Whereas some people play extreme sports, many nuclear physicists seek the thrill of extreme isotopes, by finding, for each chemical element, the largest possible number of neutrons that can be held by an atom. This boundary of nuclear existence, called the neutron drip line, has not been fully mapped — although the construction of rare-isotope facilities1 will bring the goal closer. Moreover, even the theoretical location of the drip line is uncertain2,3. In a paper in Nature, Tsunoda et al.4 argue that the mechanism responsible for the drip line is more subtle than previously understood and is related to deformation, a hallmark of much of ordinary nuclear physics.
The strong nuclear force that binds protons and neutrons together favours equal numbers of each particle. By contrast, weaker but longer-range electrostatic repulsion discourages the accumulation of protons in atoms. Competition between these two forces produces the valley of stability — the V-shaped surface that corresponds to stable nuclei when the energy per nucleus is plotted as a function of the number of protons and neutrons. The bottom of this valley is associated with the most stable isotopes, which have just the right mix of protons and neutrons. Add neutrons to these isotopes, and you move up the valley walls.
However, neutrons cannot be added forever. It takes energy to pull atomic nuclei apart, because their total energy is less than that of their components. This deficit is called the binding energy (shown as negative energies in Figure 4 of the paper4). If adding a neutron increases the binding energy, the neutron sticks. Otherwise, the energetically disfavoured neutron ‘drips off’. Note that there is also a proton drip line, driven by increasing electrostatic repulsion.
Nuclear physicists have long assumed that the neutron drip line is governed by the preference of the strong nuclear force for proton–neutron symmetry. In the nuclear-shell model, protons and neutrons occupy quantum shells, much like electrons in an atom, and each shell has a particular potential energy. The thinking was that if the proton–neutron symmetry decreases, the potential energy also falls, to the point at which adding a neutron lowers the binding energy.
But protons and neutrons do not stay in a single shell. Instead, driven by the strong nuclear force, they jump from shell to shell, forming different configurations (see Figure 1 of the paper4). Like a flock of birds wheeling in the sky, protons and neutrons move collectively. For example, they can pair up with dance partners, such as electrons in a superconductor, and they can produce deformed (ellipsoid) nuclei that rotate, throwing off γ-rays. It turns out that nuclei are easily deformed — especially when the energy gap between shells is small. With ‘magic’ numbers of protons or neutrons, akin to the filled electron shells that drive the chemical inertness of noble gases, this energy gap is large, and deformation is suppressed.
Aware of this picture, Tsunoda et al. calculated the various contributions to the binding energy, such as the mean (monopole) energy generated from adding a neutron to a shell, the energy from deformation and the energy from protons and neutrons pairing up. They discovered that, as neutrons are added to nuclei of elements ranging from fluorine to magnesium, the monopole contribution rises steadily. At the same time, the nuclei initially become increasingly deformed, magnifying the rise in the binding energy. But, as even more neutrons are added, the nuclei become difficult to deform, and the deformation contribution falls more quickly than does the increase in the monopole contribution. When that happens, the binding energy falls, giving rise to the drip line (Fig. 1). Intriguingly, although the pairing contribution is non-negligible, it is approximately constant and so does not drive the drip line.
The rise and fall of deformation for these neutron-rich nuclei touches on another topic: magic numbers far from nuclear stability. At the bottom of the valley of stability, 20 is a magic number. For example, calcium (with 20 protons) has many stable isotopes that are difficult to deform, and calcium-40 (with 20 protons and 20 neutrons) is ‘doubly magic’. But, moving away from stability, as the balance between protons and neutrons shifts, previous magic numbers can be replaced with new ones5, for example at 16 neutrons. Both theory4,5 and experiment6 show that, as neon and magnesium isotopes collect more than 16 neutrons, the energies of the lowest-energy states that have 2 and 4 units of angular momentum become markedly lower (see Figure 3 of the paper4), which is a typical sign of increased deformation.
Tsunoda and colleagues’ proposed mechanism draws on concepts familiar to nuclear physicists — in particular, the competition between deformation and mean shell structure. But some questions remain. For example, although the authors’ calculations were highly detailed, requiring supercomputers, they largely ignored the unbound (continuum) single-particle states that have an essential role in defining the drip line for lighter nuclei than those considered here. Moreover, although the authors drew on ab initio interactions between protons and neutrons, they made empirical tweaks to the single-particle energies, which are part of the potential energies of the shells. Such ‘by-hand’ adjustments leave the robustness of the proposed mechanism unclear.
But the biggest question concerns the drip-line mechanism for even heavier elements than those considered here. This mechanism could be driven either by the mean shell structure and leaking of neutrons into the continuum, or by the competition between deformation and the mean potential-energy profiles that define magic numbers — or some combination of the two. For these heavier elements, the drip line is associated with rapid neutron-capture nucleosynthesis7 — a process that forges many heavy elements, such as iodine, gold and the rare-earth metals. To fully understand this process, we have to go to nuclear extremes.
Nature 587, 40-41 (2020)