The nuclear magic numbers (2, 8, 20, 28, 50, 82, 126) differ from the atomic magic numbers (2, 10, 18, 36, 54, 86). What is the primary reason for this difference?
ANucleons obey Fermi-Dirac statistics differently than electrons, so their energy levels are populated in a different order
BThe nuclear potential is governed by the strong force (not the Coulomb force) and has much stronger spin-orbit coupling, which rearranges the energy level ordering and creates different shell gaps
CNucleons have higher mass than electrons, which shifts the zero-point energy and changes the level ordering
DThe nuclear shell model uses a harmonic oscillator potential instead of a Coulomb potential, which happens to produce the magic numbers by coincidence
The key difference is the nature of the potential and the strength of spin-orbit coupling. Atomic magic numbers come from filling shells in a Coulomb (electrostatic) potential with weak spin-orbit coupling. Nuclear magic numbers require the spin-orbit term to be large — comparable to the spacing between major shells. This strong spin-orbit splitting reorganizes the level ordering, breaking the harmonic oscillator magic numbers (2, 8, 20, 40, 70...) and shifting them to (2, 8, 20, 28, 50, 82, 126). Maria Goeppert Mayer's key insight was that only a large spin-orbit term could reproduce the observed nuclear magic numbers.
Question 2 Multiple Choice
A nucleus has Z = 50 protons and N = 82 neutrons. How would you expect its binding energy to compare to neighboring nuclei?
ALower binding energy than neighbors, because having 50 protons means high Coulomb repulsion, which destabilizes the nucleus
BAbout average for nuclei in that mass range, since binding energy mainly depends on A = Z + N
CHigher binding energy than neighbors, because both Z = 50 and N = 82 are magic numbers — this is a doubly magic nucleus with both proton and neutron shells closed
DHigher binding energy only if N is also magic; Z = 50 alone provides no stability advantage
This is a doubly magic nucleus: Z = 50 (tin) is a proton magic number and N = 82 is a neutron magic number. Both the proton shell and the neutron shell are closed, meaning the next nucleon of either type would have to occupy a much higher energy level. This makes the nucleus exceptionally tightly bound. Doubly magic nuclei show anomalously high binding energies compared to liquid-drop model predictions, extra-low neutron-capture cross sections, and unusual abundance. Tin (Z = 50) has 10 stable isotopes — far more than its neighbors — partly because the magic proton number makes many neutron-number configurations stable.
Question 3 True / False
In the nuclear shell model, the magic number 28 arises from the same energy-level filling pattern that gives the atomic magic number 18 (argon).
TTrue
FFalse
Answer: False
The nuclear and atomic magic numbers are fundamentally different because the underlying potentials are different. Atomic magic number 18 (argon) corresponds to filling the 3p subshell in a Coulomb potential with weak spin-orbit coupling. Nuclear magic number 28 arises from a shell gap created by strong spin-orbit splitting in the nuclear (Woods-Saxon) potential — the 1f_{7/2} subshell fills and the next level is far above. The level-filling sequences are entirely different. Without the large nuclear spin-orbit term, the nuclear magic numbers would be the harmonic oscillator magic numbers (2, 8, 20, 40, 70...), not the observed ones.
Question 4 True / False
A nucleus at or near a magic number has lower neutron-capture cross sections than its neighbors, meaning it is less likely to absorb an additional neutron.
TTrue
FFalse
Answer: True
This is correct and is one of the key experimental signatures of nuclear shell closures. When a magic nucleus absorbs a neutron, that neutron would have to occupy the next shell — which is much higher in energy than the closed shell. The transition matrix element for this process is small, giving a low cross section. This was one of the empirical facts that motivated the development of the nuclear shell model: magic-number nuclei are systematically less reactive to neutron capture than their neighbors, a pattern that cannot be explained by the liquid-drop model but follows naturally from shell structure.
Question 5 Short Answer
What role did spin-orbit coupling play in the development of the nuclear shell model, and why was its inclusion necessary to reproduce the observed magic numbers?
Think about your answer, then reveal below.
Model answer: Without spin-orbit coupling, nuclear energy levels follow a pattern similar to a harmonic oscillator or square well, producing shell gaps after 2, 8, 20, 40, 70 nucleons — not the observed magic numbers. Maria Goeppert Mayer realized that including a strong spin-orbit term — which shifts levels by an amount proportional to the dot product of orbital and spin angular momenta — rearranges the level ordering. Specifically, it lowers the energy of states with spin aligned with orbital angular momentum (j = l + 1/2) relative to anti-aligned states (j = l - 1/2). This splitting is large enough in nuclei to move high-j states from one major shell into the one below, creating the large gaps after 28, 50, 82, and 126 that define the magic numbers.
The spin-orbit term in nuclei is much larger relative to other energy scales than in atoms — this is why the atomic and nuclear shell models, while analogous in structure, produce different magic numbers. In atoms, spin-orbit coupling is a small relativistic correction. In nuclei, it is comparable in size to the spacing between major shells, which is why it can dramatically rearrange the level ordering. Mayer's insight was recognized with the Nobel Prize in Physics (1963), shared with J.H.D. Jensen who independently reached the same conclusion.