Other explanations in this thread are really great. To add to this, it often isn’t the radiation itself that does the job. You are about 2/3 water, so it is overwhelmingly likely that what a gamma ray will hit (external radiation is almost always gammas, because betas and alphas have very poor skin penetration) is the electron clouds of water molecules, knocking those loose. Those accelerated electrons in turn cascade into other electrons, etc, and for every primary gamma interaction, you can have 10,000+ secondary events, creating all sorts of short-lived, very nasty particles called free radicals and superoxides, like H2O+, H202, OH-, etc. Those nasty critters tear and shred all the complex molecules they encounter, like proteins, which can immediately kill the cell, and DNA, which can kill the cell via self-regulation failure, or by failure to divide. (Source: nuclear medicine specialist)
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