Nuclear Spin, Magnetic Dipole, and Electric Quadrupole Moments

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.¹

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 $\sqrt{I(I+1)}\hslash$ 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 ${\frac{1}{2}}^{+}$ and ${}^{17}\mathrm{O}$ has ${\frac{5}{2}}^{+}$.

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:

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:

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 $j=\ell \pm \frac{1}{2}$. 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:

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 $j=\ell \pm \frac{1}{2}$ of the odd nucleon.

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.

The vanishing for $I\le \frac{1}{2}$ 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,

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, $Q\sim -\frac{2j-1}{2j+2}⟨r^2⟩$, 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.

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

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 taken up later.

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.