---
title: Hyperfine Structure and the 21 cm Line
module: QED Corrections and Hyperfine Structure
moduleNumber: 4
lessonNumber: 2
order: 402
summary: >
  The proton carries a magnetic moment, and it interacts with the magnetic field
  the electron produces at the nucleus. For s-states that interaction is the
  Fermi contact term, proportional to the electron density at the origin and to
  the dot product of the nuclear and electronic spins. Coupling I and J into
  F = I + J splits each level by a Landé interval rule; in hydrogen's ground
  state it produces the F = 0/F = 1 doublet whose 1420 MHz, 21 cm transition maps
  neutral hydrogen across the galaxy.
topics: [QED Corrections and Hyperfine Structure]
sources:
  - book: Foot
    ref: "Ch. 6 — Hyperfine Structure and Isotope Shift; §6.1–6.3"
  - book: Griffiths & Schroeter
    ref: "Ch. 7 — Time-Independent Perturbation Theory; §7.5 Hyperfine Splitting"
  - book: Bransden & Joachain
    ref: "Ch. 5 — One-Electron Atoms; §5.5 Hyperfine Structure"
draft: false
---

The [fine structure](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession)
and the [Lamb shift](/atomic-physics/qed-corrections-and-hyperfine-structure/lamb-shift-qed)
treat the nucleus as a fixed point charge. The proton is more than a charge: it
is a spin-$\tfrac12$ particle with a magnetic moment, and that moment sits in the
magnetic field the orbiting, spinning electron generates at the origin. The
interaction energy is tiny — smaller than the fine structure by roughly the
ratio of the electron to the proton mass — but it splits every level into a
**hyperfine** multiplet, and one such splitting, the ground-state doublet of
hydrogen, produces the $21\ \mathrm{cm}$ line that is the single most important
probe in radio astronomy.

## The nuclear magnetic moment

A nucleus of spin $\vec I$ carries a magnetic dipole moment proportional to it,

$$
\vec\mu_I = g_I\,\mu_N\,\frac{\vec I}{\hbar},
\qquad
\mu_N = \frac{e\hbar}{2 m_p} = 5.0508\times 10^{-27}\ \mathrm{J\,T^{-1}},
$$

where $\mu_N$ is the **nuclear magneton** and $g_I$ the nuclear $g$-factor. The
nuclear magneton is smaller than the Bohr magneton $\mu_B = e\hbar/2m_e$ by the
mass ratio $m_e/m_p \approx 1/1836$, which is why hyperfine energies fall three
orders of magnitude below fine-structure energies. For the proton the measured
moment is

$$
\vec\mu_p = 2.7928\,\mu_N\,\frac{\vec I}{\hbar}
\quad\Longrightarrow\quad g_p = 5.5857,
$$

anomalously large — a Dirac point particle would have $g_p = 2$ — because the
proton is a composite of quarks. Atomic hyperfine structure measures $g_p$, and
the value is an input to the atomic problem, not something the atomic theory
predicts.

## The magnetic-dipole interaction and the Fermi contact term

The general magnetic hyperfine Hamiltonian couples $\vec\mu_I$ to the field
$\vec B_e$ the electron produces at the nucleus. That field has three pieces: a
term from the electron's orbital current, a dipolar term from the electron's spin
moment at a distance, and a contact term from the spin moment at the origin. For
a state with $\ell=0$ the orbital and dipolar pieces vanish — the orbital current
is zero and the spatial average of the dipolar field over a spherically symmetric
$s$-state is zero — and only the **Fermi contact interaction** survives:[^gs-contact]

$$
\hat H_{\text{hf}}
= \frac{\mu_0}{3}\,g_e\mu_B\,g_I\mu_N\,
\frac{1}{\hbar^2}\,(\vec S_e\cdot\vec I)\,
\bigl|\psi(0)\bigr|^2\cdot(4\pi)\,\Big/\,(4\pi)
\;=\;
\frac{2\mu_0}{3}\,\vec\mu_e\cdot\vec\mu_I\,\delta^3(\vec r),
$$

the interaction of two point dipoles evaluated at zero separation, weighted by
the probability the electron is found at the nucleus. Writing the electron
moment as $\vec\mu_e = -g_e\mu_B\vec S_e/\hbar$ and taking the expectation in a
hydrogenic $ns$ state with $|\psi_{ns}(0)|^2 = 1/(\pi n^3 a_0^3)$ (for $Z=1$)
gives a level shift proportional to $\vec S_e\cdot\vec I$.

