Angular momentum effects and barrier modification in sub-barrier fusion reactions using the proximity potential in the Wong formula

Abstract

Using the capture cross-section data from $^{48}\mathrm{Ca}+^{238}\mathrm{U}$, $^{48}\mathrm{Ca}+^{244}\mathrm{Pu}$, and $^{48}\mathrm{Ca}+^{248}\mathrm{Cm}$ reactions in the superheavy mass region, and fusion-evaporation cross sections from $^{58}\mathrm{Ni}+^{58}\mathrm{Ni}$, $^{64}\mathrm{Ni}+^{64}\mathrm{Ni}$, and $^{64}\mathrm{Ni}+^{100}\mathrm{Mo}$ reactions known for fusion hindrance phenomenon in coupled-channels calculations, the Wong formula is assessed for its angular momentum and barrier-modification effects at sub-barrier energies. The simple, $\ensuremath{\ell}=0$ barrier-based Wong formula is shown to ignore the modifications of the barrier due to its inbuilt $\ensuremath{\ell}$ dependence via $\ensuremath{\ell}$ summation, which is found to be adequate enough to explain the capture cross sections for all the three above-mentioned $^{48}\mathrm{Ca}$-based reactions forming superheavy systems. For the capture (equivalently, quasifission) reactions, the complete $\ensuremath{\ell}$-summed Wong formula is shown to be the same as the dynamical cluster-decay model expression, of one of us (R.K.G.) and collaborators, with the condition of fragment preformation probability ${P}_{0}^{\ensuremath{\ell}}=1$ for all the angular momentum $\ensuremath{\ell}$ values. In the case of fusion-evaporation cross sections, however, a further modification of barriers is required for below-barrier energies, affected in terms of either the barrier ``lowering'' or barrier ``narrowing'' via the curvature constant. Calculations are made for use of nuclear proximity potential, with effects of multipole deformations included up to hexadecapole, and orientation degrees of freedom integrated for both the coplanar and noncoplanar configurations.