A flow that piles up at the magic numbers

Fusion stops at iron. Iron-56 and its neighbours are the most tightly bound nuclei, and fusing anything heavier costs energy rather than releasing it; a star’s core that reaches iron has run out of fuel. And yet the Solar System contains every element up to uranium, and its gold, barium, lead and tin had to come from somewhere. In 1957 Margaret and Geoffrey Burbidge, William Fowler and Fred Hoyle — building on Hoyle’s argument from carbon to a nuclear level — showed where. Heavy nuclei are built by adding neutrons, which feel no electric repulsion and can be captured by a nucleus at any temperature, one at a time, and then waiting for the new nucleus to beta-decay into the next element up.

Two versions of the process are needed to explain the abundances, distinguished by how fast the neutrons arrive compared with how fast the unstable nuclei decay. In the rapid process, neutrons flood in during an explosive event and nuclei capture many before they can decay. In the slow process — the s-process — neutrons arrive rarely, a nucleus waits months or years between captures, and every unstable nucleus along the way has time to decay first. The s-process builds about half of the heavy elements, it runs in stars now, and its signature is one of the most elegant regularities in the whole of astrophysics.

A path up the valley of stability #

A nucleus that captures a neutron becomes the next isotope of the same element, one mass number heavier. If that isotope is stable, it waits for another neutron. If it is unstable to beta decay — a neutron in it turning into a proton, an electron and an antineutrino — and its half-life is short compared with the time between captures, it decays first, becoming an isotope of the next element up. The s-process therefore climbs the chart of nuclei along a path that hugs the valley of stable isotopes, adding neutrons to each element until it reaches an unstable isotope, then stepping up one element and continuing.

The time between captures is what decides the path, and in the s-process it is long — years to decades for a typical nucleus in the neutron densities of an ageing star, around $10^7$ to $10^8$ neutrons per cubic centimetre. Almost every unstable isotope on the way decays faster than that. The path is therefore a single, well-defined sequence of nuclei from iron to lead, and every nucleus on it is made by capture on the one before and destroyed by capture itself.

How readily a nucleus captures #

The rate at which a nucleus captures a neutron is set by its cross section, and cross sections vary enormously along the path.

The closed shells are the magic numbers of nuclear physics — 2, 8, 20, 28, 50, 82 and 126 — at which nuclei are exceptionally stable, in the way that atoms with closed electron shells are the unreactive noble gases. A nucleus with a magic number of neutrons is tightly bound, has a large gap before the next available neutron level, and captures neutrons badly. The cross sections of the magic nuclei are ten to a hundred times smaller than their neighbours’, and that single fact of nuclear structure shapes the heavy-element abundances of every star.

A flow in local equilibrium #

The abundance of each nucleus on the path changes at a rate equal to its production, by capture on its lighter neighbour, minus its destruction, by capture itself:

$$\frac{dN_A}{d\tau} = \sigma_{A-1}N_{A-1} - \sigma_A N_A,$$

where $\tau$ is the neutron exposure — the neutron flux integrated over time, measured conveniently in inverse millibarns. Where the cross sections are large, the nuclei respond quickly to any imbalance, and the flow settles into a state in which production equals destruction: $\sigma_A N_A = \sigma_{A-1}N_{A-1}$. The product of cross section and abundance is the same for neighbouring nuclei, and an abundance is inversely proportional to its cross section. A nucleus that captures neutrons readily is scarce, because it is quickly turned into the next one; a nucleus that captures badly accumulates.

The steps are where the flow is held up. The small cross sections at a closed shell mean that material arriving there waits a long time before it can pass on, and with a limited total exposure, much of it never does. Downstream of each magic number the flow is reduced. The flat stretches are local equilibrium; the steps are bottlenecks, and the whole curve is a picture of a flow through a pipe with three narrow places in it.

The steps measure the exposure #

The heights of the steps depend on how much neutron exposure the material received, and that is what makes the $\sigma N$ curve a measurement rather than a description.

The Solar System’s s-process abundances — separated from the rapid-process contribution by using the isotopes that only the s-process can make — follow a $\sigma N$ curve of exactly this form, with flat stretches and three steps. Fitting it requires more than a single exponential exposure: the abundances up to about strontium need a weak component with a small mean exposure, made in massive stars during helium and carbon burning, and those beyond need a main component with a larger one, made in stars of one to three solar masses late in their lives. The step heights are how the two components were found, before anyone had identified the stars that make them.

Peaks at strontium, barium and lead #

Dividing the flow by the cross sections turns the steps into the abundance peaks that are the s-process’s visible signature.

