Nuclear Size, Moments, and Isotope Shifts

The hyperfine structure treated the nucleus as a point magnetic dipole. Three further departures from a structureless point charge shift atomic levels measurably: the nucleus has a finite spatial extent, its mass differs from one isotope to the next, and if its spin is $I\ge 1$ it carries an electric quadrupole moment. Each effect is small, but each maps a distinct nuclear property onto the spectrum, and laser spectroscopy has become a standard tool for measuring nuclear charge radii and moments — including for short-lived isotopes produced one atom at a time.

The finite nuclear size

A point charge produces the Coulomb potential $-Z{e}^2/4\pi {\epsilon }_{0}r$ all the way to $r=0$. A real nucleus of radius $R$ spreads its charge over a small ball, so an electron that penetrates inside sees less charge enclosed. Modelling the nucleus as a uniformly charged sphere, the potential inside is a parabola that stays finite at the origin,

matching the point potential at $r=R$. The difference $\delta V=V_{\text{in}}-V_{\text{point}}$ is positive (the real potential is less deep) and confined to $r<R$. Only $s$-electrons have appreciable density there. Treating $|\psi (0){|}^2$ as constant across the tiny nuclear volume, first-order perturbation theory gives a level shift¹

where $⟨r_N^2⟩$ is the mean-square nuclear charge radius ($⟨r_N^2⟩=\frac{3}{5}{R}^2$ for the uniform sphere). The shift is positive: finite size reduces the binding of $s$-states, because the electron spends part of its time where the attraction is weaker than a point charge would supply.

For hydrogen the volume shift is about $1\mathrm{MHz}$ in the $1s$ level — a minute effect that nevertheless enters the precision determination of the Rydberg constant. It grows steeply with nuclear charge, roughly as $Z^4$ through the electron density and the growing radius, so in heavy atoms it dominates the isotope shift.

The isotope shift

Comparing the same spectral line in two isotopes of one element reveals a small frequency difference, the isotope shift, with two additive origins.

• Mass shift. The atomic energies depend on the nucleus only through the reduced mass $\mu =m_{e}M/(m_e+M)$; a heavier isotope has a slightly larger $\mu$ and therefore deeper levels. The normal mass shift follows directly from $E_n\propto \mu$, giving a fractional shift between isotopes of masses $M$ and $M^{\prime }$, $$\frac{\delta {\nu }_{\text{NMS}}}{\nu }=\frac{m_e}{M}\cdot \frac{M^{\prime }-M}{M^{\prime }}\approx \frac{m_e\delta M}{M^2}.$$ In many-electron atoms a specific mass shift (mass polarization) adds a cross term $\frac{1}{M}{\sum }_{i<j}{\vec{p}}_i\cdot {\vec{p}}_j$ of the same $1/M^2$ scale but of either sign and much harder to compute. Both fall off as $1/M^2$, so the mass shift dominates the isotope shift in light elements.
• Field (volume) shift. The difference in nuclear charge radius between the isotopes changes the volume shift derived above, $$\delta {\nu }_{\text{FS}}\propto |\psi (0){|}^2\delta ⟨r_N^2⟩,$$ which grows as $Z^2$ (or faster) through the electron density and so dominates the isotope shift in heavy elements.

Because the mass and field shifts scale differently with mass and with the electronic transition, they can be separated. Plotting the modified isotope shift of one transition against that of another for a chain of isotopes produces a straight line — a King plot — whose slope and intercept isolate the field and mass contributions, and thus $\delta ⟨r_N^2⟩$ across the chain. This is how the charge radii of exotic isotopes far from stability are measured.

The electric quadrupole interaction

A nucleus with spin $I\ge 1$ can be non-spherical, carrying an electric quadrupole moment $Q$. A quadrupole has no interaction with a uniform field, but it couples to the gradient of the electric field the electrons produce at the nucleus. This adds a term to the hyperfine Hamiltonian beyond the magnetic dipole,

with the quadrupole coupling constant $B=eQ⟨{\partial }^{2}V/\partial z^2⟩$ set by the nuclear moment and the electronic field gradient.² Unlike the magnetic-dipole energy, which is linear in $K\propto \vec{I}\cdot \vec{J}$, the quadrupole energy is quadratic in $K$. Its presence breaks the Landé interval rule: successive hyperfine intervals are no longer proportional to $F$. Fitting the departures from the interval rule extracts $B$, and thence $Q$ once the electronic field gradient is known.

The hyperfine anomaly

The magnetic hyperfine constant $A$ is proportional to the nuclear $g$-factor, so the ratio of $A$-values for two isotopes should equal the ratio of their $g$-factors. It does not, quite. The nuclear magnetization is distributed over the nuclear volume rather than sitting at a point, and $s$-electrons sample that distribution. The resulting fractional discrepancy is the hyperfine anomaly (the Bohr–Weisskopf effect),

where ${}^1{\Delta }^2$ is typically $10^{-3}$–$10^{-2}$.³ Small as it is, the anomaly is a rare window on how magnetization is arranged inside the nucleus, complementing the charge-radius information from the field shift.

Reading nuclear structure from atomic spectra

Collecting the effects of this module, each nuclear property leaves a specific, separable signature in the atomic spectrum.

The precision of these measurements makes atomic spectroscopy a nuclear probe. The muonic-hydrogen Lamb shift — where the muon's small orbit magnifies the volume shift by $(m_{\mu }/m_e{)}^3\approx 10^7$ — measured the proton charge radius to $r_p=0.8409(4)\mathrm{fm}$, and the initial disagreement with the value from ordinary hydrogen (the proton radius puzzle) drove a decade of re-measurement before converging.⁴ Optical spectroscopy of trapped radioactive isotopes now maps $\delta ⟨r_N^2⟩$ along entire isotope chains, turning the atom into an instrument for nuclear physics. The hierarchy established across this module — gross structure, fine structure, the Lamb shift, hyperfine structure, and now the nuclear corrections — is the full ledger of what sets an atomic energy level, ordered from the electronvolt down to the fraction of a megahertz.