In order to get more reasonable values for the nuclear masses in the liquid drop model, it is necessary to include shell effects. Soviet physicist Vilen Strutinsky proposed such a method using "shell correction" and corrections for nuclear pairing to the liquid drop model. In this method, the total energy of the nucleus is taken as the sum of the liquid drop model energy, , the shell, , and pairing, , corrections to this energy as:
The shell corrections, just like the liquid drop energy, are functions of the nuclear deformation. The shell corrections tend to lower the ground state masses of spherical nuclei with magic or near-magic numbers of neutrons and protons. They also tend to lower the ground state mass of mid shell nuclei at some finite deformation thus accounting for the deformed nature of the actinides. Without these shell effects, the heaviest nuclei could not be observed, as they would decay by spontaneous fission on a time scale much shorter than we can observe.
This combination of macroscopic liquid drop and microscopic shell effects predicts that for nuclei in the U-Pu region, a double-humped fission barrier with equal barrier heights and a deep secondary minimum will occur. For heavier nuclei, like californium, the first barrier is predicted to be much larger than the second barrier and passage over the first barrier is rate determining. In general, there is ample experimental and theoretical evidence that the lowest energy path in the fission process corresponds to having the nucleus, initially in an axially symmetric and mass (reflection) symmetric shape pass over the first maximum in the fission barrier with an axially asymmetric but mass symmetric shape and then to pass over the second maximum in the barrier with an axially symmetric but mass (reflection) asymmetric shape. Because of the complicated multidimensional character of the fission process, there are no simple formulas for the fission barrier heights. However, there are extensive tabulations of experimental characterizations of the fission barrier heights for various nuclei.