Effect of isospin on the fusion reaction cross section using various nuclear proximity potentials within the Wong model

Abstract

Various versions of proximity potentials, with different isospin dependence, are used within the Wong model for fitting the fusion-evaporation cross sections from ${}^{58}$Ni + ${}^{58}$Ni, ${}^{64}$Ni + ${}^{64}$Ni, and ${}^{64}$Ni + ${}^{100}$Mo reactions in an intermediate mass region and capture cross sections from ${}^{48}$Ca + ${}^{238}$U, ${}^{244}$Pu, and ${}^{248}$Cm reactions in a superheavy mass region, known for fusion hindrance phenomena in coupled-channel calculations. It is found that proximity 1988 gives results close to experimental data in a comparison of proximity 1977 and proximity 2000 within the Wong formula, so this Proximity 1988 (Prox 1988) is then used within the extended Wong model. This combination can fit the cross sections for the ${}^{58}$Ni + ${}^{58}$Ni and ${}^{64}$Ni + ${}^{64}$Ni reactions but deviates a little from the experimental data for the ${}^{64}$Ni + ${}^{100}$Mo reaction at a center-of-mass energy below the Coulomb barrier. But the variation of ${\ensuremath{\ell}}_{\text{max}}$ with ${E}_{\text{c.m.}}$ at sub-barrier energies is absurd for these three reactions. So, a different type of strong nuclear interaction is required that accounts for isospin effect and asymmetry of the colliding nuclei, which is obtained here by a slight adjustment of the coefficient ${\ensuremath{\gamma}}_{0}$ of the nuclear surface energy constant of the Prox 1988 potential, in order to fit the experimental data for this reaction and named it mod-Prox 1988. The surface energy constant $\ensuremath{\gamma}$ depends on the neutron/proton excess, and change in one of its constants, ${\ensuremath{\gamma}}_{0}$, used in Prox 1988, introduces the required barrier modification for fitting the experimental data. Thus, change in $\ensuremath{\gamma}$ leads to change in barrier characteristics, i.e., barrier height (${V}_{B}^{\ensuremath{\ell}}$), position (${R}_{B}^{\ensuremath{\ell}}$), and oscillator frequency ($\ensuremath{\hbar}{\ensuremath{\omega}}_{\ensuremath{\ell}}$). Using this modified proximity potential within the extended Wong model, experimental data for all three Ni-induced reactions is fitted with a smooth variation of deduced ${\ensuremath{\ell}}_{\text{max}}({E}_{\text{c.m.}})$ at above- as well as below-barrier energies. The effect of isospin is more prominent at below-barrier energies. As an application, mod-Prox 1988 is then employed for the capture cross-section data from ${}^{48}$Ca + ${}^{238}$U, ${}^{244}$Pu, and ${}^{248}$Cm reactions in superheavy mass regions and it successfully fits the data point to point with a smooth variation of ${\ensuremath{\ell}}_{\text{max}}$ with ${E}_{\text{c.m.}}$. Thus, mod-Prox 1988 within the extended Wong model leads to a simultaneous explanation of several reactions displaying a hindrance to fusion. Thus, below the Coulomb barrier a stronger nuclear interaction is required as the $N/Z$ ratio increases and colliding nuclei become more and more asymmetric. The effects of multipole deformations are included here up to hexadecapole in the calculations, and orientation degrees of freedom are integrated for the coplanar configuration.