GATE 2022 PH – Question 18
If nucleons in a nucleus are considered to be confined in a three-dimensional cubical box, then the first four magic numbers are
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Correct answer: (C) 2, 8, 14, 20
Explanation
In a three-dimensional cubical box of side $L$, the energy levels are
$$E=\frac{\pi^2\hbar^2}{2mL^2}\left(n_x^2+n_y^2+n_z^2\right),\qquad n_i=1,2,3,\dots$$
**The lowest values of $n_x^2+n_y^2+n_z^2$ and their spatial degeneracy:**
| $n_x^2+n_y^2+n_z^2$ | States | Degeneracy |
|---|---|---|
| 3 | (1,1,1) | 1 |
| 6 | (2,1,1) and permutations | 3 |
| 9 | (2,2,1) and permutations | 3 |
| 11 | (3,1,1) and permutations | 3 |
Each spatial state holds 2 nucleons (spin up and down). The cumulative numbers of nucleons that fill the shells are
$$2,\quad2+6=8,\quad8+6=14,\quad14+6=20 .$$
The first four magic numbers are **2, 8, 14, 20** (option C).