Nuclear Forces and Models Questions and Answers

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.

Answer: C

The nuclear force is independent of the nucleons' charge.

Mirror nuclei, such as \(^{3}\)H and \(^{3}\)He or \(^{13}\)C and \(^{13}\)N, have their proton and neutron numbers interchanged. Once the Coulomb energy is allowed for, their energy levels are nearly identical, which shows that the nn and pp interactions are the same: the strong force does not distinguish a proton from a neutron. This is charge symmetry, part of the wider idea of charge independence.

Swapping protons and neutrons leaves spins and parities of the nucleons unchanged, so mirror nuclei say nothing about dependence on spin or parity; and the tiny proton-neutron mass difference is not what the comparison tests.

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 1s_1/2 (2) and 1p_3/2 (4) completely. The protons put 2 in 1s_1/2 and 3 in 1p_3/2, so the odd proton is in 1p_3/2: spin \( \frac{3}{2} \).
• \( ^{27}\text{Al} \): 13 protons and 14 neutrons. The neutrons fill up to 1d_5/2 exactly (2 + 4 + 2 + 6). The protons put 8 in the first three levels and 5 in 1d_5/2, so the odd proton is in 1d_5/2: 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.

Answer: B

Nuclear forces are spin dependent and have a non-central part.

The deuteron is bound only in the spin-1 state, and neutron-proton scattering differs for singlet and triplet spins, which shows spin dependence. The deuteron also has an electric quadrupole moment, meaning it is not spherical, which requires a non-central tensor force that depends on the orientation of the spins relative to the line joining the nucleons.

Each of the other options denies one of these two experimentally established features.

Answer: D

Spin-orbit interaction was introduced in the nuclear shell model.

Maria Goeppert Mayer and, independently, Haxel, Jensen and Suess (1949) added a strong spin-orbit term to the shell model potential. It splits each level of orbital angular momentum l into \(j = l + \frac{1}{2}\) and \(j = l - \frac{1}{2}\), with the higher j pushed down, and this reproduces the observed magic numbers 2, 8, 20, 28, 50, 82 and 126. Mayer and Jensen shared the 1963 Nobel Prize for it.

The liquid drop model treats the nucleus as a charged fluid with no single-particle levels, the collective model adds rotations and vibrations, and the optical model describes nuclear scattering with a complex potential.