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In a neutral atom, can neutrons be ejected, and if so, by what (nuclear) process or reaction? I have read Neutron emission and Can a proton be ejected from an atom, but I don't understand how and if that would work for removing neutrons from a heavy atom such as lead-208. Can neutrons be ejected from lead-208 so that it is lead-197, and if not, why?

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The typical binding energy per nucleon $(E_\mathrm B/A)$ of most nuclides is about $5{-}8\ \mathrm{MeV}$. Such values are higher than the released energy of a typical radioactive decay. Therefore, emission of neutrons doesn’t occur during most usual radioactive decay processes.

In order to make a neutron source, you have to find a suitable target (not $^{208}\mathrm{Pb}$) and a corresponding nuclear reaction that releases a neutron and that is also energetically favourable, for example $$^9\mathrm{Be}\ (\alpha,\mathrm n)\ ^{12}\mathrm C$$ $$^9\mathrm{Be}\ (\gamma,\mathrm n)\ 2\ ^{4}\mathrm{He}$$ Note that $^{12}\mathrm C$ and $^{4}\mathrm{He}$ are very stable nuclides with a relatively high binding energy per nucleon so that these reactions can release a large amount of energy, but still you need to irradiate the beryllium targets with high-energy radiation (high-energy $\gamma$ radiation of more than $E=1.665\ \mathrm{MeV}$ e.g. of $^{124}\mathrm{Sb}$ or $\alpha$ radiation) to release neutrons.

Nevertheless, neutron emission can occur during radioactive decay if the neutron excess is large and the beta decay energy is high enough, i.e. at the edge of the valley of beta stability. At very high values, spontaneous emission of neutrons becomes possible.

More important, however, are so-called neutron precursors, e.g. $^{87}\mathrm{Br}$ or $^{137}\mathrm{I}$, which have a very high decay energy so that the product nuclides ($^{87}\mathrm{Kr}$ and $^{137}\mathrm{Xe}$, respectively) can get enough energy to make neutron emission possible in the next step. $${}^{87}\mathrm{Br}\xrightarrow[{E_\mathrm{max}=6.83\ \mathrm{MeV}}]{\beta^-}{}^{87}\mathrm{Kr}^*\xrightarrow[\quad]{}{}^{86}\mathrm{Kr}+\mathrm n$$ $${}^{137}\mathrm{I}\xrightarrow[{E_\mathrm{max}=5.89\ \mathrm{MeV}}]{\beta^-}{}^{137}\mathrm{Xe}^*\xrightarrow[\quad]{}{}^{136}\mathrm{Xe}+\mathrm n$$ Such reactions are responsible for the so-called delayed neutrons, which are very important for most nuclear reactors – without this effect, controlling a nuclear chain reaction would be difficult.

Of course, the most important neutron sources are nuclear reactors, which generate neutrons (about 2–4 neutrons per fission) in a controlled nuclear chain reaction.

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