{208}\mathrm{Pb}$ as targets, and ${}^{64}\mathrm{Ni}$ as the projectile is explored within the relativistic mean field (RMF) formalism. The main aim of choosing these nuclei is to explore the correlation between the nuclear incompressibility and the fusion cross section. The nucleus-nucleus interaction potential is calculated by folding the axially deformed nuclear densities and the relativistic R3Y nucleon-nucleon ($NN$) potential obtained for the nonlinear ${\mathrm{NL}3}^{*}$, hybrid, and NL1 parameter sets, which yield different values for various characteristics of nuclear matter at saturation. The fusion barrier characteristics obtained for different RMF parametrizations are further used to calculate the cross section within the $\ell$-summed Wong model. We found a decrease in the barrier height and, consequently, an increase in the cross section with a decrease in the incompressibility for all sets of parameters considered. Furthermore, comparing the barrier heights obtained for ${\mathrm{NL}3}^{*}$ and the hybrid parameters, it is observed that the barrier height decreases with decreasing symmetric energy and incompressibility value. Moreover, a lower barrier height and, consequently, a higher cross section at below-barrier energies is observed for the NL1 parameter set, which gives a soft equation of state (EoS) having a lower value of nuclear matter incompressibility. The calculated cross section is satisfactorily consistent with the available experimental data for ${}^{64}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ system. In contrast, the nuclear potentials obtained for ${\mathrm{NL}3}^{*}$ and hybrid parameter sets underestimate the cross section at below-barrier energies for ${}^{64}\mathrm{Ni}+{}^{112–124,132}\mathrm{Sn}$ reactions. This discrepancy between the experimental data and the theoretical results for ${}^{64}\mathrm{Ni}+{}^{112–124,132}\mathrm{Sn}$ reactions can be correlated with the soft behavior of the Sn isotopes. The compressible nature of Sn isotopes is inferred to lower the barrier height, which further leads to enhancement of the experimental fusion and/or capture cross section at below-barrier energies. Thus, the NL1 parameter set with a comparatively soft EoS is observed to be a better choice to describe the sub-barrier nuclear fusion dynamics of reactions involving the Sn isotopes.

Figure 1

(a) The effective R3Y $NN$ potential, (b) radial distribution of total nuclear densities, and (c) total nuclear densities at the surface region calculated using ${\mathrm{NL}3}^{*}$ (solid black line), hybrid (dashed blue lines), and NL1 (dash double dotted orange lines) parameter sets. See text for details.

Figure 2

Nuclear interaction potential $V_n$ (MeV) obtained using different RMF parameter sets as a function of internuclear separation $R$ (fm) for (a) ${}^{64}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ and (b) ${}^{64}\mathrm{Ni}+{}^{112}\mathrm{Sn}$ and ${}^{64}\mathrm{Ni}+{}^{132}\mathrm{Sn}$ reactions.

Figure 3

The capture cross section ${\sigma }_{\mathrm{int}}$ (mb) calculated using ${\mathrm{NL}3}^{*}$ (black), hybrid (blue), and NL1 (orange) parameter sets within the $\ell$-summed Wong model as a function of center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}$ (MeV) for the ${}^{64}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ reaction. The dashed line signifies the cross section calculated for the spherical case (${\beta }_2=0$) and solid lines are for the cross sections obtained with the inclusion of target quadrupole deformations (${\beta }_2>0$). The corresponding experimental data are taken from [59].

Figure 4

The fusion cross section ${\sigma }_{\mathrm{int}}$ (mb) calculated using ${\mathrm{NL}3}^{*}$ (black), hybrid (blue), and NL1 (orange) parameter sets within the $\ell$-summed Wong model as a function of center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}$ (MeV) for the even-even ${}^{64}\mathrm{Ni}+{}^{112–124,132}\mathrm{Sn}$ systems. The dashed signifies the cross section calculated for the spherical case (${\beta }_2=0$) and solid lines are for the cross section obtained with the inclusion of target quadrupole deformations (${\beta }_2>0$). The corresponding experimental data are taken from [42, 43, 60].

Figure 5

The fusion cross section ${\sigma }_{\mathrm{int}}$ (mb) calculated using ${\mathrm{NL}3}^{*}$ (black) and NL1 (orange) parameter sets along with the barrier modification (solid lines) within the $\ell$-summed Wong model as a function of the center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}$ (MeV) for the ${}^{64}\mathrm{Ni}+{}^{124}\mathrm{Sn}$ reaction. The dashed lines signify the calculations done without the incorporation of nuclear deformations and dashed double-dotted lines signify the cross section calculated with the inclusion of target deformation. The experimental data are taken from [42].