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I was reading Atomic Structure and Spectral Lines by Arnold Sommerfeld for historical purposes. This book is a classic from the 1920s. He writes

If, in accordance with the sense of Prout’s hypothesis, H-nuclei are the real elementary “bricks” of which all gravitational matter is built up, it must cause surprise that in the radioactive transformations “ H-rays ” have never been observed. Why does not the hydrogen nucleus occur as a decay product of the higher elements just as well as the less simple He-nucleus ? According to what law of displacement would such an “ H-transformation ” take place? Since the H-nucleus is endowed with a simple positive charge and since it has the atomic weight 1, the law must clearly be: Displacement in the periodic system by one unit to the left and simultaneously a decrease of the atomic weight by one unit. Actually, such H-transformations have never been observed among the spontaneous radioactive processes, however much they may have been sought.

For reference, Prout hypothesized that hydrogen is the building block of elements. Prout was so shy of this idea that he published it anonymously. The Wikipedia article on photon emission was found as early as 1969. This is quite late in the field, which was so intensively studied for 50 years.

Why is proton emission such a rare phenomenon, i.e., during nuclear distintegration, a proton does not leave the nucleus but rather a "helium nucleus" comes out?

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Just to be clear, when you say proton emission, I assume you mean a nucleus in ground state decay out a proton in a similar fashion of alpha decay, since beta-delay proton emission is not really that rare. For the ground state proton decay, by definition the proton separation energy is negative and therefore only unbound nuclei outside the proton dripline are allowed. However, within these nuclei the competitions of proton decay with positron decay, electron capture, and alpha decay are also strong. Finally, creating these proton-rich nuclei in accelerator and measuring their extremely fast decay are very challenging. These reason explain why the first proton decay from ground state nucleus was found rather recently (1982) compared to other forms of decay and only around 40 of them (with mass number more than 100) are found as of 2023. For more detail references, see M. Pfutzner et al., Review of Modern Physics 84, 567 (2012).

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