The line that takes eleven million years

The line that is really two splits sodium’s yellow line by the magnetic energy of the electron’s spin in the field it sees as it moves round the nucleus. That is the fine structure, and it lives in levels where the electron has orbital angular momentum. The ground state of hydrogen has none — the 1s electron has no orbit to speak of — and on the fine-structure account its single level is exactly single.

It is not. The proton is itself a small magnet, and the electron’s spin can point along its field or against it. The two arrangements differ in energy by six millionths of an electronvolt, and an atom that finds itself in the higher one will eventually flip its spin and emit a photon with a wavelength of twenty-one centimetres. “Eventually” is the whole story. The average wait is eleven million years, and yet the 21 cm line of hydrogen is the brightest radio line in the sky and the instrument by which the Galaxy’s spiral arms were first mapped. How a transition that slow can be that bright — and why its brightness turns out to be a count of atoms rather than a measure of their temperature — follows from three numbers that can each be computed from constants.

Three scales of one atom #

The first scale is the one the spectrum is a subtraction builds from differences of the Bohr energies: ten electronvolts between the ground state and the first excited level, an ultraviolet photon. The second is the fine structure, $\alpha^2/16$ of a rydberg for the n = 2 level: forty-five microelectronvolts, a microwave photon of eleven gigahertz. The third, the ground-state hyperfine splitting, is smaller again: 5.874 microelectronvolts, less than a millionth of the Lyman α energy.

The hyperfine split has two states above and one below, and the counting is familiar from four states, and one of them is odd. Two spin-½ particles — here the electron and the proton — combine into a triplet with total spin 1 and a singlet with total spin 0. In hydrogen’s ground state the triplet is the higher: the electron’s magnetic moment points opposite to its spin, and the energy is lowest when the two magnetic moments are anti-aligned, which puts the spins in the singlet. Three states above, one below, and the ratio of three to one will turn out to matter more than anything else about the line.

The same two levels respond to a magnetic field applied from outside, and the scale at which they do sets another of the ground state’s numbers. The field an atom calls strong finds that a field is strong or weak only relative to the energy the atom already has for the thing the field acts on. For the hyperfine levels that energy is 5.9 microelectronvolts, and the electron’s spin reaches it in a field of about five hundredths of a tesla — the field of a small permanent magnet. Below that, the triplet splits into three evenly spaced levels while the singlet stays put; above it, the electron’s spin decouples from the proton’s and aligns with the field, and the four states reorganise into two pairs. That crossover, worked out by Gregory Breit and Isidor Rabi in 1931, is how the hyperfine levels are selected and probed in atomic beams, and it is why a hydrogen maser sits inside magnetic shields that keep its field below a millionth of a tesla. The electron’s spin is the angular momentum that is not a rotation, and here it is being turned by fields far too weak to disturb the electron’s motion round the proton at all.

The frequency from constants #

The energy of the split is the energy of one magnet in the field of another. The proton’s magnetic moment makes a field that is largest at the proton itself, and the 1s electron has a finite probability of being exactly there. Enrico Fermi’s formula of 1930 gives the energy from that contact:

$$\Delta E = \tfrac{4}{3}\,g_p\,\frac{m_e}{m_p}\,\alpha^4\,m_ec^2,$$

with $g_p = 5.586$ the proton’s magnetic $g$-factor. The factor $\alpha^4 m_ec^2$ is the scale of the fine structure; the ratio of the electron’s mass to the proton’s is why the proton’s moment is so much weaker than the electron’s.

Fermi’s formula alone gives 1421.160 megahertz, half a part in a thousand above the measured value. The corrections are each a physical effect with a name. The electron’s magnetic moment is not exactly one Bohr magneton but larger by a tenth of a per cent, the anomaly that quantum electrodynamics predicts and that is one of the best-measured numbers in physics; that correction makes the agreement worse. The proton is not infinitely heavy, so the electron orbits their common centre of mass and its density at the proton is slightly reduced; that brings the calculation back to within fifty-six parts per million. Relativity raises the electron’s density at the nucleus slightly, and the electron’s interaction with its own radiation field lowers it; after those two the calculation stands forty parts per million above the measurement.