Those relative heights are observable in stellar spectra, element by element, because strontium, barium and lead each have strong absorption lines. Stars differ in them in ways that follow their history. Metal-poor stars that received s-process material have far more lead relative to barium than stars like the Sun, because in a metal-poor star there are fewer iron seed nuclei to share the neutrons among, so each seed receives a larger exposure and the flow runs further — a prediction of the model confirmed in the 2000s by the discovery of lead-rich, metal-poor stars. The ratio of lead to barium in an old star is a measure of how many neutrons each iron nucleus received, set by the iron content of the star that made them.

Where the flow ends #

The s-process cannot run past lead and bismuth, and the way it stops is a small loop. Bismuth-209 captures a neutron and becomes bismuth-210, which beta-decays to polonium-210, which emits an alpha particle and becomes lead-206. The flow that reaches the top of the chart is sent back down to lead, and further captures cycle it round again. Lead therefore accumulates at the end of the path, and in the most neutron-rich conditions — metal-poor stars with few seeds — a large share of all the s-process material ends there.

That end point is one of a series of places where nuclear structure halts a build-up. The big bang made almost nothing beyond helium because there are no stable nuclei with five or eight nucleons, and the gaps could not be bridged in the few minutes available; stars bridge the gap at eight with the triple-alpha reaction and its resonance, fusion then carries matter to iron and stops, and neutron capture carries it past iron to lead and stops again. Each stage of element building is bounded by a feature of nuclear physics, and the abundance pattern of the Universe is the sum of what each stage could reach before its barrier.

Where the neutrons come from, and the element that should not be there #

The main s-process runs in stars of one to three solar masses on the asymptotic giant branch, the last stage of their lives, when a thin shell of helium burns in unstable flashes every few tens of thousands of years round an inert carbon–oxygen core. Between flashes, protons from the envelope are mixed into the top of the carbon-rich helium layer, and a thin pocket of carbon-13 forms. The reaction of carbon-13 with helium nuclei, $^{13}\mathrm{C}(\alpha,n)^{16}\mathrm{O}$, releases neutrons slowly over thousands of years, at exactly the low densities the s-process needs. The flashes then mix the processed material outward, and the star’s convective envelope brings it to the surface.

The proof that this happens came from an element with no stable isotopes. In 1952 Paul Merrill found absorption lines of technetium in the spectra of several red giants of the S type. The longest-lived isotope made by the s-process, technetium-99, has a half-life of about 210,000 years, far shorter than the age of any star, so no technetium could have survived from the gas the star formed from. It had to have been made inside the star, recently, and brought to its surface — the first direct evidence that heavy elements are being made in stars now, rather than having been made once at the beginning. Merrill’s observation preceded the 1957 paper that explained it and was one of the facts that paper was built on.

The same material is found in the Solar System in solid form. Microscopic grains of silicon carbide, extracted from primitive meteorites, carry isotopic compositions of barium, strontium and other elements that are pure s-process — unlike anything in the Solar System’s average material — because they condensed in the outflowing wind of a particular asymptotic-giant-branch star before the Sun formed and survived intact. They are samples of individual stars that can be held in a laboratory, and their isotopes match the s-process model’s predictions, isotope by isotope, including the effects of the branchings where the flow splits.

The weak component, in massive stars #

The main component’s neutrons come from carbon-13 in low-mass giants; the weak component’s come from neon-22 in massive stars, and they tie this process to the last stages of burning.

During core helium burning in a star of more than about ten solar masses, the nitrogen-14 left over from hydrogen burning’s CNO cycle is converted by helium captures into neon-22. Near the end of helium burning, when the core has reached about 300 million kelvin, the reaction $^{22}\mathrm{Ne}(\alpha,n)^{25}\mathrm{Mg}$ begins to release neutrons, and it releases more in the carbon-burning shell that follows. The exposure is small — the neutron source is limited by how much nitrogen the star began with — and the flow runs only as far as the first bottleneck. The weak component makes most of the Solar System’s s-process copper, gallium, germanium and the lighter elements up to strontium, and almost nothing beyond, and it is ejected when the star explodes, within a few million years of the star’s birth.

The two components therefore have very different clocks. The weak component enriches a galaxy almost as soon as massive stars form; the main component waits for stars of one to three solar masses to reach the end of their lives, hundreds of millions to billions of years later, and then leaves them in slow winds that strip the envelope before the core becomes a white dwarf. The heavy elements of a stellar population arrive in the order the two kinds of star die.

