Discussion
If nucleons in a nucleus are considered to be confined in a three-dimensional cubical box, then the first four magic numbers are
Answer: C
The first four magic numbers are 2, 8, 14, 20.
In a cubical box the levels are \(E \propto n_{x}^{2} + n_{y}^{2} + n_{z}^{2}\) with each \(n \ge 1\). Count the distinct arrangements (degeneracy), and double for the two spin states:
- (1,1,1), sum 3: 1 state, 2 nucleons, total 2.
- (2,1,1) and permutations, sum 6: 3 states, 6 nucleons, total 8.
- (2,2,1), sum 9: 3 states, 6 nucleons, total 14.
- (3,1,1), sum 11: 3 states, 6 nucleons, total 20.
Shell closures, the magic numbers in this model, are therefore 2, 8, 14, 20. The sequence 2, 8, 20, 28 is the observed set from the real nuclear shell model with spin-orbit coupling, which a simple box does not reproduce.
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