The Impossible Rung Inside a Star: The Birth of Carbon

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Every carbon atom in your body got here by climbing a rung inside a star that looked impossible to climb: what comes out when two helium nuclei join lives for about a quadrillionth of a second and falls apart. So for carbon to form, a third nucleus had to arrive in exactly that gap before anything had come apart. In nineteen fifty-three an astronomer said that for this to be possible there had to exist, in nature, an energy level no one had yet seen — and he said where it was, too.

At the center of a star, when the hydrogen runs out, a strange silence settles in. For billions of years, millions of tons of hydrogen have been turning into helium in that core every second, and the energy pressing outward has held the star's own weight at bay. When the fuel is spent, the balance breaks. The core collapses onto itself; as it collapses it compresses, and as it compresses it heats. The temperature climbs to 100 million degrees. There is no hydrogen left now. What remains is the helium ash the star made for itself. And it is precisely here that the most ordinary-looking question in the universe arises: can anything heavier be built out of this ash?

The ladder of the elements looks, at first glance, like an easy ladder to climb. You add a proton or a neutron to a nucleus and you step up one rung. From hydrogen to helium, from helium to lithium, then to beryllium, to boron, to carbon. On paper, a flawless sequence of steps. But nature never built two of this ladder's rungs.

The first is rung number 5. A helium-4 nucleus is an extraordinarily stable, tightly packed lump of four particles; that is why it is given a name of its own and called an alpha particle. Add a proton to it and you would have lithium-5. Add a neutron and you would have helium-5. Neither of them exists. More precisely, they barely even attempt to exist: once formed, within about a billionth of a trillionth of a second — a span as short as the time light needs to cross an atomic nucleus from one side to the other — they spit the extra particle back out. There is no stable nucleus with mass number 5. On the ladder, there is no rung.

So you could skip 5 and go straight to 8. Smash two alpha particles together; eight particles in total, beryllium-8. This nucleus really does form. It has been measured in the laboratory, its energy is known, its identity is beyond dispute. It has only one flaw: it prefers to sit as two separate alpha particles rather than as a single combined nucleus, by about 92 kiloelectronvolts. Beryllium-8, in other words, is less stable than the two pieces that make it. The outcome is inevitable. It forms and it falls apart. Its lifetime is roughly one ten-quadrillionth of a second. There is no way to picture a span that short. The only thing one can say is this: in the time it takes you to hear this sentence, trillions upon trillions of lifetimes have come and gone for a beryllium-8 nucleus.

Rung number 8 is missing as well. And that means the ladder is severed outright. Because in a stellar core filled with helium ash there are no protons left over; the route upward by adding one particle at a time is closed. If all you hold in your hand are alpha particles, then joining two of them brings you to 8, and 8 does not hold.

This same break raised the same wall in the first moments of the universe. In the first few minutes after the Big Bang, the temperature and density were right for nuclear reactions; the universe made hydrogen, helium, and a trace of lithium. Then expansion pulled everything apart, the temperature fell, and the window closed. There was not enough time to cross the gap between 5 and 8. The cosmic kitchen shut down before the cooking had properly begun.

The conclusion that follows, viewed coldly, is this: carbon should not exist. Oxygen should not exist. Iron, calcium, phosphorus, none of them should exist. The universe should have been a place made of hydrogen and helium, staying plain forever. Stars could have flared and faded, but nothing would ever have been left behind to build a planet, no silicate to build rock, no carbon skeleton to build a molecule.

And yet carbon is the fourth most abundant element in the universe by mass, after hydrogen, helium, and oxygen. It is everywhere we turn our eyes. In the spectra of stars, in interstellar clouds, in the ice of comets, in ink, in coal, in the backbone of DNA. There is carbon in the cartilage of the ear hearing these words right now, and in the membranes of the nerve cells making sense of that sound. About 18 percent of your body's mass is carbon.

So somewhere, somehow, something jumped across that broken rung. That is the puzzle: how was a bridge crossed that falls apart at 8?

