What happens to neutrons after they are radiated?

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During certain processes, there can be free neutrons. If an accelerator beam hits into a target, it can spray neutrons into a cloud, called neutron skyshine for example. What happens to these neutrons that float around?

My guess would be that they float around until they beta decay into a proton and an electron. What happens to these protons and electrons?

Sorry for my very limited knowledge on this.
 
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Some neutrons may hit other nuclei and produce some radioactive isotopes. The majority decays into protons and electrons. The protons eventually capture an electron and become hydrogen atoms that diffuse away. The quantities of hydrogen are so minute that they are undetectable.
 
Depends what materials are around them, to some extent. They bounce around and loose their KE to the nucleii they bounce into. Bodies that bump into like-masses transfer the greatest magnitudes of energy, so water, or fully saturated hydrocarbons like polyethylene, will slow down the neutrons until they become 'thermal' the quickest. This process is called 'moderation'. Usually, the neutrons will then just float around, like hydrogen in a sponge, until a nucleus absorbs the thermal neutron and undergoes 'activation'.

'Activation' products are [obviously] more neutron-heavy that they were before, so usually decay by electron emission, leaving behind the proton and causing that nucleus to transform into the next element up. Some nucleii will absorb thermal neutrons so readily that effectively none will remain in a very short period of time after moderation.

However, they may also moderate into other materials that do not readily absorb neutrons, and those neutrons would then decay with a half-life of around 20 mins into a hydrogen atom. This can get pretty serious for nuclear reactors because what effectively happens, with enough neutrons, is that surrounding metals such as stainless steel exposed to neutron flux will become 'sponge-like' - filled with interstitial hydrogen which embrittles the metal.
 
Thanks for your answers; very helpful!
 
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