> **Definition (Fermi contact interaction).** The part of the magnetic hyperfine
> coupling that acts only where the electron overlaps the nucleus,
> $\hat H_F = \tfrac{2\mu_0}{3}\,\vec\mu_e\cdot\vec\mu_I\,\delta^3(\vec r)$. It is
> nonzero only for $s$-states, whose wavefunction does not vanish at the origin,
> and it is the dominant hyperfine term in every alkali and hydrogenic
> ground state.

$$
% caption: Only s-electrons have nonzero density at the nucleus, so only they feel
% the Fermi contact interaction; a p-electron wavefunction vanishes at the origin
% and its ground-state hyperfine coupling is suppressed.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% axes
\draw[->] (0,0) -- (5.2,0) node[right, black] {$r$};
\draw[->] (0,0) -- (0,3.1) node[above, black] {density};
% s-state: peaks at origin
\draw[acc, very thick, domain=0.0:5.0, samples=80] plot (\x, {2.7*exp(-0.9*\x)});
\node[acc, anchor=west] at (0.7,2.3) {$s$-state};
% p-state: zero at origin, rises then falls
\draw[black, very thick, dashed, domain=0.0:5.0, samples=80] plot (\x, {5.0*\x*\x*exp(-1.3*\x)});
\node[black, anchor=west] at (2.0,1.35) {$p$-state};
% nucleus marker
\fill[black] (0,0) circle (2.2pt);
\node[black, anchor=north] at (0.0,-0.1) {nucleus};
\end{tikzpicture}
$$

## Coupling I and J into F

Because $\hat H_{\text{hf}}\propto \vec S_e\cdot\vec I$ (more generally
$\propto\vec J\cdot\vec I$ once the full electronic angular momentum is used),
neither $\vec I$ nor $\vec J$ is separately conserved, but their sum is. The good
quantum number is the **total** angular momentum

$$
\vec F = \vec J + \vec I,
\qquad
|J - I| \le F \le J + I,
$$

with $F$ ranging over integer steps. Squaring $\vec F = \vec J + \vec I$ isolates
the operator that appears in the Hamiltonian,

$$
\vec I\cdot\vec J = \tfrac12\bigl(\vec F^{\,2} - \vec J^{\,2} - \vec I^{\,2}\bigr)
\;\Longrightarrow\;
\langle \vec I\cdot\vec J\rangle
= \tfrac{\hbar^2}{2}\bigl[F(F+1) - J(J+1) - I(I+1)\bigr].
$$

The hyperfine energy of a level therefore takes the form

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

where $A$ is the **magnetic hyperfine constant**, a single number for a given
electronic level that packages $g_I$, $g_e$, and the electron density (or
$\langle 1/r^3\rangle$ for $\ell\neq 0$). Successive members of the multiplet
obey a spacing rule.

> **Theorem (Landé interval rule).** In a hyperfine multiplet the energy gap
> between adjacent levels is proportional to the larger $F$:
> $$
> E_{\text{hf}}(F) - E_{\text{hf}}(F-1) = A\,F.
> $$
> Measuring the ratios of successive intervals therefore fixes $F$ and, with the
> electronic $J$ known, the nuclear spin $I$.

$$
% caption: The Landé interval rule. A level with J and nuclear spin I splits into
% components F = |J−I| … J+I, and the gap above each component is proportional to
% its F, so the spacings grow linearly up the multiplet.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% unperturbed level
\draw[black, dashed] (-0.3,0) -- (2.0,0);
\node[anchor=east] at (-0.35,0) {level};
% split components with growing intervals
\draw[acc, very thick] (3.0,-1.2) -- (5.0,-1.2) node[right, black] {$F=1$};
\draw[acc, very thick] (3.0,0.4) -- (5.0,0.4) node[right, black] {$F=2$};
\draw[acc, very thick] (3.0,2.4) -- (5.0,2.4) node[right, black] {$F=3$};
% interval brackets
\draw[<->, black] (2.6,-1.2) -- (2.6,0.4);
\node[black, anchor=east] at (2.5,-0.4) {$2A$};
\draw[<->, black] (2.6,0.4) -- (2.6,2.4);
\node[black, anchor=east] at (2.5,1.4) {$3A$};
\end{tikzpicture}
$$