The first clue came from looking at beryllium-8's embarrassingly brief lifetime from an entirely different angle. In 1951 and 1952, two researchers working independently of one another — one the Estonian astronomer Ernst Öpik, the other Edwin Salpeter, working in America — noticed the same thing. One ten-quadrillionth of a second is nothing on a human scale; but not on the scale of an atomic nucleus. Compared with the time it takes two alpha particles to pass by each other, the lifetime of beryllium-8 is hundreds of thousands of times longer. For a nucleus, that is not a fleeting instant. That is ample time.

Salpeter's transformative move was to stop seeing beryllium-8 as a product. It had to be thought of as an equilibrium population instead. At that temperature, in a stellar core full of helium, alpha particles collide ceaselessly. Every second an unimaginable number of beryllium-8 nuclei form, and every one of them promptly falls apart. But because the rates of formation and decay balance each other, at every instant there is a small but steady reservoir of beryllium-8 waiting inside the core. A dilution on the order of one part in a billion; so little as to be almost nothing. Yet almost nothing is not nothing. And before a beryllium-8 nucleus comes apart, within that tiny interval of time, a third alpha particle can arrive and strike it. Three alphas, twelve particles. Carbon-12.

That is how the bridge was built: not in a single move, but by stepping onto a collapsing rung before the collapse was complete.

The trouble was that the arithmetic did not support it. When Fred Hoyle put the rate of this reaction into stellar models in the early 1950s, the amount of carbon that came out was laughably small next to the amount of carbon observed in the universe. Something somewhere was missing.

The reasoning Hoyle then carried out is one of the boldest backward inferences in the history of physics. He did not start from theory and then look at nature; he looked at nature and declared what theory was obliged to say. Carbon exists, he said, and in this quantity. Therefore the fusion of three alphas has to be many times faster than a straightforward calculation would predict. And in nuclear physics only one mechanism is known that can speed up a reaction like that: resonance.

To understand resonance, think of a swing. Push the swing at random moments and it hardly rises; push it at the moments that match its own natural rhythm and, with the very same force, it climbs far higher. Atomic nuclei, likewise, can exist only at certain discrete energy steps. If the energy the colliding particles bring with them happens to land on one of the target nucleus's permitted steps, the probability of fusion rises by hundreds or thousands of times. If it does not land there, the colliding partners simply scatter apart.

Hoyle saw that he could calculate the required energy, because all the inputs were known. The sum of the masses of a beryllium-8 nucleus and an alpha particle lies 7.367 megaelectronvolts above the most stable state of carbon-12. The thermal motion in a stellar core adds a few hundred kiloelectronvolts more. Therefore, Hoyle said, the carbon-12 nucleus must have an excited energy level at roughly 7.68 megaelectronvolts. What is more, the rotational properties of that level had to be of a particular kind, or the reaction still would not work.

Among the known energy levels of carbon-12 in those years, no such thing existed.

In 1953, Hoyle went to the California Institute of Technology and told the experimenters doing nuclear measurements there, Ward Whaling in particular, that they needed to go looking for this level. The request was extraordinary: an astronomer, on the grounds that carbon exists in the universe, was asking a nuclear physicist to find an energy step that no one had ever seen. Whaling and his colleagues Dunbar, Pixley, and Wenzel set up the experiment at the Kellogg Radiation Laboratory. They bombarded nitrogen-14 nuclei with deuterons; the products of that reaction are a carbon-12 nucleus and an alpha particle. By measuring carefully the energies of the emerging alpha particles, it was possible to read off which energy levels the carbon left behind had been placed in.

The level was there. The result, published that same year, reported an excited state at 7.68 megaelectronvolts with an uncertainty of 0.03 megaelectronvolts. Where Hoyle wanted it, and as sharply as he wanted it.

The story was completed a few years later. In 1957, Cook, Fowler, and the two Lauritsens showed that this level, when it decays, breaks into three alpha particles, and that it carries the rotational properties being sought. It was not merely in the right place; it was of the right kind. Today this energy step is called the Hoyle state, and modern measurements put it at 7.654 megaelectronvolts.

With that, the bridge was complete. Two alphas take on a debt of 92 kiloelectronvolts for an instant and assemble beryllium-8. The third alpha arrives before the debt comes due. Carbon-12 is born in a trembling, strained excited state, and most of the time it falls back into three alphas; but once in thousands of attempts, it emits a gamma ray, sheds its excess energy, and settles into stable carbon. One carbon out of three heliums, releasing 7.275 megaelectronvolts of energy in total.

