Discussion
In the energy levels predicted by shell model, the labelled energy states and the corresponding nucleon number, starting from the lowest to higher energy levels are as follows :
(i) 1s1/2 − 2 Nucleons
(ii) 1p3/2 − 4 Nucleons
(iii) 1p1/2 − 2 Nucleons
(iv) 1d5/2 − 6 Nucleons
The estimated groundstate spins of \( {}^{27}_{13}\text{Al} \) and \( {}^{11}_{5}\text{B} \) nuclei are :
Answer: B
The spins are \( \frac{3}{2} \) for \( ^{11}_{5}\text{B} \) and \( \frac{5}{2} \) for \( ^{27}_{13}\text{Al} \).
In the shell model, paired nucleons contribute nothing, so an odd-A nucleus takes the spin of its single unpaired nucleon.
- \( ^{11}\text{B} \): 5 protons and 6 neutrons. The neutrons fill 1s1/2 (2) and 1p3/2 (4) completely. The protons put 2 in 1s1/2 and 3 in 1p3/2, so the odd proton is in 1p3/2: spin \( \frac{3}{2} \).
- \( ^{27}\text{Al} \): 13 protons and 14 neutrons. The neutrons fill up to 1d5/2 exactly (2 + 4 + 2 + 6). The protons put 8 in the first three levels and 5 in 1d5/2, so the odd proton is in 1d5/2: spin \( \frac{5}{2} \).
The other options give spins of levels the odd proton does not occupy, or, like \( \frac{11}{2} \), values not available in these shells.
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