---
title: Nuclear Size, Moments, and Isotope Shifts
module: QED Corrections and Hyperfine Structure
moduleNumber: 4
lessonNumber: 3
order: 403
summary: >
  A real nucleus has a finite size, a mass that changes between isotopes, and,
  when its spin is at least one, an electric quadrupole moment. Each leaves a
  fingerprint in the atomic spectrum: the volume shift from s-electrons sampling
  the charge distribution, the mass and field isotope shifts that separate on a
  King plot, the quadrupole interaction that breaks the Landé interval rule, and
  the hyperfine anomaly from the magnetization distribution. Atomic spectroscopy
  reads nuclear properties out of these shifts.
topics: [QED Corrections and Hyperfine Structure]
sources:
  - book: Foot
    ref: "Ch. 6 — Hyperfine Structure and Isotope Shift; §6.4–6.5"
  - book: Bransden & Joachain
    ref: "Ch. 5 — One-Electron Atoms; §5.6 Nuclear Effects"
  - book: Demtröder
    ref: "Ch. 5 — The Hydrogen Atom and Atomic Structure; isotope and hyperfine effects"
draft: false
---

The [hyperfine structure](/atomic-physics/qed-corrections-and-hyperfine-structure/hyperfine-structure-21cm)
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 $-Ze^2/4\pi\varepsilon_0 r$ 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,

$$
V_{\text{in}}(r) = -\frac{Ze^2}{4\pi\varepsilon_0}\,\frac{1}{2R}
\left(3 - \frac{r^2}{R^2}\right),
\qquad r \le R,
$$

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 shift[^bj-size]

$$
\delta E_{\text{vol}} = \int |\psi(0)|^2\,\delta V\,\d^3 r
= \frac{2\pi}{3}\,\frac{Ze^2}{4\pi\varepsilon_0}\,|\psi(0)|^2\,\langle r_N^2\rangle,
$$

where $\langle r_N^2\rangle$ is the mean-square nuclear charge radius
($\langle r_N^2\rangle = \tfrac35 R^2$ 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.

$$
% caption: A point nucleus (dashed) gives the full 1/r Coulomb well; a finite
% nucleus of radius R (solid) flattens the potential inside, so a penetrating
% s-electron is bound slightly less tightly. The gap acts only for r < R.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% axes
\draw[->] (0,0) -- (5.4,0) node[right, black] {$r$};
\draw[->] (0,-3.0) -- (0,0.5) node[above, black] {$V(r)$};
% point Coulomb (dashed)
\draw[black, dashed, very thick, domain=0.55:5.2, samples=80] plot (\x, {-1.5/\x});
% finite: parabola inside R, coulomb outside
\def\Rn{1.3}
\draw[acc, very thick, domain=1.3:5.2, samples=80] plot (\x, {-1.5/\x});
\draw[acc, very thick, domain=0.0:1.3, samples=40]
  plot (\x, {-1.5/(2*1.3)*(3 - (\x*\x)/(1.3*1.3))});
% mark R
\draw[black, dashed] (1.3,0) -- (1.3,-1.15);
\node[black, anchor=south] at (1.3,0.02) {$R$};
\node[acc, anchor=west] at (1.6,-1.55) {f\/inite nucleus};
\node[black, anchor=west] at (2.3,-0.55) {point};
\end{tikzpicture}
$$

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 $M'$,
  $$
  \frac{\delta\nu_{\text{NMS}}}{\nu} = \frac{m_e}{M}\cdot\frac{M' - M}{M'}
  \approx \frac{m_e\,\delta M}{M^2}.
  $$
  In many-electron atoms a **specific mass shift** (mass polarization) adds a
  cross term $\tfrac{1}{M}\sum_{i<j}\vec p_i\cdot\vec p_j$ 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,
  $$
  \delta\nu_{\text{FS}} \propto |\psi(0)|^2\,\delta\langle r_N^2\rangle,
  $$
  which grows as $Z^2$ (or faster) through the electron density and so dominates
  the isotope shift in heavy elements.

$$
% caption: The same line measured in two isotopes appears at slightly different
% frequencies. The heavier isotope (larger reduced mass, larger charge radius) is
% displaced from the lighter one; the split is the isotope shift.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% frequency axis
\draw[->] (0,0) -- (7.4,0) node[right, black] {frequency};
% two lines (Gaussians)
\draw[acc, very thick, domain=1.0:3.4, samples=60]
  plot (\x, {2.4*exp(-9*(\x-2.2)*(\x-2.2))});
\draw[black, very thick, dashed, domain=3.4:5.8, samples=60]
  plot (\x, {2.4*exp(-9*(\x-4.6)*(\x-4.6))});
\node[acc, anchor=south] at (2.2,2.45) {isotope $A$};
\node[black, anchor=south] at (4.6,2.45) {isotope $A${'}};
% shift bracket
\draw[<->, black] (2.2,-0.3) -- (4.6,-0.3);
\node[black, anchor=north] at (3.4,-0.35) {isotope shift};
\end{tikzpicture}
$$

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 $\delta\langle r_N^2\rangle$ across the chain. This
is how the charge radii of exotic isotopes far from stability are measured.

