Question regarding spectroscopic notation

[Telemachus]

I think there are different conventions. Once I worked with XPS, and there we used the spin quantum number, the letter for the orbital, and a number for the energy leverl I think, to characterize photoelectrons.

Another notation uses, to characterize an atomic state, the convention $^{2S+1}L_J$, where S is the total spin, L is the orbital angular momentum (characterized with a capital letter, S,P,D,F, etc), and J is the total angular momentum.

However, now I was trying to interpret this paper: http://adsabs.harvard.edu/full/1983ApJ...275L..19M

There are some atomic transitions involved, where relaxation of an atom delivers a photon, for example, in the abstract it is mentioned $^5S_2\rightarrow \left.^3P_{2,1}\right.$. I'm not sure on how to interpret this, because if it where $2S+1=5$, on the right hand side I would have $S=2$, and as is an S orbital, the orbital angular momentum should be 0, which would give $J=2$, but it should be $J=3$. How should I interpret this?EDIT: I think I have realized what is going on. I have forgotten after a while about angular momentum addition in quantum mechanics, I think the issue has to do with that.

 

[BigJDubs]

So, for the example in the abstract, you have $^5S_2\rightarrow \left.^3P_{2,1}\right.$. What this means is that the angular momentum of the state on the left hand side is 2, and the angular momentum of the right hand side is either 2 or 1, depending on the transition. So, for the right hand side you have a P orbital, which has an angular momentum of 1, and then you add the total angular momentum (which is 2) to get either 3 or 2, depending on the transition. Is this correct?