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While doing some studying for an exam in introductory nuclear physics, I stumbled upon a question I can't answer. I'm supposed to explain the ground state and the three lowest energy excited states of $^{13}C$ using the shell model. The ground state has spin and parity $1/2^-$. The excited state with $E = 3.09 \text{ MeV}$ corresponds to $1/2^+$, the $3.68 \text{ MeV}$ state to $3/2^-$ and $3.85 \text{ MeV}$ to $5/2^+$. I know that $^{13}C$ has one unpaired neutron, which will determine the spin and parity of the nucleus. $1/2^-$ must then correspond to a $p_{1/2}$-level, $1/2^+$ to an $s_{1/2}$-level, $3/2^-$ to $p_{3/2}$ and $5/2^+$ to $d_{5/2}$. (See table below.) The unpaired nucleon must reside in the level corresponding to each state.

\begin{align} 0 \text{ MeV} && 1/2^- && p_{1/2} \\ 3.09 \text{ MeV} && 1/2^+ && s_{1/2} \\ 3.68 \text{ MeV} && 3/2^- && p_{3/2} \\ 3.85 \text{ MeV} && 5/2^+ && d_{5/2} \\ \end{align}

In order to transition from $1/2^-$ to $1/2^+$, we can either let the unpaired neutron in $1p_{1/2}$ jump to $2s_{1/2}$ or let a paired neutron jump from $1s_{1/2}$ to $1p_{1/2}$. Other options seem out of the question, since the $3s_{1/2}$-level lies much higher and would therefore require too much energy to achieve. How would I determine which of the two would happen?

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Unfortunately there is no easy rule to unambiguously determine whether the valence neutron is in 2s1/2 or 1s1/2 orbital. The situation is even more complex if you consider the interacting shell model, which include 2-body interaction between those single-particle states. Usually if you consider some specific model space in shell model (for this case is spsdpf) you can roughly estimate based on the single-particle energy but still it is a nontrivial task for many cases. A proper shell model calculation will give you the (fractional) occupation number for each orbital or the spectroscopic factor for specific configuration, which are the information you need.

For the specific case you ask, from the particle-vibration coupling (PVC) model by N. Vinh Mau, Nuclear Physics A 592, 33 (1995), the 1/2+ is mainly (91.5%) from a valence neutron 2s1/2 + 12C core in ground state. It is interesting to note that it also include a small 8.5% fraction from the configuration of 1d5/2 + 12C core in 2+ excited state.

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