This doesn’t normally happen. Radiation *almost* never makes something radioactive.
There are two situations in which it does. The first, you already seem to have a decent idea of. Things exposed to *specifically* neutron radiation can become radioactive through nuclear transmutation (a physical change in the composition of the nucleus, in this case adding a neutron). However, most of the time, they will emit a different kind of radiation, and so the process cannot be repeated.
The other situation is the more common one – absorption or adhesion. If you hang out around some radioactive stuff, odds are good that some of it will stick to your clothes or be absorbed by you through breathing. The radiation itself doesn’t make you radioactive, but the radioactive shit that sticks to you and gets inside of you does. This is why marie curie’s body is so radioactive.
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