Barium on the wrong star #

Some stars show the s-process signature without being able to have made it. Barium stars are ordinary giants, too young and too low in mass to have reached the asymptotic giant branch, whose spectra show strong lines of barium, strontium and other s-process elements. Every one that has been monitored long enough turns out to be a binary with an invisible companion: a white dwarf, the remnant of a star that was once more massive, went through its own asymptotic-giant-branch stage, and transferred its s-enriched envelope onto the star now seen.

The barium in a barium star is a fossil of its companion’s nucleosynthesis, and the orbital periods of barium stars, from hundreds to thousands of days, trace how mass transfer works in wide binaries. The same mechanism explains many of the metal-poor stars rich in carbon and s-process elements found in the Galaxy’s halo, which are among the oldest objects carrying the signature.

A delayed enrichment, read in old stars #

Because the main s-process waits for low-mass stars to age, its products appear in a galaxy late, and the delay is written into the composition of the stars that formed along the way.

The rapid process, by contrast, is prompt: whatever its sites — mergers of neutron stars, rare kinds of supernova — the elements it makes, such as europium, appear in the oldest metal-poor stars. The ratio of barium, mostly s-process in the Sun, to europium, almost entirely rapid-process, is therefore a clock. In the oldest stars of the Galaxy’s halo it is low, at the value a pure rapid process gives; as the iron content rises it climbs towards the solar value, as the low-mass stars of each generation reach the asymptotic giant branch and add their barium. The rise is spread over billions of years, as every delayed enrichment in a galaxy is, because stars of different masses reach the end of their lives at different times.

That record depends on the gas retaining what the stars returned to it, and the metals a galaxy keeps measure what it threw away: a dwarf galaxy that lost its gas in winds shows a different barium-to-europium history from the Milky Way’s disc. The s-process abundances of old stars are a record of when a galaxy’s low-mass stars were dying, read through a nuclear flow whose shape was fixed by the magic numbers.

Magic numbers from the sky #

The connection between the abundance peaks and the magic numbers runs in both directions, and historically in the less expected one.

In the late 1940s Maria Goeppert Mayer, working on the origin of the elements with Edward Teller, noticed that certain numbers of protons or neutrons recurred in the most abundant nuclides — the abundances compiled from meteorites and the Sun showed excesses at nuclei with 50, 82 and 126 neutrons. Together with the patterns of nuclear binding energies and stabilities, that led her, and independently Hans Jensen and colleagues, to the nuclear shell model in 1949, with a strong spin–orbit force that reproduces exactly the magic numbers observed. They shared the Nobel Prize in 1963. The abundance peaks that the s-process leaves were part of the evidence from which nuclear structure was deduced, and the essay’s figures run that argument in reverse: take the shell model’s small cross sections at the magic numbers, run the flow, and the peaks come out where the abundances show them.

What the model simplifies #

The cross sections are a smooth curve with three dips. Real cross sections vary by factors of several between neighbouring nuclei, with odd–even effects and many individual resonances; the measured values, from time-of-flight experiments at accelerator neutron sources, are what a real $\sigma N$ analysis uses, and they make the curve between the steps noticeably less smooth than the one drawn.

The path has no branches. At a dozen unstable nuclei — krypton-85, samarium-151, thulium-170 and others — the beta-decay half-life is comparable with the time between captures, and the flow splits. The split depends on the neutron density and the temperature, and the abundances on each side of a branching are among the best thermometers and densitometers of the stellar interior where the s-process ran.

The exposure distribution is a single exponential, the classical approximation. In a real asymptotic-giant-branch star the material is exposed in repeated pulses with mixing between them, which produces a distribution that an exponential fits approximately but not exactly, and the modern analyses use full stellar models instead.

And only the s-process is drawn. The Solar System’s heavy elements are roughly half s-process and half rapid process, and the rapid process leaves its own peaks, displaced to lower mass numbers, because it passes the magic neutron numbers far from stability where the nuclei are lighter.

Still open: how the carbon-13 pocket forms #

The main s-process depends on the pocket of carbon-13 that supplies its neutrons, and nobody knows in detail how that pocket forms. It requires protons from the hydrogen-rich envelope to penetrate a small distance into the carbon-rich layer below, in just the right amount — too few and there are not enough neutrons, too many and the protons capture the neutrons themselves — and the mixing that does it happens at a boundary between convective and radiative regions, on scales far smaller than any stellar model resolves. Models insert the pocket with an adjustable size and profile, calibrated against the abundances. The spread of s-process patterns among stars of the same mass and metallicity, measured in barium stars and in the presolar grains, suggests the pocket varies from star to star; whether the variation comes from rotation, magnetic fields, or the physics of convective boundaries is what the next generation of stellar models is trying to find out, and the abundance record of the Galaxy is the test they will have to pass.