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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).

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  • $\begingroup$ related: physics.stackexchange.com/questions/454246/… $\endgroup$ – Ben Crowell May 24 at 16:54
  • $\begingroup$ 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 $\pi p_{3/2}\otimes \nu p_{3/2}$. $\endgroup$ – Ben Crowell May 24 at 16:59
  • $\begingroup$ 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 $\endgroup$ – Dalton Bentley May 27 at 20:19
  • $\begingroup$ 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 $\endgroup$ – Dalton Bentley May 27 at 20:25

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