Barrier for cold-fusion production of superheavy elements

Abstract

We estimate the fusion-barrier height ${B}_{\mathrm{fu}}^{\text{(two-body)}}$ for approaching ions in cold-fusion reactions in a model where the projectile deformation and quadrupole zero-point vibrational energy are taken into account. This barrier height is defined as the barrier energy at the target and projectile separation distance where an original oblate deformation of projectile and/or target caused by a repulsive Coulomb force turns into a large prolate deformation caused by the attractive nuclear force as the target and projectile come closer. The instability develops before touching because the attractive short-range nuclear force overcomes the repulsive Coulomb force and the shape-stabilizing effect of shell structure. The shell structure of the doubly magic $^{208}\mathrm{Pb}$ target is sufficiently strong that its shape remains very close to spherical in all cases studied here. The fusion potential for approaching ions in the two-body channel is calculated in the macroscopic-microscopic model with the quadrupole vibrational zero-point energy obtained in the WKB approximation. We compare our results with data from 10 experimental cold-fusion reactions and with the Bass barriers. Differences and similarities between the Yukawa-plus-exponential model and the Bass model are discussed. We also calculate five-dimensional potential-energy surfaces for the single compound system and show that well-established fission and fusion valleys are both present. For heavy systems, ${B}_{\mathrm{fu}}^{\text{(two-body)}}$ becomes lower than the fission barrier just beyond the ground state of the compound system. In the vicinity of this transition, the optimum collision energy for formation of evaporation residues can be expected to depend in a delicate fashion on the interplay among ${B}_{\mathrm{fu}}^{\text{(two-body)}}$, the fusion valley, the fission barrier of the compound system, and the one- and two-neutron separation energies ${S}_{1n}$ and ${S}_{2n}$. We discuss these issues in detail and calculate fission-barrier heights. Except for reactions in which the projectile is doubly magic or near doubly magic, the calculated quantities are consistent with the observed optimal energies for evaporation-residue formation.