Nuclear Models — Question 6

Question 6

Shell degeneracies explain closed subshell populations

For one nucleon species, take ordered orbitals $1s_{1/2},1p_{3/2},1p_{1/2},1d_{5/2},2s_{1/2},1d_{3/2}$. Each orbital j holds $2j+1$ identical nucleons. Protons and neutrons fill separate sets.

1. Calculate capacities and cumulative closures through these orbitals.

2. For N=20 and Z=20, determine whether both species fill the listed set and the total A. If one neutron is added, how many particles occupy the next shell?

Question 6 - Solution

Shell degeneracies explain closed subshell populations

(a) Method and calculation

$$\boxed{g_j=2j+1:\quad2,4,2,6,2,4}.$$$$\boxed{\text{cumulative populations}:\quad2,6,8,14,16,20}.$$

(b) Interpretation and further calculation

$$\boxed{N=Z=20:\ \text{both sets filled},\quad A=N+Z=40}.$$$$\boxed{N=21:\ \text{one neutron occupies the next available shell}}.$$

Physical check

Pauli exclusion operates separately for protons and neutrons. Actual shell gaps require a nuclear potential, including spin-orbit coupling; degeneracy counting alone does not predict all magic numbers.