---
title: Nuclear Spin, Magnetic Dipole, and Electric Quadrupole Moments
module: Nuclear Properties
moduleNumber: 1
lessonNumber: 5
order: 105
summary: >
  The ground state of a nucleus carries a definite spin and parity, a magnetic dipole
  moment of order the nuclear magneton, and, when its spin exceeds one-half, an electric
  quadrupole moment that measures its shape. The single-particle Schmidt lines predict
  the magnetic moment of an odd-A nucleus from the last unpaired nucleon, and the
  measured moments fall between them. The quadrupole moment distinguishes prolate from
  oblate deformation, and hyperfine structure is the experimental handle that fixes the
  spin and the moments from an atomic spectrum.
topics: [Nuclear Properties]
sources:
  - book: Krane
    ref: "Ch. 3 — Nuclear Properties; §3.5 Angular Momentum and Parity, §3.6 Nuclear Magnetic Dipole Moments, §3.7 Nuclear Electric Quadrupole Moments"
  - book: Wong
    ref: "Ch. 7 — The Shell Model; single-particle moments and the Schmidt lines"
  - book: Tipler & Llewellyn
    ref: "Ch. 11 — Nuclear Physics; §11-2 nuclear spin and hyperfine structure"
draft: false
---

Beyond its mass and size, a nucleus in its ground state has three static moments that
fix its quantum numbers and shape: the angular momentum, the magnetic dipole moment, and
the electric quadrupole moment. Each is an expectation value in the ground state, each is
measured through the hyperfine coupling to atomic electrons, and each tests the
single-particle picture directly.[^krane-moments]

## Spin and parity

The total angular momentum of a nucleus, $\vec I$, is the vector sum of the orbital and
spin angular momenta of all its nucleons. It is conventionally called the **nuclear
spin**, though it includes orbital motion. Its magnitude is $\sqrt{I(I+1)}\,\hbar$ and it
is quantized along any axis in $2I+1$ steps.

Three regularities fix $I$ without a detailed calculation.

- **Even-even nuclei** have $I = 0$ in the ground state: every proton and every neutron
  pairs with a partner of opposite $m_j$, and the pairs cancel.
- **Odd-$A$ nuclei** have half-integer $I$, carried by the single unpaired nucleon.
- **Odd-odd nuclei** have integer $I$, from the coupling of one unpaired proton to one
  unpaired neutron.

The parity $\pi = \pm 1$ is the sign the wavefunction acquires under coordinate inversion.
It is fixed by the orbital angular momenta of the unpaired nucleons, $\pi = (-1)^{\ell}$
for a single particle, and is written as a superscript on the spin, $I^\pi$; the proton
has $\tfrac12^+$ and $^{17}\mathrm{O}$ has $\tfrac52^+$.

$$
% caption: The spin vector precesses on a cone about the field axis, its projection
% taking the 2I+1 allowed values from plus I to minus I in integer steps; the cones for
% a spin of three-halves are shown.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % field axis
  \draw[black, ->] (0,-2.4) -- (0,2.7) node[anchor=south, black!70] {$B$};
  % four projection levels for I = 3/2
  \foreach \y in {2.0,0.7,-0.7,-2.0} \draw[black, dashed] (-2.3,\y) -- (2.3,\y);
  \node[anchor=west, black, font=\scriptsize] at (2.35,2.0) {$m = +\tfrac32$};
  \node[anchor=west, black, font=\scriptsize] at (2.35,0.7) {$m = +\tfrac12$};
  \node[anchor=west, black, font=\scriptsize] at (2.35,-0.7) {$m = $ half down};
  \node[anchor=west, black, font=\scriptsize] at (2.35,-2.0) {$m = $ three down};
  % vectors of fixed length to each cone
  \def\R{2.55}
  \draw[very thick, ->] (0,0) -- (1.58,2.0);
  \draw[thick, ->] (0,0) -- (2.45,0.7);
  \draw[thick, ->] (0,0) -- (2.45,-0.7);
  \draw[thick, ->] (0,0) -- (1.58,-2.0);
  % cone hints (ellipses at top projection)
  \draw[black] (0,2.0) ellipse (1.58 and 0.22);
  \draw[black] (0,0.7) ellipse (2.45 and 0.30);
  \node[black, anchor=east] at (-0.1,1.2) {$I$};
\end{tikzpicture}
$$

## The nuclear magneton and magnetic moments

A charged particle with angular momentum carries a magnetic moment. The natural scale
for the nucleus uses the proton mass:

> **Definition (Nuclear magneton).**
> $$
> \mu_N = \frac{e\hbar}{2 m_p} = 3.152\times10^{-8}\ \mathrm{eV/T} = 5.051\times10^{-27}\ \mathrm{J/T}.
> $$
> It is smaller than the Bohr magneton by the mass ratio $\mu_N/\mu_B = m_e/m_p = 1/1836$,
> so nuclear magnetic effects are about three orders of magnitude weaker than atomic ones.

