Editing Nuclear shell model (section)

===Including a spin-orbit interaction===
We next include a [[spin–orbit interaction]]. First, we have to describe the system by the [[Quantum number#Quantum numbers with spin–orbit interaction|quantum numbers]] ''j'', ''m<sub>j</sub>'' and [[Parity (physics)|parity]] instead of ''ℓ'', ''m<sub>l</sub>'' and ''m<sub>s</sub>'', as in the [[Hydrogen-like atom#Including spin–orbit interaction|hydrogen–like atom]]. Since every even level includes only even values of ''ℓ'', it includes only states of even (positive) parity. Similarly, every odd level includes only states of odd (negative) parity. Thus we can ignore parity in counting states. The first six shells, described by the new quantum numbers, are
* level 0 (''n'' = 0): 2 states (''j'' = {{sfrac|2}}). Even parity.
* level 1 (''n'' = 1): 2 states (''j'' = {{sfrac|2}}) + 4 states (''j'' = {{sfrac|3|2}}) = 6. Odd parity.
* level 2 (''n'' = 2): 2 states (''j'' = {{sfrac|2}}) + 4 states (''j'' = {{sfrac|3|2}}) + 6 states (''j'' = {{sfrac|5|2}}) = 12. Even parity.
* level 3 (''n'' = 3): 2 states (''j'' = {{sfrac|2}}) + 4 states (''j'' = {{sfrac|3|2}}) + 6 states (''j'' = {{sfrac|5|2}}) + 8 states (''j'' = {{sfrac|7|2}}) = 20. Odd parity.
* level 4 (''n'' = 4): 2 states (''j'' = {{sfrac|2}}) + 4 states (''j'' = {{sfrac|3|2}}) + 6 states (''j'' = {{sfrac|5|2}}) + 8 states (''j'' = {{sfrac|7|2}}) + 10 states (''j'' = {{sfrac|9|2}}) = 30. Even parity.
* level 5 (''n'' = 5): 2 states (''j'' = {{sfrac|2}}) + 4 states (''j'' = {{sfrac|3|2}}) + 6 states (''j'' = {{sfrac|5|2}}) + 8 states (''j'' = {{sfrac|7|2}}) + 10 states (''j'' = {{sfrac|9|2}}) + 12 states (''j'' = {{sfrac|11|2}}) = 42. Odd parity.
where for every ''j'' there are {{gaps|2''j''|+|1}} different states from different values of ''m<sub>j</sub>''.

Due to the spin–orbit interaction, the energies of states of the same level but with different ''j'' will no longer be identical. This is because in the original quantum numbers, when <math>\scriptstyle \vec{s}</math> is parallel to <math>\scriptstyle \vec{l}</math>, the interaction energy is positive, and in this case ''j'' = ''ℓ'' + ''s'' = ''ℓ'' + {{sfrac|2}}. When <math>\scriptstyle \vec{s}</math> is anti-parallel to <math>\scriptstyle \vec{l}</math> (i.e. aligned oppositely), the interaction energy is negative, and in this case {{gaps|''j''|{{=}}|''ℓ''|−|''s''|{{=}}|''ℓ''|−|{{sfrac|2}}}}. Furthermore, the strength of the interaction is roughly proportional to ''ℓ''.

For example, consider the states at level 4:
* The 10 states with ''j'' = {{sfrac|9|2}} come from ''ℓ'' = 4 and ''s'' parallel to ''ℓ''. Thus they have a positive spin–orbit interaction energy.
* The 8 states with ''j'' = {{sfrac|7|2}} came from ''ℓ'' = 4 and ''s'' anti-parallel to ''ℓ''. Thus they have a negative spin–orbit interaction energy.
* The 6 states with ''j'' = {{sfrac|5|2}} came from ''ℓ'' = 2 and ''s'' parallel to ''ℓ''. Thus they have a positive spin–orbit interaction energy. However, its magnitude is half compared to the states with ''j'' = {{sfrac|9|2}}.
* The 4 states with ''j'' = {{sfrac|3|2}} came from ''ℓ'' = 2 and ''s'' anti-parallel to ''ℓ''. Thus they have a negative spin–orbit interaction energy. However, its magnitude is half compared to the states with ''j'' = {{sfrac|7|2}}.
* The 2 states with ''j'' = {{sfrac|2}} came from ''ℓ'' = 0 and thus have zero spin–orbit interaction energy.