A nucleus decays by giving
- Alpha particles
- Beta particles
- Gamma particles
My question is, why does nucleus decay in only 3 ways? Why don't we see proton emission or neutron emission etc.?
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The comment by @james-large contains the essential answer. I repeat it here and give a bit more context.
why does nucleus decay in only 3 ways?
The nucleus decays in more ways than 3. You refer to alpha, beta, and gamma radiation but there are many other ways a nucleus can decay as shown in the wikipedia link in the comments, again here.
Alpha, beta, and gamma radiations are the most commonly referenced because they are the most common on Earth and they were the first three to be discovered (they are in fact named for the first three Greek letters). The main particles emitted in these decays have long since been identified - the "alpha" particle a Helium-4 nucleus, the "beta" particle is and electron or positron, and the "gamma" particle is a photon. Nevertheless, the original names have remained.