Barrier for Cold-Fusion Production of Superheavy Elements

We estimate the fusion-barrier height { m fu}^{ ext{(2-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. The fusion potential for approaching ions is calculated in the macroscopic-microscopic model and the quadrupole vibrational zero-point energy is obtained in the WKB approximation. With this model, we obtain a large reduction of { m fu}^{ ext{(2-body)}}$ relative to the conventional model for the 'Coulomb barrier' in heavy-ion reactions. 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. Although { m fu}^{ ext{(2-body)}}$ is higher than the fission barrier for lighter systems, it becomes lower for heavier systems. In the vicinity of this transition the optimum collision energy for formation of evaporation residues depends in a delicate fashion on the interplay between { m fu}^{ ext {(2-body)}}$, the inner fusion barrier, the fission barrier of the compound system, and on the one and two neutron separation energies { m 1n}$ and { m 2n}$. We discuss these issues in detail and compare theoretical results with experimental cold-fusion data for 10 reactions. 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.