Physicists in Germany and the US have discovered that certain nuclei exist close to a quantum phase transition, which dictates whether a nucleus resembles a loose collection of alpha particles or looks more like a single tightly bound object. The team found that whether a nucleus is on one side or the other of this phase divide is very sensitive to the specific interactions between individual protons and neutrons. The physicists say that their work could improve the understanding of heavy-element production inside stars.

Physicists know that protons and neutrons bind together inside nuclei via the strong force, an attractive interaction that is far stronger at small scales than is electromagnetism. However, despite decades of research, they still don’t fully understand the internal structure of certain simple light nuclei: those having even and equal numbers of protons and neutrons, and which can be described as clusters of helium-4 nuclei called alpha particles.

The properties of most other nuclei can be successfully reproduced by modelling a nucleus as if it were a liquid in which each proton and neutron feels the collective pull of all the other protons and neutrons. But because alpha particles are particularly stable, nobody has been able to show how a “gas” of non-interacting self-contained alpha particles can transform into a “liquid” nucleus.

Two forces

In the latest work, Ulf Meißner of the University of Bonn and colleagues have come up with an answer. They did so after modelling the effect of two types of strong interaction on the properties of several different light nuclei. Those interactions are made up of a number of sub-interactions that have varying degrees of “locality” – the extent to which they act at a point rather than over a finite distance. One of those interactions – A – is like the strong force that acts within a gas of alpha particles, while the other – B – resembles the strong force in a liquid nucleus.

The researchers’ aim was to show that the relatively simple formulae used to represent these interactions could be used to replace the complex, multi-order expansions usually employed to describe nuclear forces. In this, they succeeded. They incorporated A and B into lattice effective field theory, a type of modelling that represents space and time as a network of lattice points. They found that for beryllium-8, carbon-12, oxygen-16 and neon-20 interactions, B yielded ground-state energies within a few per cent of experimental values, while A gave values of the same parameter that were integer multiples of the alpha-particle ground-state energy – reminiscent of a Bose–Einstein gas of particles.

Family of interactions

Meißner’s team then built a “family of interactions”, with each family member located somewhere on a sliding scale defined by the parameter λ. With λ equal to zero the interaction is A, and when equal to one it is B. For each of the four nuclei, the researchers plotted how the nucleus’s ground-state energy relative to that of an alpha particle varies with λ. The result is a phase diagram with a quantum transition – a diagonal line – for each nucleus. To the left of the line, at small values of λ, the nucleus is a gas, and to the right, at high values, it is a liquid, within which alpha particles interact with one another so that they form the nuclear liquid.

Unlike a classical phase transition, such as the conversion of steam into liquid water, a quantum phase transition takes place at zero temperature. It is driven by quantum fluctuations, which arise as a result of Heisenberg’s uncertainty principle. Meißner’s team points out that because a nucleus’s position on the phase diagram is sensitive to the exact form of the interaction between its protons and neutrons, a more sophisticated version of the calculations they have carried out might potentially knock that nucleus over the transition. In that sense, they wrote, “nature is near a quantum phase transition.”

Testing the Hoyle state

According to Meißner, their phase diagram can be used as a “diagnostic tool” to work out the structure of certain nuclei. In particular, he says, it could be used to investigate the nature of the Hoyle state, an excited state of carbon-12 that is an important step in the production of heavy elements inside of red-giant stars. The idea is to “tune” λ to find out whether the Hoyle state resides on the left or the right of the phase transition. “Some people believe that the Hoyle state of carbon consists of three alpha particles,” he says. “We can now put that idea to the test,” says Meißner, adding that there is the “intriguing possibility” that the state sits exactly on the line.

Other researchers are impressed by the latest work. David Jenkins of the University of York in the UK says there are “a number of fascinating aspects” to the way in which alpha clustering arises naturally from the fundamental interactions of effective field theory. Oliver Kirsebom of Aarhus University in Denmark agrees, suggesting that the insights could help to guide future research. “It would be very exciting,” he says, if it were possible to predict whether the Hoyle state can also decay directly into three alpha particles as opposed to decaying sequentially via single-alpha emissions – research that he and his colleagues are working on.

Witek Nazarewicz of Michigan State University, meanwhile, says that the research might also be relevant to other open quantum systems where clustering could occur, such as neutron-rich nuclei.

The research is described in Physical Review Letters.