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What type of radioactive decay involves only the release of high-energy photons and therefore does not change the mass of an atom of a radioactive isotope?

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Answer is: gamma emission or gamma decay.
During gamma emission the nucleus emits radiation without changing its composition, if for example have nucleus with six protons and six neutrons (carbon atom) and after gamma decay there is nucleus with six protons and six neutrons.
Gamma rays are the electromagnetic waves with the shortest wavelengths (1 pm), highest frequencies (300 EHz) and highest energy (1,24 MeV).

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