The Shell Model: Single-Particle States and Spin-Orbit Coupling

The Fermi-gas model treats the nucleus as a structureless box and cannot see the magic numbers 2, 8, 20, 28, 50, 82, 126, where nuclei are anomalously tightly bound. The shell model keeps the independent-particle idea but replaces the box with a realistic central potential whose quantized levels group into shells separated by large gaps. A plain oscillator or square well gives the wrong gaps above 20. Mayer and Jensen supplied the missing ingredient in 1949: a strong spin-orbit coupling, opposite in sign and far larger than the atomic one, that rearranges the levels so the large gaps fall exactly at the observed magic numbers.¹

The central potential and oscillator shells

A nucleon moves in the average potential of all the others, well approximated between the square well and the smoothly rounded Woods-Saxon form. The harmonic oscillator is analytically tractable and captures the essential level structure. Its energies depend on a single principal quantum number,

so each oscillator shell $N$ collects several orbital angular momenta of the same parity $(-1{)}^N$. Counting the $2(2\ell +1)$ nucleons each level holds and running the total gives the closed-shell occupancies,

The first three, $2,8,20$, are the observed magic numbers. The rest are wrong: nature closes shells at $28,50,82,126$, not $40,70,112$. A central potential alone, oscillator or square well, cannot produce the higher gaps.²

The spin-orbit term

The nuclear force contains a strong spin-orbit component, and its sign is opposite to the atomic case: the level with spin and orbital angular momentum parallel lies lower, not higher. Adding

to the central potential splits each level of orbital angular momentum $\ell$ into two, labelled by the total angular momentum $j=\ell \pm \frac{1}{2}$. The splitting follows from

which evaluates to $+\frac{\ell }{2}{\hslash }^2$ for $j=\ell +\frac{1}{2}$ and $-\frac{\ell +1}{2}{\hslash }^2$ for $j=\ell -\frac{1}{2}$. The energy gap between the two members is

proportional to $2\ell +1$, so the splitting grows with $\ell$ and is largest for the high-$\ell$ levels near the top of each oscillator shell. The stretched $j=\ell +\frac{1}{2}$ member is driven down. For a high-$\ell$ level the drop is so large that this member sinks out of its own shell and joins the shell below, creating a new gap. The $1f_{7/2}$ level falls to close a gap at 28; the $1g_{9/2}$ falls to close 50; the $1h_{11/2}$ to close 82; the $1i_{13/2}$ to close 126. The observed magic numbers emerge in full.

Level ordering and ground-state spins

The levels are labelled $n{\ell }_j$: $n$ counts nodes, $\ell$ is written $s,p,d,f,g,h,\dots$, and $j$ is the subscript. Their order, filling from the bottom, is

with the double bars marking the magic gaps at $2,8,20,28,50$. Each filled level, holding $2j+1$ nucleons in pairs of opposite $m_j$, couples to total angular momentum zero and even parity. A closed shell contributes nothing to the spin. The consequences are a set of sharp predictions.

• A doubly closed-shell nucleus has $J^{\pi }=0^{+}$. Examples: ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, ${}^{40}\mathrm{Ca}$, ${}^{208}\mathrm{Pb}$.
• One nucleon outside closed shells takes the spin and parity of that nucleon: $J=j$, parity $(-1{)}^{\ell }$. In ${}^{17}\mathrm{O}$ the ninth neutron sits in $1d_{5/2}$, predicting ${\frac{5}{2}}^{+}$, as observed.
• One nucleon short of a closed shell (a hole) takes the spin and parity of the missing nucleon. In ${}^{15}\mathrm{N}$ the last proton hole is in $1p_{1/2}$, predicting ${\frac{1}{2}}^{-}$, as observed.
• Odd-odd nuclei couple the odd proton's $j_p$ and odd neutron's $j_n$ through the Nordheim rules; the result is not fixed by a single level.

Single-particle magnetic moments and the Schmidt lines

If one nucleon outside closed shells carries the whole angular momentum, it also carries the whole magnetic moment. The moment of a single nucleon combines its orbital and spin contributions with gyromagnetic factors $g_{\ell }$ and $g_s$, projected onto $j$. The result, in nuclear magnetons, takes two forms:

The free-nucleon factors are $g_{\ell }=1$, $g_s=+5.586$ for the proton and $g_{\ell }=0$, $g_s=-3.826$ for the neutron. For each $j$ the two formulas give two values, and plotting them against $j$ traces the pair of Schmidt lines, one for $j=\ell +\frac{1}{2}$ and one for $j=\ell -\frac{1}{2}$. The lines bracket the measured moments of odd-$A$ nuclei, but almost every measured moment lies strictly between them, not on either.

Limits of the extreme single-particle model

The extreme single-particle model, which assigns all properties to one valence nucleon, works best near closed shells, where a single level dominates. Away from them it breaks down in predictable ways.

• Configuration mixing: the ground state is a superposition of several arrangements of the valence nucleons, not a single $n{\ell }_j$ configuration, so moments drift off the Schmidt lines toward the interior.
• Pairing: like nucleons pair to $J=0$, so for several identical nucleons in a level the spin is again set by the last unpaired one, but the binding gains a pairing energy the single-particle picture omits.
• Collective deformation: mid-shell nuclei with many valence nucleons deform and develop rotational and vibrational spectra beyond any single-particle scheme, treated in the collective model.

The shell model and the liquid-drop model divide the labor: the drop captures the smooth bulk binding and deformation, the shell model captures the quantized level structure, spins, parities, and moments riding on top. Together they anchor the more complete collective description.