Semiclassical Calculation of Shell Effects in Deformed Nuclei

The explanation of the asymmetry in the fragment mass distributions of many actinide nuclei has long been a stumblestone in fission theory. Shortly after the advent of the shell model, Lise Meitner related the mass asymmetry to shell structure, in particular to magic nucleon numbers in one of the fragments. Indeed, since the classical liquid drop model (LDM) prefers symmetric shapes, the mass asymmetry had to be a quantum shell effect. Johansson attempted to calculate fission barriers by summing the single-particle energies of a Nilssontype shell model including left-right asymmetric deformations. However, due to their wrong average behaviour at large deformations, he could not obtain finite barriers. Only after the introduction of the ingenious shell-correction method by Strutinsky this became possible, launching large-scale investigations of the shell structure in fission barriers. The resulting picture of the doublehumped fission barrier was nicely confirmed in many detailed experiments. One interesting outcome was that the onset of the left-right asymmetry of the nuclear shapes starts quite early during the fission process: in nuclei like Pu, already the outer fission barrier is unstable against octupole-type deformations. This quantum shell effect could be related to a few specific diabatic single-particle states which were particularly sensitive to the asymmetric deformations. Today, we know that symmetric and asymmetric fission modes may coexist in the same nucleus. Of course, the mass distributions of the fission fragments can only be predicted from a dynamical theory involving inertial parameters. But interestingly enough, the most probable experimental mass ratios were found to be roughly equal to those of the nascent fragments obtained statically at the asymmetric outer barrier. The onset of the mass asymmetry in nuclear fission was thus established as a quantum shell effect due to specific single-particle states in the deformed average potentials of the nucleons, which could not be understood in the classical liquid drop model.