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It's not atoms that are involved in nuclear fusion, rather, as the name implies, it's nuclei that are involved. Therefore it's easier to talk about what nuclear isotopes are involved in fusion. Well, with enough energy supplied, they all could be, but I bet you mean exothermic fusion reactions (release energy), in which case the isotopes involved have an atomic number of around 26 (iron) or lower.

On a subatomic level, the explanation can get complicated. Basically, though, as one isotope gets closer and closer to another, they become more and more electromagnetically repelled, since nuclei only consist of protons (charge +1) and neutrons (charge 0). However, there is a distance, known as the "Coulomb barrier," where the force of electromagnetic repulsion gets overtaken by the strong nuclear force, and the two recently separate isotopes bind together. Breaking this barrier can take quite a bit of energy though.

Ultimately then, the main particle involved in nuclear fusion is the gluon, since it mediates the strong nuclear force.

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