From “Nuclear structure from a simple perspective” by Richard F. Casten, Chapter 4. The first half discusses the coupling between pp or nn T=1 isovector pair under δ-interaction, which is a good approximation to short-range interaction.

I found that the book makes it a bit complicated. The only 3 cases matter

The logic is follow. As the spatial part of the total wavefunction must be symmetric for the δ-interaction to be effective. The spin-isospin part must be antisymmetric as the total wavefunction must be antisymmetric due to identical fermions system. Also, the isospin part must be isovector or symmetric. Thus, the spin part must be antisymmetric or S = 0.

- In other word, the δ-interaction has no effect on the S =1 state.
- Also, the book said that .
- Most parallel states are either

Using these 3 reasons, and consider these 3 cases, which actually covered all possible combinations and all cases. Point 2 also means that only half of the coupled states are affected. Since the number of state are even, when then or via versa. If is formed by pure S = 1 state, thus, the most affected state is as is the other most parallel states, in which the overlap of the wavafunctions is maximum.

In the book, the example is d5/2 f7/2, J = 1, 2, 3, 4, 5, 6. This is belong to case 1, thus, the lowest state is J = 1. another example is d5/2 g7/2, which is case 2, the lowest state is J = 6.

For equivalent orbit, either case 1 or case 3. The lowest state is J = 0.

There are one interesting case. when coupling the s1/2. Since the s-orbit is isotropic, thus, no matter how the other orbit, the overlap is always maximum. Since pure S = 1 state is unaffected, thus, we can somehow, imagine the nucleon pair is forming S = 0 pair. In fact, for example, when s1/2 couples with d5/2,

The J = 3 state is unaffected by δ-interaction