$$
% caption: A King plot. For an isotope chain the mass-scaled shift of one line
% plotted against that of a second line falls on a straight line; the slope fixes
% the ratio of field-shift factors and the intercept the mass contribution.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->] (0,0) -- (5.4,0) node[right, black] {line 1 (scaled)};
\draw[->] (0,0) -- (0,3.6) node[above, black] {line 2 (scaled)};
% best-fit line
\draw[black, very thick, dashed] (0.6,0.7) -- (4.8,3.2);
% data points
\foreach \p in {(0.9,0.9),(1.7,1.4),(2.5,1.95),(3.3,2.4),(4.1,2.9)}
  \fill[acc] \p circle (2.4pt);
\node[black, anchor=north west] at (2.9,2.0) {King line};
\end{tikzpicture}
$$

## 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,

$$
E_Q = \frac{B}{4}\,
\frac{3K(K+1) - 4\,I(I+1)J(J+1)}{2 I(2I-1)\,J(2J-1)},
\qquad
K = F(F+1) - I(I+1) - J(J+1),
$$

with the quadrupole coupling constant $B = e\,Q\,\langle\partial^2 V/\partial
z^2\rangle$ set by the nuclear moment and the electronic field gradient.[^foot-q]
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.

$$
% caption: A prolate nucleus (quadrupole moment Q) in the electric field gradient
% of the electrons. The energy depends on the orientation of the nuclear spin
% relative to the gradient axis, adding a term quadratic in I·J.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% field-gradient axis
\draw[->, black, thick] (0,-1.3) -- (0,2.2) node[above] {gradient axis};
% prolate nucleus (ellipse elongated along axis, tilted)
\begin{scope}[rotate=28]
\draw[black, very thick, fill=black!8] (0,0) ellipse (0.5 and 1.15);
\end{scope}
% spin arrow along nuclear symmetry axis
\draw[->, acc, very thick] (0,0) -- ({1.15*sin(28)},{1.15*cos(28)});
\node[acc, anchor=south] at (0.35,1.4) {spin $I$};
\node[black, anchor=north] at (0,-1.55) {nucleus ($Q$)};
% field-gradient hint: converging field lines
\foreach \x in {-1.6,-1.2,1.2,1.6}
  \draw[black] (\x,-1.8) -- (\x,1.8);
\end{tikzpicture}
$$

## 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),

$$
\frac{A_1}{A_2} = \frac{g_1}{g_2}\bigl(1 + {}^1\!\Delta^2\bigr),
$$

where ${}^1\!\Delta^2$ is typically $10^{-3}$–$10^{-2}$.[^foot-anom] 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.

| Nuclear property | Atomic signature | Extracted from |
| --- | --- | --- |
| Spin $I$ | number of hyperfine components | counting multiplet lines |
| Magnetic moment $\mu_I$ | magnetic constant $A$ | Landé interval spacings |
| Quadrupole moment $Q$ | quadrupole constant $B$ | interval-rule departures |
| Mean-square radius $\langle r_N^2\rangle$ | field (volume) shift | King plot of isotope shifts |
| Magnetization distribution | hyperfine anomaly | $A$-ratio vs $g$-ratio |

> **Definition (Isotope shift).** The difference in a spectral line's frequency
> between two isotopes of an element, the sum of a **mass shift** (from the
> changed reduced mass and mass polarization, scaling as $1/M^2$) and a **field
> shift** (from the changed nuclear charge radius, scaling as $Z^2$ and
> $\delta\langle r_N^2\rangle$). A King plot separates the two.

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.[^prp] Optical spectroscopy of trapped
radioactive isotopes now maps $\delta\langle r_N^2\rangle$ 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.

[^bj-size]: **Bransden & Joachain**, §5.6 — the finite-nuclear-size (volume) correction from a uniformly charged sphere, $\delta E_{\text{vol}} = \tfrac{2\pi}{3}\tfrac{Ze^2}{4\pi\varepsilon_0}|\psi(0)|^2\langle r_N^2\rangle$; see also **Foot**, §6.4.
[^foot-q]: **Foot**, §6.5 — the electric-quadrupole hyperfine energy and the breaking of the Landé interval rule; the quadrupole coupling constant $B = eQ\,\langle\partial^2 V/\partial z^2\rangle$. Also **Demtröder**, Ch. 5.
[^foot-anom]: **Foot**, §6.5 — the hyperfine anomaly (Bohr–Weisskopf effect) from the finite distribution of nuclear magnetization. Bohr, A. & Weisskopf, V. F. (1950), _Phys. Rev._ **77**, 94.
[^prp]: Antognini, A. et al. (2013), "Proton Structure from the Measurement of 2S–2P Transition Frequencies of Muonic Hydrogen," _Science_ **339**, 417: $r_p = 0.840\,87(39)\ \mathrm{fm}$. CODATA 2018 proton rms charge radius $r_p = 0.8414(19)\ \mathrm{fm}$, [physics.nist.gov/cgi-bin/cuu/Value?rp](https://physics.nist.gov/cgi-bin/cuu/Value?rp).