---
title: "The Shell Model: Single-Particle States and Spin-Orbit Coupling"
module: Nuclear Models
moduleNumber: 3
lessonNumber: 3
order: 303
summary: >
  A harmonic-oscillator well reproduces the first three magic numbers but fails
  above twenty; adding a strong inverted spin-orbit term that drives the
  stretched j equals l plus one-half level down closes the gaps at 28, 50, 82,
  and 126. The filled shells couple to zero, so the last unpaired nucleon fixes
  the ground-state spin and parity, and its single-particle magnetic moment falls
  on the Schmidt lines. Configuration mixing sets the limits of the extreme
  single-particle model.
topics: [Nuclear Models]
sources:
  - book: Krane
    ref: "Ch. 5 — Nuclear Models; §5.1 The Shell Model, §5.2 Single-Particle Energies"
  - book: Wong
    ref: "Ch. 7 — The Shell Model; §7-2, §7-3"
  - book: Tipler & Llewellyn
    ref: "Ch. 11 — Nuclear Physics; §11-6 The Shell Model"
draft: false
---

The
[Fermi-gas model](/nuclear-physics/nuclear-models/fermi-gas-model)
treats the nucleus as a structureless box and cannot see the
[magic numbers](/nuclear-physics/nuclear-force-deuteron/nuclear-force-shell-overview)
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.[^krane-sm]

## 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,

$$
E_N = \hbar\omega\left(N + \tfrac{3}{2}\right),
\qquad
N = 2(n-1) + \ell,
$$

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,

$$
2,\ 8,\ 20,\ 40,\ 70,\ 112,\ \dots
$$

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.[^wong-sm]

$$
% caption: Harmonic-oscillator shells fix the first three magic numbers at 2, 8,
% and 20 through the running occupancy, but the higher closures fall at 40, 70,
% 112 instead of the observed 28, 50, 82, 126.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0.2) -- (0,6.4) node[above, black] {energy};
  \foreach \y/\lev/\tot in {0.8/{1s}/2, 1.7/{1p}/8, 2.7/{1d 2s}/20, 3.7/{1f 2p}/40, 4.7/{1g 2d 3s}/70, 5.7/{1h 2f 3p}/112} {
    \draw[very thick] (0.8,\y) -- (2.6,\y);
    \node[black, anchor=east, font=\scriptsize] at (0.75,\y) {\lev};
    \node[black, anchor=west, font=\scriptsize] at (2.75,\y) {closes at \tot};
  }
  % highlight the correct first three
  \foreach \y in {0.8,1.7,2.7} \node[black, anchor=west, font=\scriptsize] at (4.6,\y) {magic};
\end{tikzpicture}
$$

## 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

$$
V_{\text{so}}(r)\,\vec{\ell}\cdot\vec{s},
\qquad V_{\text{so}} < 0,
$$

to the central potential splits each level of orbital angular momentum $\ell$
into two, labelled by the total angular momentum $j = \ell \pm \tfrac12$. The
splitting follows from

$$
\vec{\ell}\cdot\vec{s} = \tfrac12\bigl[j(j+1) - \ell(\ell+1) - s(s+1)\bigr]\hbar^2,
$$

which evaluates to $+\tfrac{\ell}{2}\hbar^2$ for $j = \ell + \tfrac12$ and
$-\tfrac{\ell+1}{2}\hbar^2$ for $j = \ell - \tfrac12$. The energy gap between the
two members is

$$
\Delta E_{\text{so}} = \frac{2\ell+1}{2}\,\bigl\langle V_{\text{so}}\bigr\rangle\hbar^2,
$$

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
$j = \ell + \tfrac12$ 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.

$$
% caption: The spin-orbit term splits each level into j equals l plus one-half
% and j equals l minus one-half; the stretched member drops far enough to join
% the shell below and open a new gap at 28, 50, 82, and 126.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % oscillator level (left) splitting into two (right)
  \draw[black, very thick] (0.6,3.0) -- (2.0,3.0);
  \node[black, anchor=east, font=\scriptsize] at (0.55,3.0) {$1f$};
  % split lines
  \draw[black, dashed] (2.0,3.0) -- (3.4,3.9);
  \draw[black, dashed] (2.0,3.0) -- (3.4,1.6);
  % upper member j=l-1/2
  \draw[black, very thick] (3.4,3.9) -- (4.8,3.9);
  \node[black, anchor=west, font=\scriptsize] at (4.85,3.9) {$1f_{\tfrac{5}{2}}$, raised};
  % lower member j=l+1/2 dropped
  \draw[acc, very thick] (3.4,1.6) -- (4.8,1.6);
  \node[acc, anchor=west, font=\scriptsize] at (4.85,1.6) {$1f_{\tfrac{7}{2}}$, dropped};
  % gap marker
  \draw[acc, ->] (4.0,2.4) -- (4.0,1.75);
  \node[acc, anchor=west, font=\scriptsize] at (3.4,2.5) {new gap at 28};
\end{tikzpicture}
$$

## 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

$$
1s_{1/2}\,\|\,1p_{3/2}\,1p_{1/2}\,\|\,1d_{5/2}\,2s_{1/2}\,1d_{3/2}\,\|\,
1f_{7/2}\,\|\,2p_{3/2}\,1f_{5/2}\,2p_{1/2}\,1g_{9/2}\,\|\,\dots
$$

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 $\tfrac52^+$, 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 $\tfrac12^-$, 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.

