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I am reading about several processes in nuclear physics that put letters in parentheses next to the names of the isotopes.

Usually two letters, separated by a comma.

Besides $d$ and $p$, I have seen a (or alpha?) and n.

I first thought d and p stood for energy (sub)levels, but now I'm thinking p is proton, n is neutron, d is 'daughter' (nuclide?) Or deuterium? And a or 'alpha' is alpha particle....

Are these protons and neutrons and deuterons and such part of the initial reaction? Or end result?

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That notation confused me when I first encountered it, too. The interpretation is

$$ \rm target (incoming, outgoing) residual $$

So your example, $\rm^{28}Si(d,p)$, suggests a silicon target exposed to a beam of deuterons (deuterium nuclei), where the particles which are detected after the interaction are protons. The mostly likely interpretation is neutron transfer,

$$ \rm ^{28}Si + d \to p + {}^{29}Si $$

but there are other possibilities, too. Depending on the energies involved the $\rm^{29}Si$ nucleus might emit a gamma cascade, or re-emit the captured neutron, or fission. But if all you have is a proton detector, then you detect $\rm^{28}Si(d,p)$, and speculating about all the different final states occupies a long section in your paper.

This notation also allows you to talk about beam reactions independently of their targets. I did this already when I referred to your $\rm(d,p)$ as a "neutron transfer reaction." You will also find $(n,\gamma)$ for radiative neutron capture and $\rm(n,f)$ for neutron-induced fission.

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