The magnetic moment of a nucleus is quoted as its maximum projection,
$\mu = g_I\,\mu_N\,I$, with $g_I$ the nuclear g-factor. The free-nucleon values are
themselves anomalous:

$$
\mu_p = +2.793\,\mu_N, \qquad \mu_n = -1.913\,\mu_N.
$$

A structureless Dirac proton would have $\mu_p = +1\,\mu_N$ and a neutral point neutron
$\mu_n = 0$. The observed values, in particular the neutron's large negative moment,
are direct evidence that the nucleons have internal quark structure.

**Schmidt lines.** In the extreme single-particle model an odd-$A$ nucleus has all but
one nucleon paired to zero, and the moment is that of the last odd nucleon in an orbital
of definite $\ell$ and $j = \ell \pm \tfrac12$. Adding the orbital and spin contributions
with the appropriate g-factors ($g_\ell = 1$, $g_s = +5.586$ for a proton; $g_\ell = 0$,
$g_s = -3.826$ for a neutron) gives the two **Schmidt formulas**:

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

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

Plotting these against $j$ traces two curves, the **Schmidt lines**, one for each
spin-orbit coupling. Almost every measured odd-$A$ moment falls between the two lines
rather than on them, because configuration mixing and the polarization of the paired core
dilute the pure single-particle value. The lines nonetheless bracket the data and
identify the coupling $j = \ell \pm \tfrac12$ of the odd nucleon.

$$
% caption: Measured magnetic moments of odd-proton nuclei lie between the two Schmidt
% lines, the upper for spin aligned with orbital angular momentum and the lower for
% anti-aligned; the data cluster inside the bracket rather than on either limit.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {$j$};
  \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {moment};
  % upper Schmidt line (j = l + 1/2): rises with j
  \draw[very thick] (0.6,1.6) -- (6.8,4.0);
  \node[black, anchor=south, font=\scriptsize] at (5.4,3.75) {aligned};
  % lower Schmidt line (j = l - 1/2): falls / low
  \draw[black, thick, dashed] (0.9,2.4) .. controls (3.0,1.15) and (5.0,0.75) .. (6.8,0.6);
  \node[black, anchor=north, font=\scriptsize] at (5.2,0.75) {anti-aligned};
  % data points between the lines
  \foreach \x/\y in {1.4/1.95,2.3/2.05,3.2/2.15,4.1/2.35,5.0/2.55,5.9/2.9} \fill[acc] (\x,\y) circle (1.7pt);
\end{tikzpicture}
$$

## The electric quadrupole moment and nuclear shape

A spherical charge distribution has only a monopole moment. The first correction that a
nucleus can carry (the dipole vanishes by parity) is the electric quadrupole, which
measures the departure from sphericity.

> **Definition (Electric quadrupole moment).** For the ground state with maximum spin
> projection $m = I$, the spectroscopic quadrupole moment is
> $$
> Q = \frac{1}{e}\int \rho_p(\vec r)\,\bigl(3z^2 - r^2\bigr)\,\d^3 r,
> $$
> with $\rho_p$ the proton charge density and $z$ the symmetry axis. $Q$ has units of area
> (barns): $Q > 0$ for a prolate (elongated) distribution, $Q < 0$ for an oblate
> (flattened) one, and $Q = 0$ for a spherical nucleus or whenever $I = 0$ or
> $I = \tfrac12$.

The vanishing for $I \le \tfrac12$ is kinematic: a distribution with too little angular
momentum cannot present an oriented quadrupole to the laboratory, however deformed its
intrinsic shape. The **intrinsic** quadrupole moment $Q_0$ of a body-fixed deformed
nucleus relates to the measured spectroscopic $Q$ through the projection of the deformed
shape onto the laboratory axis,

$$
Q = Q_0\,\frac{I(2I-1)}{(I+1)(2I+3)},
$$

for a nucleus whose spin is the rotation of a symmetric deformed body. Small
single-particle moments arise when one proton orbits a spherical core, $Q \sim
-\tfrac{2j-1}{2j+2}\langle r^2\rangle$, of order $10^{-1}\,\mathrm{barn}$. The rare-earth
and actinide nuclei show $Q$ of several barns, far too large for a single particle: their
whole charge distribution is deformed, and $Q_0$ reaches $+7\,\mathrm{barn}$ for a strongly
prolate rotor.

