By calculating the behaviour of protons and neutrons inside carbon nuclei from first principles, physicists in Germany and the US have identified the shape of carbon’s Hoyle state – which is an important step in the production of heavy elements inside stars. The researchers found the state to have an unusual bent structure, a finding that should help identify the forces at work in carbon production.

Carbon-12 comprises six protons and six neutrons and is a key step in nucleosynthesis – the process by which heavier elements are produced inside stars. Physicists studying stellar fusion in the 1940s and 1950s reckoned that carbon-12 forms when two helium-4 nuclei fuse to produce beryllium-8 – which then fuses with a third helium-4 nucleus. There was a problem with this hypothesis, however. The energy of the fused particles is considerably higher than that of the ground state of carbon-12. This implies that the new particle is in fact extremely unlikely to form in this way – far too unlikely to account for the great abundance of carbon in the universe.

According to Hoyle

To overcome this apparent contradiction the British astronomer Fred Hoyle proposed in 1954 that carbon-12 has an excited state that had never been seen before. The idea is that carbon-12 would form readily in this state and then decay to its ground state, giving off a well defined amount of energy (7.6 MeV) in the process. This excited state was then observed three years later by researchers at the California Institute of Technology, when carrying out experiments involving beta decays of boron-12.

For the past 60 years nuclear physicists have been trying to understand the nature of this “Hoyle state”, which is not predicted by standard nuclear models. These models regard nuclei as being made up of individual protons and neutrons, and it was reckoned that the Hoyle state is better described as three helium-4 clusters. Those clusters have now been identified by Ulf Meissner of the University of Bonn and colleagues, thanks to the number-crunching power of the JUGENE supercomputer in Jülich and a new form of Steven Weinberg’s “effective field theory”, which considers protons and nucleons as individual entities rather than as bound states of three quarks.

Space–time lattice

Weinberg’s theory reduces the number of particles that can be considered to make up a carbon-12 nucleus by a factor of three – from 36 to 12. Even 12, however, is too many for an analytical description of the nucleus. Instead, Meissner’s group combined the theory with numerical methods often used to describe the interaction of individual quarks via the strong force. This approach breaks down space and time into discrete chunks, constraining particles to exist only at the vertices of a space–time lattice and so radically simplifying the possible evolution of the particle system.

In a paper published in 2011, Meissner and co-workers described how they used this hybrid approach to identify the Hoyle state. To do this they first picked out carbon-12’s ground state, setting up vast numbers of configurations of the virtual protons and neutrons within JUGENE and then watching what happened as those configurations evolved in time. The configuration that lasted the longest, being the most stable, was the ground state. Identifying the Hoyle state was a bit trickier since it involved stopping the simulation at some earlier point in time and then disentangling the various states that remained. Despite the challenges of calibrating their simulation using scattering and other data, their calculated values for the energy of the carbon-12 ground state and the Hoyle state agreed very well with experiment.

“Bent arm” shape

Now in this latest work, the team has calculated the structure of those states using a more sophisticated representation of the nuclear wavefunction. Likening the nucleons and groups of nucleons to LEGO bricks, Meissner says that “before we had bricks of just one size and now we have a whole series of different-sized bricks that we can use to construct more complex structures”. Building up those structures, the group found that in the ground state, carbon-12 consists of three helium-4 clusters arranged in a compact equilateral-triangle formation, whereas in the Hoyle state the three clusters form an obtuse triangle or “bent arm” shape. This more open configuration, the researchers explain, results from the extra energy in the system.

One exciting aspect of the research, according to Morton Hjorth-Jensen of the University of Oslo in Norway, is that it should allow scientists to understand which part of the strong force dictates the carbon-12 decay. This is important because the force in fact consists of several elements, including some that deform the shape of nuclei. “Hoyle predicted his state on the basis of the anthropic principle, arguing that if the state didn’t exist we wouldn’t be here,” he says. “But we now want to understand the structure of this state in terms of its basic constituents and forces.”

Experimental tests

Meanwhile, David Jenkins of York University in the UK points out that the latest work makes a number of explicit predictions that could, in principle, be tested experimentally, including the existence of a number of electromagnetic transitions involving the Hoyle state. But he adds that these transitions are very weak and therefore hard to measure. “Such experiments will be no less challenging than the theoretical achievement,” he says, “but renewed effort is warranted given the strong topical interest.”

According to Meissner there is also more theoretical work to be done. One job, he says, is to reduce the spacings in their virtual lattice, in order to make more precise calculations. Another is to investigate larger nuclei, such as oxygen-16, as well as the reactions that give rise to these nuclei – in this case carbon-12 combining with a helium-4 nucleus. “This is a very important reaction in the sequence that generates life-giving molecules,” he adds.

The latest work is published in Physical Review Letters.

Physics World will soon be publishing a feature article about the Hoyle state written by David Jenkins and Oliver Kirsebom, who is at TRIUMF in Canada.