## The hydrogen ground state and the 21 cm line

For the $1s$ ground state of hydrogen the electron has $J = S = \tfrac12$ and the
proton has $I = \tfrac12$, so $F = 0$ or $F = 1$. The Fermi contact energy from
the calculation above is[^gs-94]

$$
\Delta E_{\text{hf}}
= \frac{4 g_p}{3}\,\frac{m_e}{m_p}\,\alpha^2\,
\bigl(\text{Ry}\bigr)\,\frac{m_e c^2}{\ }
\;=\;
\frac{4 g_p \hbar^4}{3\, m_p m_e^2 c^2 a_0^4},
$$

the difference between the $F=1$ (triplet, spins parallel) and $F=0$ (singlet,
spins antiparallel) states. The triplet lies **above** the singlet because two
antiparallel magnetic moments — the electron's moment is opposite its spin —
correspond to parallel spins costing energy. Evaluating the constants,

$$
\Delta E_{\text{hf}} = 5.88\times 10^{-6}\ \mathrm{eV},
\qquad
\nu = \frac{\Delta E_{\text{hf}}}{h} \approx 1420\ \mathrm{MHz},
\qquad
\lambda = \frac{c}{\nu} \approx 21\ \mathrm{cm}.
$$

The full experimental value, one of the most precisely known quantities in
physics, is

$$
\nu_{\text{HI}} = 1\,420.405\,751\,768\ \mathrm{MHz},
\qquad
\lambda_{\text{HI}} = 21.106\ \mathrm{cm},
$$

measured with the hydrogen maser.[^maser] The simple contact estimate reproduces
this to the accuracy of the neglected corrections — the electron's anomalous
moment ($g_e = 2.0023$ rather than $2$), the reduced mass, and relativistic and
QED pieces each enter at the $0.1\%$ level.

$$
% caption: The hydrogen 1s ground state splits into F = 0 (singlet, spins
% antiparallel) and F = 1 (triplet, spins parallel), separated by 5.9 μeV. The
% forbidden F = 1 → F = 0 magnetic-dipole transition is the 21 cm line.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% unsplit 1s
\draw[black, dashed] (-0.6,0.6) -- (1.2,0.6);
\node[anchor=east] at (-0.65,0.6) {$1s$};
% split
\draw[acc, very thick] (3.0,1.7) -- (5.4,1.7) node[right, black] {$F=1$ (triplet)};
\draw[black, very thick] (3.0,-0.4) -- (5.4,-0.4) node[right, black] {$F=0$ (singlet)};
% transition arrow
\draw[->, black, very thick] (3.6,1.7) -- (3.6,-0.4);
\node[black, anchor=west] at (3.7,0.65) {$21\ \mathrm{cm}$};
\node[black, anchor=west, align=left] at (3.7,0.2) {1420 MHz};
% spin cartoons: triplet up-up, singlet up-down
\draw[->, acc] (2.1,1.55) -- (2.1,1.9);
\draw[->, acc] (2.4,1.55) -- (2.4,1.9);
\draw[->, black] (2.1,-0.25) -- (2.1,0.1);
\draw[->, black] (2.4,0.1) -- (2.4,-0.25);
\end{tikzpicture}
$$

## Why the transition is forbidden, and why that matters

The $F=1\to F=0$ transition connects two states of the same spatial wavefunction
($1s$ in both). An electric-dipole transition requires a change of parity and of
orbital angular momentum, $\Delta\ell = \pm 1$; here $\Delta\ell = 0$, so the
electric-dipole amplitude is exactly zero. The transition proceeds only through
the far weaker **magnetic-dipole** channel, a spin flip. Its spontaneous
emission rate is

$$
A_{10} = 2.85\times 10^{-15}\ \mathrm{s^{-1}},
\qquad
\tau = \frac{1}{A_{10}} \approx 1.1\times 10^{7}\ \mathrm{yr},
$$

a radiative lifetime of eleven million years.[^m1] For any laboratory atom this
would make the line invisible. In the interstellar medium it is a virtue: the
line is so weak that a photon crosses a galaxy without being reabsorbed, so the
$21\ \mathrm{cm}$ radiation escapes from deep inside neutral-hydrogen clouds that
are opaque to visible light. Collisions, not spontaneous emission, keep the
$F=1$ level populated in the diffuse gas, and the enormous total number of
hydrogen atoms compensates for the minute per-atom rate.