The last forty belong to the proton. Fermi’s formula treats it as a point magnet, and it is not: its charge and its magnetism are spread over nearly a femtometre, and the electron’s wavefunction, though enormous by comparison, samples that spread. The correction for it is about forty parts per million, and it depends on the details of the proton’s internal structure, which is why hydrogen’s hyperfine frequency — measured by hydrogen masers to twelve figures — is calculated only to about six. The line is so well known, and so simple, that it became the standard hydrogen masers keep time by; the clocks that must all slow together uses exactly such masers to test whether time is the same for every kind of clock.

Why the flip takes eleven million years #

An atom in an excited state emits spontaneously at a rate that a charge that turns must glow derives classically for an oscillating dipole: the power goes as the fourth power of the frequency, so the rate of emitting photons of that frequency goes as its cube, times the square of the dipole moment. The Lyman α transition has an electric dipole of the order of the electron’s charge times the Bohr radius and a frequency of 2.5 × 10¹⁵ hertz. The hyperfine transition has neither advantage.

A spin flip changes no charge distribution, so it has no electric dipole at all; it radiates through the magnetic dipole of the flipping spin, one Bohr magneton. A magnetic dipole radiates like an electric one with a moment of $\mu_B/c$, and the ratio of that to $ea_0$ is $\alpha/2$ — so magnetic transitions are weaker by $\alpha^2/4$, thirteen parts in a million. That is one factor. The other is the frequency: 1420 megahertz is 1.7 million times less than Lyman α’s, and cubed that is five parts in $10^{19}$. Together they turn a rate of $6\times10^8$ per second into one of $2.87\times10^{-15}$, computed here from the magnetic-dipole formula and agreeing with the accepted value to a third of a per cent.

A hydrogen atom in the upper hyperfine level waits, on average, eleven million years to emit. The width of the line that lifetime implies, by the width that is a lifetime, is about $10^{-16}$ of its frequency — a natural width so small that every real 21 cm line is broadened entirely by the motions of the gas.

The caesium clock line, also a ground-state hyperfine transition, sits on the same magnetic-dipole line at 9.2 gigahertz and waits about ten thousand years. That the second is defined by a transition nobody waits for to happen spontaneously is no paradox: a clock drives the transition with microwaves and measures the resonance, and a tiny spontaneous rate is exactly what makes the resonance sharp.

Why the populations forget the temperature #

Eleven million years is far longer than anything else an interstellar hydrogen atom experiences. In the gas between the stars, at a density of about one atom per cubic centimetre, an atom collides with another every few thousand years or sooner, and a collision can exchange the electrons’ spins. So the populations of the two levels are set by collisions, not by radiation, and collisions bring them to the Boltzmann distribution at the gas’s temperature. The temperature that describes the ratio of the two populations is called the spin temperature, and in most of the Galaxy’s atomic gas it is close to the gas’s kinetic temperature.

Here the smallness of the splitting takes over. The energy difference corresponds to a temperature of 68 millikelvin, so at any temperature interstellar gas actually has, the Boltzmann factor $e^{-h\nu/kT}$ is indistinguishable from one. The exponential that decides everything is here deciding nothing: the populations are simply proportional to the number of states in each level, three in the upper and one in the lower. Three-quarters of all hydrogen atoms in the Galaxy are in the upper level at any moment, waiting their eleven million years, whether the gas is at 10 kelvin or 10,000.

The same smallness nearly cancels absorption. An atom in the lower level can absorb a 21 cm photon; an atom in the upper level can be stimulated by the same photon to emit a second one. With the populations almost at their statistical weights, the two processes almost balance, and the net absorption is only the small difference, a fraction $h\nu/kT$ of what it would be with every atom in the lower level — less than a thousandth at 100 kelvin. A cloud of hydrogen is therefore very nearly transparent to its own line.

A brightness that counts atoms #

Put these together and the 21 cm line becomes a scale for weighing gas.

Each atom in the upper level emits at the same slow rate, and three-quarters of the atoms are always there, so the line’s emission is proportional to the number of atoms along the line of sight and to nothing else. Because the gas is nearly transparent to its own line, that emission escapes. The brightness temperature of a thin cloud is simply the number of atoms per square centimetre divided by a constant times the line’s width — the constant, $1.822\times10^{18}$ atoms per square centimetre per kelvin kilometre per second, computed here from the spin-flip rate and matching the value radio astronomers use. At $10^{20}$ atoms per square centimetre and a ten-kilometre-per-second spread, the brightness is about 5.3 kelvin whether the gas is at 50 kelvin or 1000. The emission counts atoms and ignores temperature.

