Total nuclear spin of deuteron

[Demon117]

Hello all, I am having trouble understanding how this works. In Krane there arises a discussion on total angular momentum I of the deuteron. While it is true it has three components, namely the individual spins of the neutron and proton, but also the orbital angular momentum l of the nucleons as they move about their common center of mass. This total angular momentum can be denoted by

$I=s_{p} + s_{n} + l$

He continues on to talk about the different ways to couple these contributions and states there are only four possibilities. I can see the first two possibilities for total angular momentum I=1, but the other two make no sense. These are the possibilities:

(1) $s_{n}$ and $s_{p}$ are parallel with $l=0$
(2) $s_{n}$ and $s_{p}$ are antiparallel with $l=1$
(3) $s_{n}$ and $s_{p}$ are parallel with $l=1$
(4) $s_{n}$ and $s_{p}$ are parallel with $l=2$

One can see why (1) and (2) hold by inspection but (3) and (4) make my brain hurt. Perhaps I am just not seeing the correct orientation. Any suggestions?

 

[Bill_K]

matumich26, Do you know how to couple two angular momenta? You don't just add them. When you couple J₁ to J₂ the combined system can have any value of J in the range from their sum J₁ + J₂ to their difference |J₁ - J₂|. You must include all of these possibilities. For example when you coupled the two spins together, 1/2 ⊗ 1/2 = 1 ⊕ 0. That's the way we write it, and it means the coupled system can have either S = 1 (parallel) or S = 0 (antiparallel).

Ok, you wanted to look at the cases with S = 1. When you further couple S with L,

1 ⊗ 0 = 1
1 ⊗ 1 = 2 ⊕ 1 ⊕ 0
1 ⊗ 2 = 3 ⊕ 2 ⊕ 1

You can see in all three of these cases, I = 1 is one of the possibilities.