Nuclear fission in various nuclei

Question

If a nucleus of U-238 has binding energy of 7.570 MeV, does that mean if a neutron hitting it has a higher energy, it'd cause it to split? If so, why are some nuclei split with thermal neutrons, and how do we determine whether a certain nucleus is fissile or not?

Answer 1

According to the Table of Nuclides, the strong interaction binding energy of U-238 is $7.570$ MeV per nucleon, not in total. Its total binding energy, i.e. the energy needed to separate U-238 nucleus into individual protons and neutrons is $7.570\times238=1802$ MeV. This not, however, the energy needed for it to split, since there are many lighter combinations of nuclei with the same total number of protons and neutrons as U-238, but with less total rest mass. In other words, it actually takes zero energy to split a U-238 nucleus, but the reason it doesn't fall apart immediately is because any fission fragments must overcome a fission barrier before they can escape.

For example, the most common decay of U-238 is via alpha emission, and the alpha particle must tunnel through a Coulomb barrier of about 25 MeV. This high barrier is why the lifetime of U-238 is billions of years.

A captured neutron can induce fission if it leaves the new nucleus in an state with an excitation energy greater than the fission barrier for that nucleus. Fissile nuclides (e.g. U-235) are those that even capturing a slow (approximately zero kinetic energy) neutron leaves the new nucleus in an excited state (e.g. U-236$^*$) above the fission barrier, so it immediately undergoes fission. Fissionable nuclides (e.g. U-238) are those that require the captured neutron to have had sufficient kinetic energy to put the new nucleus in an excited state above the fission barrier.

For example, most uranium isotopes have fission barriers $\sim5.5-6$ MeV. When a slow ($\sim$ zero kinetic energy) neutron is captured by a U-235 nucleus it produces a U-236 nucleus with about 6.5 MeV excitation energy, which is above the fission barrier so the nucleus splits. In contrast, when a slow neutron is captured by a U-238 nucleus it produces a U-239 nucleus with an excitation energy of only 4.8 MeV, which is below the fission barrier. This is why only fast neutrons with energies $\gtrsim 1$ MeV can dependably induce fission in U-238.