# Questions about specific behavior of fission daughter products

I'm trying to understand the specific behavior of the daughter products created by nuclear fission. My statements and questions below are specific to $^{235}$U fission moderated by light pressurized water.

## My current understanding

As I understand, when a $^{235}$U gets "hit" by a thermal neutron, then the resulting $^{236}$U is in an elevated energy state, so much so that it deforms (i.e. the "punching bag" effect), and then Coulomb repulsion overcomes the strong force and pushes the two lobes apart.

In binary fission, the two daughter nuclei are generally around mass 95 and 135, and an average of 2.4 fast neutrons are emitted.

The fast neutrons are moderated primarily by the hydrogen nuclei of the water molecules. Any secondary moderation occurs by elastic (or inelastic) collisions with other nuclei.

I've already reviewed the below sites:

## Questions

1. Are the 2.4 neutrons (or thereabouts) emitted at the same exact moment1 that the two daughter lobes split apart? Or, are they subsequently emitted later from the daughter nuclei because they are in an unstable state? (Or both scenarios?)

2. Various sites mention that the free neutrons are travelling at c. 10000-25000 $\frac{km}{s}$, but how fast are the larger daughter nuclei travelling?

3. Assuming 95/135 binary fission, do these nuclei fly apart at velocities inversely proportional to their mass2?

4. Since the Coulomb repulsion dominates the trajectory of the larger nuclei, then by what mechanism are the neutrons emitted at such high velocities3?

5. I vaguely remember reading somewhere (or perhaps I'm imagining it) that fast neutrons are travelling at velocities such that, if one were to hit a large nucleus (e.g. $^{235}$U, $^{238}$U, leftover daughter nuclei, decay garbage, etc.) it could conceivably "punch through" the nucleus, briefly creating a "hole" of sorts, but that the nucleus would quickly regain its composure. Is that indeed the case?

## Footnotes

1. On a quantum time scale.
2. I'm thinking of Newtonian Mechanics here.
3. I am aware of total conservation of momentum, but considering the neutral charge of neutrons, I can't imagine Coulomb forces at work here.

1. Are the fission neutrons emitted at the exact same momentum that the daughter lobes split apart? Or are they subsequently emitted later from the daughter nuclei because [the daughter nuclei] are in unstable states?

We can answer the first question with information about the energy spectrum of emitted neutrons. Let's consider a specific example reaction, just to grab some numbers: $$\rm ^{235}U + n \to {}^{236}U^* \to {}^{94}Zr + {}^{134}Xe + 3n + 200.2\,MeV$$ The energy released is the difference in the mass excess from the initial state to the final state. Initially the uranium is at some thermal-ish temperature, and the neutrons typically have thermal energies as well --- so the initial kinetic energy is about fifty milli-eV, and the momentum of the temporary $\rm U+n$ unbound state is negligible compared to the momenta of the five fragments that come out.

Now, if the fragments have no total momentum, their individual momenta must add up to (vector) zero. There are lots of solutions to that problem --- but most of them involve all five momenta having roughly the same magnitude $\left|\vec p\right|$. If the momentum of fragment $i$ is $\vec p_i$, its kinetic energy is $E_i = p_i^2/2m_i$, which gives us the expected "typical" distribution of particle energies: $$E_\mathrm{Zr} \approx E_\mathrm{Xe} \approx \frac{E_\mathrm{neutron}}{100}.$$ If all five particle were emitted at the same instant, we'd expect the neutrons to come out with an energy of sixty-ish MeV, and the heavy fission fragments to have energies in the range of rounding error.

In fact we see just the opposite (e.g. here): fission neutron energies are peaked around 1 MeV, and neutrons faster than 10 MeV are rare. That suggests instead that the fission is into two heavy fragments, which partially thermalize in their uranium environment before emitting a neutron with somewhere under the "neutron separation energy."

1. Various sites mention that the free neutrons are travelling at circa $10\,000-25\,000\rm\,km/s$, but how fast are the larger daughter nuclei travelling?
2. Assuming 95/135 binary fission, do these nuclei fly apart at velocities inversely proportional to their mass2?

I guess that takes care of your questions 1--3, actually.

1. Since the Coulomb repulsion dominates the trajectory of the larger nuclei, then by what mechanism are the neutrons emitted at such high velocities?

The mega-eV is a pretty typical energy scale for nuclear processes, just based on the momentum of a nucleon whose de Broglie wavelength is so short that it can be reasonably described as "confined to a nucleus." The mega-eV is about $10^{-3}m_\text{n} c^2$, so relativity gives typical neutron speeds like this: \begin{align} \gamma &= \frac{1}{\sqrt{1-\beta^2}} &&\approx 1 + 10^{-3} \\ \gamma^2 &= \frac{1}{1-\beta^2} \\ \beta^2 &= \frac{\gamma^2-1}{\gamma^2} \approx \gamma^2 - 1 &&\approx 2\times10^{-3} = 20\times10^{-4} \\ &&\beta &\approx 4.5\times10^{-2} =0.045 \end{align} That's the low end of your range: MeV neutrons around $c/20 \approx 10^7\rm\,m/s$.

1. "neutron punch-through"

I don't think that's a good description of the neutron-nucleus interaction.

• Thank you, sir, for the cool answer. That being said, even though the "neutron punch-through" might not be a good description, I'm CERTAIN I read that somewhere... – pr1268 Apr 2 '17 at 11:09

These are caused by the small value of the neutron lifetime, typically ranging from $10^{-8}$ to $10^{-4}$ s [...]