Hyperfine Structure and the 21 cm Line

The fine structure and the Lamb shift treat the nucleus as a fixed point charge. The proton is more than a charge: it is a spin-$\frac{1}{2}$ 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,

where ${\mu }_N$ is the nuclear magneton and $g_I$ the nuclear $g$-factor. The nuclear magneton is smaller than the Bohr magneton ${\mu }_B=e\hslash /2m_e$ 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

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:¹

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 ${\vec{\mu }}_e=-g_e{\mu }_B{\vec{S}}_e/\hslash$ 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 ${\vec{S}}_e\cdot \vec{I}$.

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

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

The hyperfine energy of a level therefore takes the form

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 $⟨1/r^3⟩$ for $\ell \ne 0$). Successive members of the multiplet obey a spacing rule.

The hydrogen ground state and the 21 cm line

For the $1s$ ground state of hydrogen the electron has $J=S=\frac{1}{2}$ and the proton has $I=\frac{1}{2}$, so $F=0$ or $F=1$. The Fermi contact energy from the calculation above is²

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,

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

measured with the hydrogen maser.³ 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.

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 radiative lifetime of eleven million years.⁴ 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.

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.

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 takes the nucleus one step further, from a point dipole to an extended body with a finite size and an electric quadrupole moment.