$$
% caption: A prolate nucleus (left) has its charge elongated along the spin axis and a
% positive quadrupole moment, while an oblate nucleus (right) is flattened and has a
% negative moment; a spherical nucleus has none.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % prolate
  \draw[black, ->] (0,-1.5) -- (0,1.7) node[anchor=south, black!70] {spin axis};
  \draw[very thick] (0,0) ellipse (0.55 and 1.15);
  \fill[black!8] (0,0) ellipse (0.55 and 1.15);
  \node[black!70, anchor=north, font=\scriptsize] at (0,-1.7) {prolate};
  \node[black, anchor=west, font=\scriptsize] at (0.65,0.7) {$Q > 0$};
  % oblate
  \begin{scope}[xshift=4.6cm]
    \draw[black, ->] (0,-1.5) -- (0,1.7) node[anchor=south, black!70] {spin axis};
    \draw[very thick] (0,0) ellipse (1.15 and 0.55);
    \fill[black!8] (0,0) ellipse (1.15 and 0.55);
    \node[black!70, anchor=north, font=\scriptsize] at (0,-1.7) {oblate};
    \node[black, anchor=west, font=\scriptsize] at (1.25,0.35) {$Q < 0$};
  \end{scope}
\end{tikzpicture}
$$

## Hyperfine structure as the measurement

The nuclear moments are read from the atom, not the nucleus. The nuclear magnetic moment
$\vec\mu_I$ couples to the magnetic field $\vec B_e$ produced at the nucleus by the atomic
electrons, adding an energy $-\vec\mu_I\cdot\vec B_e$ that depends on the relative
orientation of $\vec I$ and the electronic angular momentum $\vec J$. The coupled total
$\vec F = \vec I + \vec J$ is quantized, and the magnetic hyperfine energy follows the
interval rule

$$
E_{\text{hf}} = \frac{A}{2}\bigl[F(F+1) - I(I+1) - J(J+1)\bigr],
$$

with $A \propto \mu_I$ the magnetic hyperfine constant. A level of electronic angular
momentum $J$ splits into $2I + 1$ components when $J \ge I$, or $2J + 1$ when $J < I$.
Counting the components fixes $I$ directly, and the spacing (through the interval rule)
gives the ratio of successive intervals $F : F-1$ and hence $A$ and $\mu_I$. A residual
electric quadrupole coupling of $Q$ to the electronic field gradient perturbs the interval
rule and yields $Q$. Driving the same nuclear moment to resonance in an external field is
nuclear magnetic resonance, the basis of the
[imaging techniques](/nuclear-physics/radiation-matter-applications/nuclear-applications-dating-medicine)
taken up later.

$$
% caption: A fine-structure level of electronic angular momentum J splits by the coupling
% to the nuclear spin into hyperfine components labelled by F, whose 2I plus 1 count fixes
% the nuclear spin and whose spacings follow the interval rule.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % unsplit level
  \draw[black, very thick] (0,3.0) -- (2.2,3.0);
  \node[black!70, anchor=east] at (-0.1,3.0) {level $J$};
  % split components (interval rule: increasing spacing)
  \draw[very thick] (3.4,3.65) -- (5.6,3.65);
  \draw[very thick] (3.4,3.15) -- (5.6,3.15);
  \draw[very thick] (3.4,2.45) -- (5.6,2.45);
  \draw[very thick] (3.4,1.55) -- (5.6,1.55);
  \node[black, anchor=west, font=\scriptsize] at (5.7,3.65) {$F = I + J$};
  \node[black, anchor=west, font=\scriptsize] at (5.7,3.15) {next $F$};
  \node[black, anchor=west, font=\scriptsize] at (5.7,2.45) {lower $F$};
  \node[black, anchor=west, font=\scriptsize] at (5.7,1.55) {lowest $F$};
  % connectors
  \foreach \y in {3.65,3.15,2.45,1.55} \draw[black] (2.2,3.0) -- (3.4,\y);
  % interval-rule brace hint
  \draw[black] (3.1,3.65) -- (3.1,3.15);
  \draw[black] (3.1,3.15) -- (3.1,2.45);
  \node[black, anchor=east, font=\scriptsize] at (3.05,3.4) {$A$};
  \node[black, anchor=east, font=\scriptsize] at (3.05,2.8) {$2A$};
\end{tikzpicture}
$$

The moments close the description of the nuclear ground state begun with size and mass.
The spin, parity, and magnetic moment already point past the liquid drop toward a
single-particle structure, since the last unpaired nucleon controls all three; the
quadrupole moment points toward collective deformation. Both threads are taken up in the
nuclear-models module, and the deviation of the measured moments from the Schmidt lines
is the first quantitative test the shell model must meet.

[^krane-moments]: **Krane**, _Introductory Nuclear Physics_, §3.5 (Angular Momentum and Parity), §3.6 (Nuclear Magnetic Dipole Moments), and §3.7 (Nuclear Electric Quadrupole Moments): the nuclear magneton, free-nucleon moments, the Schmidt-line single-particle formulas, the spectroscopic and intrinsic quadrupole moments, and hyperfine structure. The single-particle moment treatment follows **Wong**, _Introductory Nuclear Physics_, Ch. 7. Measured moments are the CODATA/NIST and NNDC values, [https://physics.nist.gov/cuu/Constants/](https://physics.nist.gov/cuu/Constants/) and [https://www.nndc.bnl.gov/](https://www.nndc.bnl.gov/).