The count stops being a count only when the cloud is thick enough to absorb its own line, and then each curve levels off at its own spin temperature: a thick cloud cannot be brighter than its own temperature. The same arithmetic that makes thin gas a mass balance makes thick, cold gas a thermometer, and radio astronomers use both, telling them apart by observing the same gas in absorption against a bright background source.

Here the smallness of everything conspires. The line is weak per atom and there are enormous numbers of atoms: a column through the Galaxy’s disc holds $10^{21}$ of them in every square centimetre. The line is weak, but it is also transparent, so the whole column contributes. And the level spacing is tiny, so the emission does not care about temperature. Each of the facts that made the line seem hopeless is what makes it a clean measurement.

A line predicted before it was seen #

Hendrik van de Hulst predicted in 1944, in occupied Leiden, that interstellar hydrogen should be detectable at 21 centimetres, having estimated the transition probability and realised that the Galaxy’s hydrogen was plentiful enough to make up for it. Harold Ewen and Edward Purcell detected the line in 1951 with a horn antenna poking out of a Harvard laboratory window, and Dutch and Australian groups confirmed it within weeks. Within a few years the line’s Doppler shifts had traced the spiral arms of the Milky Way through the dust that hides them at optical wavelengths, because 21 centimetre radiation passes through dust almost unaffected.

What made the prediction possible was exactly the chain of this essay: a rate small enough that collisions win, a splitting small enough that the populations sit at their weights, and a quantity of hydrogen large enough that the total emission is measurable. None of it needed a telescope to establish. It needed Fermi’s formula, the magnetic-dipole rate and a Boltzmann factor.

Where the two-level picture stops #

Collisions set the spin temperature only where the gas is dense enough. Between galaxies, at a millionth of an atom per cubic centimetre, collisions are too rare to compete even with an eleven-million-year lifetime, and the spin temperature is set instead by the cosmic microwave background, which bathes every atom in 21 cm photons, and by ultraviolet starlight, which can flip spins by exciting atoms and letting them decay back. In that regime the line can appear in absorption against the background, and its detection from the era before the first galaxies is one of the targets of the largest radio arrays now being built.

Optically thin gas. The count is exact only when the cloud is transparent. In the densest, coldest clouds the line saturates, and part of the hydrogen is in molecules, which do not emit at 21 centimetres at all; there the count is a lower limit.

One velocity component. The figures treat a cloud as having one spin temperature and one velocity spread. A real line of sight crosses many clouds at different velocities and temperatures, and separating them is the work of a 21 cm survey, not a single formula.

A point proton, at the last digit. The frequency calculation stops at the proton’s structure. To go further needs the distributions of the proton’s charge and magnetism, measured by electron scattering, and the calculation is then limited by those measurements rather than by atomic theory.

Two spins turning about each other #

Every figure here treats the atom as two levels, and none shows the flip itself: an electron spin and a proton spin precessing about each other, with the transition corresponding to a change in their relative orientation that happens with no change in where the electron is. The rates figure places a handful of lines by their measured rates and shows none of the selection rules that decide which transitions exist; the 21 cm line is allowed as a magnetic transition and forbidden as an electric one, and that distinction, not drawn, is the factor of $\alpha^2/4$. The brightness figure is a single cloud at one temperature. The Galaxy’s actual 21 cm sky, with its spiral arms, its warped disc and its high-velocity clouds falling in from outside, is a map, and no single curve is one.

Still open: when the first stars lit the hydrogen #

Before the first stars formed, the universe’s hydrogen was neutral and its 21 cm line, redshifted by the expansion to metre wavelengths, carries a record of the gas’s temperature. As the first stars switched on, their ultraviolet light coupled the spin temperature to the gas, the gas cooled below the background radiation’s temperature, and the line should appear as a faint absorption dip in the radio sky’s spectrum at around 70 to 100 megahertz. In 2018 one experiment reported such a dip, twice as deep as expected; others have not confirmed it, and separating a signal of a tenth of a kelvin from galactic foregrounds ten thousand times brighter is the difficulty. Whether the dip is real, and if so what made the early gas so cold, is unsettled.

The habit worth carrying away is to ask what a small quantity buys as well as what it costs. A slow transition is a faint one per atom, and it is also one that collisions control, that the gas cannot absorb, and whose populations do not depend on temperature — and each of those turns a weak line into a clean measurement. The 21 cm line is bright because everything about it is small except the amount of hydrogen.