Induced fission

Induced fission is a mode of decay in which a heavy nucleus is struck by a neutron, causing it fuse with the nucleus and producing a highly unstable excited isotope which then immediately undergoes nuclear fission.

The nuclear reaction reads

where $\ce{X}$ is the original nuclide, $\ce{^{A+1}X^{*}_{N+1}}$ is its isotope with one more neutron, left in an excited state, $\ce{X'}$ and $\ce{X''}$ are the fission fragments and $\nu n$ are the $\nu$ additional neutrons that are emitted as part of the fission. It is a two-step process in which the nucleus is first bombarded with a neutron, then the resulting isotope undergoes spontaneous fission. As such, induced fission is essentially a brute-force method to make spontaneous fission happen when it can't happen naturally, or when it doesn't happen sufficiently fast for the intended use case. For this reason, the physics behind induced fission are almost the same as the spontaneous case, with the only major difference being the first step.

Since induced fission relies on creating a highly unstable isotope that will spontaneously split, not all nuclides are a good fit for nuclear fuel. Materials that undergo fission after being struck with a low-energy neutron (thermal neutron, $\sim 0.025\text{ eV}$) are called fissile materials. The most common fissile materials is $\ce{^{235}U}$. Materials that undergo fission only when struck by a high-energy neutron (fast neutron, $\sim 1\text{ MeV}$) are called fissionable materials. The most common one is $\ce{^{238}U}$. The big difference is that fissile materials can sustain a thermal-neutron chain reaction, whereas fissionable ones cannot. Finally, materials that don't undergo fission when struck, but produce fissile or fissionable material are called fertile materials. The most common are $\ce{^{234}U}$ and, maybe surprisingly, $\ce{^{238}U}$ again. $\ce{^{234}U}$ produces $\ce{^{235}U}$, which we mentioned is a fissile. Meanwhile, $\ce{^{238}U}$ can momentarily become $\ce{^{239}U^{*}}$ and then split, but unless the neutron was very energetic, it generally becomes $\ce{^{239}U}$ (not excited) and then beta decays twice to become $\ce{^{239}Pu}$, which is fissile.

For example, one possible induced fission of $^{235}\text{U}$ reads

Energy#

In induced fission, there's two kinds of energy we're interested in. On one hand, we are interested in the Q value, since that's what the fission finally yields in usable energy. But on the other hand, we also need to consider how much energy we need to give to the nucleus to trigger the fission. The latter is a bit more complicated and there are actually two metrics for it.

The excitation energy is the difference in rest energy between the excited and non-excited state of the isotope that's created by the neutron absorption:

where the excited isotope's mass is easily calculated by just adding a neutron to the original isotope's mass: $M(\ce{^{A+1}X^{*}})=M(\ce{^{A}X})+m_{n}$. This is different from the Q value, which we don't care about for the neutron collision. This intentionally does not include the kinetic energy contribution of the neutron because...

...the activation energy is the energy that is actually required to induce fission. It's the energy that's needed overcome the potential barrier that prevents the nucleus from splitting. Critically, this is different from excitation energy and this has important consequences on induced fission. If the activation energy is higher than the excitation energy, that means that just exciting that nucleus is not enough to induce fission. In that case, not only must you excite the nucleus, but the neutron you excite it with must also transfer enough kinetic energy to cover the difference between excitation and activation energy. Only then will the nucleus have enough to undergo fission. However, if the activation energy is lower than the excitation energy, that means that any excitation will result in a splitting of the nucleus. As such, materials that have a low activation energy are much more likely to be useful for induced fission. In fact, this is the difference between fissile and fissionable materials: if the activation energy is lower than excitation energy, it's fissile; if it's higher, it's fissionable.

Activation energy is often measured empirically, but we can use the nuclear shell model to make some theoretical predictions.

If you do manage to supply the activation energy, fission happens. This is where energy is actually released and where we care about the $Q$ value. Since this process is spontaneous fission, you may refer to that article for discussion on the $Q$ value.

The difference between these two is important and just exciting $\ce{^{239}U}$ will not cause it to split. The neutron must supply at least $E_\text{act}-E_\text{exc}=1.8\text{ MeV}$ of energy for it split. This is why $\ce{^{238}U}$ is considered fissionable.