Finding the Hoyle state closed one question and immediately opened a far more unsettling one. Because the position of that level was not merely right; it was placed with a strange and startling fineness.

The energy that a beryllium-8 nucleus and an alpha particle bring between them is 7.367 megaelectronvolts. The Hoyle state sits at 7.654. The difference is 287 kiloelectronvolts. The thermal motion inside the star closes exactly that gap. The triple-alpha reaction is exponentially sensitive to the size of that difference: the probability that the particles can find that much extra energy varies sharply with temperature. Indeed, the energy production of helium burning depends on roughly the fortieth power of the temperature, one of the steepest dependences known. A difference of 10 degrees shifts the rate measurably.

To see what this implies, you have to move the level in your imagination. Since the late 1980s more than one research group has done precisely that: shifting the position of the Hoyle state a few hundred kiloelectronvolts up or down and running stellar models again. The results, though the uncertainties are large, all point the same way. Move the level a few hundred kiloelectronvolts up, and the reaction slows so much that stars can barely convert helium into carbon at all. Move it down, and carbon is produced so easily that nearly all of it is swept on to the next step, oxygen. In either case the universe becomes a place where carbon-based chemistry cannot develop. The window being discussed is roughly 100 kiloelectronvolts wide; very small on the scale of particle physics, very large in its cosmic consequences. In 2013, lattice calculations starting from quark masses suggested that this window corresponds to changes of a few percent in the fundamental constants.

Here one has to be very careful. These calculations are genuine physics results, and they teach us something: how sensitive the abundance of carbon is to the fine details of nuclear physics. But they say nothing whatsoever about why that sensitivity is what it is. To draw a purpose, a design, or an intention out of this belongs not to physics but to philosophy, and no agreed answer exists there either. The one solid observation we have is this: in a universe capable of making carbon, beings curious about carbon can also appear. Everything known about the fineness of the Hoyle state ends there; the rest is not yet science and should not be presented as such.

The less familiar, and perhaps subtler, half of the story lies on the next rung. Once carbon-12 has been assembled, it can capture one more alpha particle and turn into oxygen-16. If this second reaction also had a resonance sitting in exactly the right place, stars would convert all the carbon they make into oxygen, and no carbon would be left. There is no such resonance. The energy levels of oxygen-16 fall outside the places that would accelerate this reaction, and so the conversion of carbon into oxygen limps along slowly, and when the helium is exhausted the star is left holding both carbon and oxygen. The precise rate of this reaction is still considered the most important unmeasured number in nuclear astrophysics; in a handful of accelerator laboratories around the world, including one set up underground, that value has been pursued for years.

The Hoyle state itself is not a closed file either. The stable state of carbon-12 is a nearly spherical, tightly bound structure. The Hoyle state is not. The calculations and lattice simulations of the past two decades describe this excited state as a structure in which three alpha clusters are arranged like a loose, swollen triangle with unequal corners, its volume several times that of stable carbon. Some theorists read this as a dilute nuclear gas in which the alpha particles move almost independently; others insist on a bent triangular geometry. That debate continues and is nowhere near settled. Which is to say that no one yet knows for certain what the energy step responsible for the existence of carbon in the universe actually looks like.

The last link is how the carbon coming off that step escapes the star. A star of the Sun's mass, as it begins to exhaust its helium, swells outward and becomes a red giant. The carbon produced in the core is carried toward the surface by deep and violent convective currents. The star's outer layers are now held only loosely; dense stellar winds blow this carbon-rich material slowly out into space. At the end of its life the star releases its outer layers entirely and leaves behind a core like an ember. As the ejected material cools, it forms soot and microscopic diamond-like grains; these dusts mix into interstellar clouds, where they enter the dough of new stars and new planets.

About 99 percent of the natural carbon on Earth is carbon-12. Nearly all of those nuclei — in a proportion high enough to call it without exception — have passed through these three steps at the center of a star: a beryllium-8 nucleus doomed to fall apart, a third alpha particle arriving just in time, and an energy step waiting in that narrow interval of 287 kiloelectronvolts. Every single carbon atom in the graphite of your pencil, in the carbon dioxide of the air you breathe, in the cells forming this sentence, has stepped on that impossible rung and crossed.

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