Fission barrier
It is important to note that efforts to understand fission processes are still an ongoing and have been a very difficult problem to solve since fission was first discovered by Lise Meitner, Otto Hahn, and Fritz Strassmann in 1938. While nuclear physicists understand many aspects of the fission process, there is currently no encompassing theoretical framework that gives a satisfactory account of the basic observations. Scission. The fission process can be understood when a nucleus with some equilibrium deformation absorbs energy (through neutron capture, for example), becomes excited and deforms to a configuration known as the "transition state  or saddle point" configuration. As the nucleus deforms, the nuclear Coulomb energy decreases while the nuclear surface energy increases. At the saddle point, the rate of change of the Coulomb energy is equal to the rate of change of the nuclear surface energy. The formation and eventual decay of this transition state nucleus is the rate-determining step in the fission process and corresponds to the passage over an activation energy barrier to the fission reaction. When this occurs, the neck between the nascent fragments disappears and the nucleus divides into two fragments. The point at which this occurs is  called the "scission point". Liquid drop model. From the description of the beginning of the fission process to the "scission point," it is apparent that the change of the shape of the nucleus is associated with a change of energy of some kind. In fact, it is the change of two types of energies: (1) the macroscopic energy related to the nuclear bulk properties as given by the liquid drop model and (2) the quantum mechanical energy associated with filling the shell model orbitals. For the nuclear bulk properties with small distortions, the surface, formula_1, and Coulomb, formula_2, energies are  given by: where formula_5 and formula_6 are the surface and Coulomb energies of the undistorted spherical drops, respectively, and formula_7 is the quadrupole distortion parameter. When the changes in the Coulomb and surface energies (formula_8, formula_9) are equal, the nucleus becomes unstable with respect to fission. At that point, the relationship between the undistorted surface and Coulomb energies becomes: where formula_11 is called the fissionability parameter. If formula_12, the liquid drop energy decreases with increasing formula_7, which leads to fission. If formula_14, then the liquid drop energy decreases with decreasing formula_7, which leads to spherical shapes of the nucleus. The  Coulomb and surface energies of a uniformly charged sphere can be approximated by the following expressions: where formula_18 is the atomic number of the nucleus, formula_19 is the mass number of the nucleus, formula_20 is the charge of an electron, formula_21 is the radius of the undistorted spherical nucleus, formula_22 is the surface tension per unit area of the nucleus, formula_23 and formula_24. The equation for the fissionability parameter then becomes: where the ratio of the constant formula_26 is referred to as formula_27. The fissionability of a given nucleus can then be categorized relative to formula_28. As an example, plutonium-239  has a formula_28 value of 36.97, while less fissionable nuclei like bismuth-209 have a formula_28 value of 32.96. For all stable nuclei, formula_11 must be less than 1. In that case, the total deformation energy of nuclei undergoing fission will increase by an amount formula_32, as the nucleus deforms towards fission. This increase in potential energy can be thought of as the activation energy barrier for the fission reaction. However, modern calculations of the potential energy of deformation for the liquid drop model involve many deformation coordinates aside from formula_7 and represent major computational tasks. Shell corrections. 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, formula_34, the shell, formula_35, and pairing, formula_36, 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.