Fusion by diffusion. II. Synthesis of transfermium elements in cold fusion reactions

We describe a method of estimating cross sections for the synthesis of very heavy nuclei by the fusion of two lighter ones. The cross section is considered to be the product of three factors: the cross section for the projectile to overcome the Coulomb barrier, the probability that the resulting composite nucleus reaches the compound nucleus configuration by a shape fluctuation treated as a diffusion of probability in one dimension, and the probability that the excited compound nucleus survives fission. Semi-empirical formulas for the mean Coulomb barrier height and its distribution around the mean are constructed. After overcoming the Coulomb barrier the system is assumed to be injected into an asymmetric fission valley by a rapid growth of the neck between the target and projectile at approximately frozen asymmetry and elongation. Diffusion in the elongation coordinate in this valley can occasionally bring the system over the saddle separating the injection point from the compound nucleus configuration. This is the stage that accounts for the hindrance to fusion observed for very heavy reacting systems. The competition between deexcitation of the compound nucleus by neutron emission and fission is treated by standard methods, but an interesting insight allows one to predict in an elementary way the location of the maximum in the resulting excitation function. Adjusting one parameter in the theory causes the calculated peak cross sections to agree within about a factor of 2 or so with 12 measured or estimated values for cold one-neutron-out reactions where targets of Pb-208 and Bi-209 are bombarded with projectiles ranging from Ca-48 to Zn-70. The centroids of the excitation functions agree with theory to within 1 or 2 MeV for the six cases where they have been determined, and their widths are reproduced. Hot fusion reactions, where several neutrons are emitted, are not treated, except that a comparison is made between the hindrance factors in cold and hot reactions to make elements with atomic numbers 112 to 118. The calculated diffusive hindrances in the hot reactions are less unfavorable by 4 to 5 orders of magnitude.