# What are the nuclear shell assignments for the odd neutron and proton in $^8B$?

I am having some trouble assigning the odd proton and neutron to the appropriate nuclear shells in the $$^8B$$ nuclide formed in the $$^7\mathrm{Be}(p, \gamma)^8\mathrm{B}$$ reaction in the solar plasma.

The $$1s_{1/2}$$ shell is filled by two $$n$$ and two $$p$$. It seems appropriate to place the remaining three protons in the next level $$1p_{3/2}$$ since that is the lower energy of the two splits at $$\mathcal{l}=1$$. I would think the odd remaining neutron should go in $$1p_{3/2}$$ as well, since it is lower energy than $$1p_{1/2}$$, but I can't obtain the published $$J^{\pi}=2^+$$ for the resultant nuclear spin with that assignment, i.e., $$J = j_1 + j_2 = 3/2 + 3/2 = 6/2 = 3$$.

If I kick the odd neutron up to $$1p_{1/2}$$ then technically I am supposed to subtract the two spins, $$J = j_1 - j_2$$, since they are in different Schmidt groups. That would give resultant $$J= 3/2 - 1/2 = 2/2 = 1$$.

If I go ahead and add the two spins as $$j_1 + j_2 = 3/2 + 1/2$$ then I get the published $$J = 2$$ and the parity, being the product of two even parity, is the desired positive parity for $$J^{\pi} = 2^+$$.

Am I correct in this assignment? I haven't been able to find any mention of this nuclide as an exception to the rules as I set out above, perhaps because it is odd-odd and not going to be stable (it $$\beta^+$$ decays to $$^8\mathrm{Be}$$ which almost immediately disintegrates to two $$\alpha$$ particles).

• – user4552
May 24, 2019 at 16:54
• I'm a nuclear physicist, and I've never heard the term Schmidt group. What does it refer to? When you combine two spins $j_1$ and $j_2$, you don't necessarily get the sum and difference. You can get the values in between as well. The ground state does not have to be a state in which the odd particles have good $j$. It can be a mixture of the configurations $\pi p_{1/2}\otimes \nu p_{3/2}$, $\pi p_{3/2}\otimes \nu p_{1/2}$, and $\pi p_{3/2}\otimes \nu p_{3/2}$.
– user4552
May 24, 2019 at 16:59
• Nordheim and Mayer published an internal document on nuclear shell model in 1951 using Schmidt group term for $j_1 = \mathcal{l}_1 \pm 1/2$ and $j_2 = \mathcal{l}_2 \mp 1/2$. It was declassified in 2004: osti.gov/servlets/purl/4420949 May 27, 2019 at 20:19
• The $^8\mathrm{B}$ problem I posted is solved using the Brennan-Bernstein rules. I wanted to post those since they are helpful in the odd-odd case more often than not (and difficult to find online). Oregon U. discusses them in their Shell Model Lecture 6 at oregonstate.edu/instruct/ch374/ch418518 May 27, 2019 at 20:25