$$
% caption: The nlj neutron levels filling to oxygen-17: the first eight neutrons
% pair off in the levels through 1p-one-half, and the ninth sits alone in
% 1d-five-halves (accent), fixing the ground-state spin and parity at five-halves
% plus.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0.2) -- (0,5.7) node[above, black] {energy};
  \foreach \y/\lab in {0.8/{$1s_{\tfrac{1}{2}}$}, 1.7/{$1p_{\tfrac{3}{2}}$}, 2.5/{$1p_{\tfrac{1}{2}}$}, 3.6/{$1d_{\tfrac{5}{2}}$}, 4.4/{$2s_{\tfrac{1}{2}}$}, 5.0/{$1d_{\tfrac{3}{2}}$}} {
    \draw[black, very thick] (1.5,\y) -- (3.3,\y);
    \node[black, anchor=east, font=\scriptsize] at (1.4,\y) {\lab};
  }
  % occupancy: 2 + 4 + 2 = 8 paired neutrons
  \foreach \x in {2.1,2.7} \fill[black] (\x,0.8) circle (1.4pt);
  \foreach \x in {1.9,2.3,2.7,3.1} \fill[black] (\x,1.7) circle (1.4pt);
  \foreach \x in {2.1,2.7} \fill[black] (\x,2.5) circle (1.4pt);
  % ninth (valence) neutron in accent on 1d5/2
  \fill[acc] (2.1,3.6) circle (1.7pt);
  % annotation to the valence neutron, placed in the clear gap
  \draw[acc, ->] (4.6,3.0) -- (2.25,3.55);
  \node[acc, anchor=west, font=\scriptsize] at (4.6,2.9) {ninth neutron};
  \node[black, anchor=west, font=\scriptsize] at (3.6,1.25) {eight paired};
\end{tikzpicture}
$$

## 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:

$$
j = \ell + \tfrac12:\quad
\mu = \Bigl[g_\ell\bigl(j - \tfrac12\bigr) + \tfrac12 g_s\Bigr],
$$

$$
j = \ell - \tfrac12:\quad
\mu = \frac{j}{j+1}\Bigl[g_\ell\bigl(j + \tfrac32\bigr) - \tfrac12 g_s\Bigr].
$$

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 $j = \ell + \tfrac12$ and one for $j = \ell - \tfrac12$. The lines bracket
the measured moments of odd-$A$ nuclei, but almost every measured moment lies
strictly between them, not on either.

> **Worked example (Magnetic moment of oxygen-17).** The odd neutron of
> $^{17}\mathrm{O}$ is in $1d_{5/2}$, so $j = \ell + \tfrac12$ with $\ell = 2$.
> With $g_\ell = 0$ and $g_s = -3.826$ for the neutron,
> $$
> \mu = \tfrac12 g_s = \tfrac12(-3.826) = -1.91\ \mu_N,
> $$
> against the measured $-1.89\ \mu_N$. The single-particle estimate is close but
> not exact, the usual outcome away from doubly closed shells.

$$
% caption: The Schmidt lines bound the single-particle magnetic moment against j:
% one line for spin parallel to orbit (j equals l plus one-half), one for spin
% antiparallel (j equals l minus one-half); measured odd-neutron moments (points)
% fall between the two lines rather than on either.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0.6,0.4) -- (7.0,0.4) node[right, black!70] {angular momentum $j$};
  \draw[->, black] (0.6,0.2) -- (0.6,4.3) node[above, black!70] {moment};
  % spin-parallel schmidt line (solid, upper)
  \draw[very thick] (1.2,3.5) -- (6.3,3.2);
  \node[black, anchor=south, font=\scriptsize] at (3.8,3.35) {spin parallel};
  % spin-antiparallel schmidt line (dashed, lower)
  \draw[very thick, dashed] (1.2,1.2) -- (6.3,1.7);
  \node[black, anchor=north, font=\scriptsize] at (3.8,1.4) {spin antiparallel};
  % measured moments between the two lines
  \foreach \x/\y in {1.7/2.5, 2.7/2.3, 3.6/2.6, 4.6/2.2, 5.5/2.4}
    \fill[black] (\x,\y) circle (1.6pt);
  \node[black, anchor=west, font=\scriptsize] at (5.7,2.35) {measured};
\end{tikzpicture}
$$

## 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](/nuclear-physics/nuclear-models/collective-model-rotations-vibrations).

The shell model and the
[liquid-drop model](/nuclear-physics/nuclear-models/liquid-drop-collective-coordinates)
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.

| Nuclide | Odd nucleon level | Predicted $J^\pi$ | Observed $J^\pi$ |
| --- | --- | --- | --- |
| $^{17}\mathrm{O}$ | neutron $1d_{5/2}$ | $\tfrac52^+$ | $\tfrac52^+$ |
| $^{15}\mathrm{N}$ | proton $1p_{1/2}$ hole | $\tfrac12^-$ | $\tfrac12^-$ |
| $^{39}\mathrm{K}$ | proton $1d_{3/2}$ hole | $\tfrac32^+$ | $\tfrac32^+$ |
| $^{41}\mathrm{Ca}$ | neutron $1f_{7/2}$ | $\tfrac72^-$ | $\tfrac72^-$ |
| $^{207}\mathrm{Pb}$ | neutron $3p_{1/2}$ hole | $\tfrac12^-$ | $\tfrac12^-$ |

[^krane-sm]: Krane, "Introductory Nuclear Physics," Wiley (1988), §5.1–5.2.
    Ground-state spins and moments from NNDC,
    [https://www.nndc.bnl.gov/](https://www.nndc.bnl.gov/).
[^wong-sm]: Wong, "Introductory Nuclear Physics," 2nd ed., Wiley-VCH (1998),
    §7-2, §7-3; Tipler & Llewellyn, "Modern Physics," 5th ed., §11-6.