# Nucleus shell-model problem

## Homework Statement

The low-lying levels of C-13 are ground state, $$\frac{1^-}{2}$$; 3,09 MeV, $$\frac{1^+}{2}$$; 3,68 MeV, $$\frac{3^-}{2}$$; 3,85 MeV, $$\frac{5^+}{2}$$. Interpret these four states according to the shell model.

## Homework Equations

Negative parity --> $$\ell$$ = odd; the valence nucleon must occupy a level with the spectroscopic symbol p, f, h etc.
Positive parity --> $$\ell$$ = even; the valence nucleon must occupy a level with the spectr. symbol s, d, g etc.

## The Attempt at a Solution

In the ground state, the 7th neutron must be in the $$1p_{\frac{1}{2}}$$ level. All levels below are filled.

In the first excited state, that is, the $$\frac{1^+}{2}$$ state, I think the 7th neutron is excited from the $$1p_{\frac{1}{2}}$$ level to the $$2s_{\frac{1}{2}}$$ level. The remaining neutrons occupy the same levels as in the ground state.

In the $$\frac{3^-}{2}$$ state, one of the two pairs in the $$1p_{\frac{3}{2}}$$ level is broken, and a neutron is excited to the $$1p_{\frac{1}{2}}$$ level, where it forms a pair with the former valence neutron. The remaining neutrons occupy the same levels as in the ground state.

In the $$\frac{5^+}{2}$$ state, the single neutron in the $$1p_{\frac{1}{2}}$$ level is excited to the $$1d_{\frac{5}{2}}$$ level. The remaining neutrons occupy the same levels as in the ground state.

Is this correct? Do I seem to understand the shell-model somewhat?

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