$$
% caption: Neutral hydrogen (HI) throughout a galaxy's disk emits the 21 cm line.
% Its Doppler shift along the line of sight maps the rotation curve, and the total
% intensity maps the column density of atomic hydrogen.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% galactic disk (ellipse) with spiral hint
\draw[black, very thick] (0,0) ellipse (3.4 and 1.2);
\fill[black] (0,0) circle (2.2pt);
\node[black, anchor=south] at (0,0.12) {center};
% HI clouds
\foreach \p in {(-2.4,0.3),(-1.3,-0.5),(1.1,0.6),(2.3,-0.2),(0.4,-0.7),(1.7,0.35)}
  \fill[acc!35, draw=acc] \p circle (3.5pt);
\node[acc, anchor=south] at (1.7,1.5) {HI clouds};
% observer + line of sight
\draw[->, black, very thick] (5.2,-1.6) -- (2.0,-0.5);
\node[black, anchor=west] at (5.0,-1.7) {to observer};
% doppler labels (approaching / receding)
\node[black, anchor=north] at (-2.6,-1.4) {blueshift};
\node[black, anchor=north] at (2.6,-1.4) {redshift};
\end{tikzpicture}
$$

## What the line measures

The $21\ \mathrm{cm}$ line carries three independent pieces of astrophysical
information.

- **Column density.** The line intensity, integrated over the profile, is
  proportional to the number of hydrogen atoms along the line of sight (in the
  optically thin limit), so a $21\ \mathrm{cm}$ survey is a direct map of where
  neutral hydrogen sits.
- **Velocity.** The Doppler shift of the line centre gives the line-of-sight
  velocity of each cloud. Applied across a spiral galaxy this reconstructs the
  **rotation curve** $v(r)$; the observation that $v(r)$ stays flat far beyond
  the visible disk is a principal piece of evidence for dark matter.
- **Spin temperature.** The ratio of atoms in $F=1$ to $F=0$ defines a spin
  temperature through the Boltzmann factor $\exp(-h\nu/k_B T_s)$. Because
  $h\nu/k_B = 0.068\ \mathrm{K}$ is tiny, the two levels are nearly equally
  populated at any realistic temperature, and departures from that ratio probe
  the radiation field and collisions in the early universe.

> **Definition (21 cm line).** The magnetic-dipole transition between the
> $F=1$ and $F=0$ hyperfine levels of the hydrogen $1s$ ground state, at
> $1420.405\,751\,768\ \mathrm{MHz}$ (wavelength $21.106\ \mathrm{cm}$).
> Predicted by van de Hulst in 1944 and detected in 1951, it is the standard
> tracer of neutral atomic hydrogen in the interstellar and intergalactic medium.

The hyperfine splitting is the smallest structure treated in this course and the
one with the largest reach. The same contact interaction that shifts a hydrogen
level by six parts in ten million lets radio telescopes weigh galaxies and probe
the cosmic dawn. The
[next lesson](/atomic-physics/qed-corrections-and-hyperfine-structure/nuclear-effects-isotope-shift)
takes the nucleus one step further, from a point dipole to an extended body with
a finite size and an electric quadrupole moment.

[^gs-contact]: **Griffiths & Schroeter**, §7.5 — the magnetic hyperfine Hamiltonian and the isolation of the contact term for $\ell=0$; see also **Foot**, §6.1, and **Bransden & Joachain**, §5.5. The delta-function form follows from $\nabla^2(1/r) = -4\pi\delta^3(\vec r)$ applied to the vector potential of a point dipole.
[^gs-94]: **Griffiths & Schroeter**, §7.5, eq. for the ground-state hyperfine splitting $E_{\text{hf}} = \tfrac{4 g_p \hbar^4}{3 m_p m_e^2 c^2 a^4}$; evaluates to $5.88\times10^{-6}\ \mathrm{eV}$. Constants: **Foot**, §6.2.
[^maser]: Hellwig, H. et al. (1970), "Measurement of the Unperturbed Hydrogen Hyperfine Transition Frequency," _IEEE Trans. Instrum. Meas._ **19**, 200: $\nu_{\text{HI}} = 1\,420\,405\,751.768\ \mathrm{Hz}$. NIST Atomic Spectra Database, [nist.gov/pml/atomic-spectra-database](https://www.nist.gov/pml/atomic-spectra-database).
[^m1]: The spontaneous $M1$ rate $A_{10} = 2.85\times10^{-15}\ \mathrm{s^{-1}}$ gives $\tau \approx 1.1\times10^{7}\ \mathrm{yr}$. Predicted by van de Hulst (1944); detected by Ewen, H. I. & Purcell, E. M. (1951), _Nature_ **168**, 356. See **Foot**, §6.3.