The bundle of 78 nucleons in a single nickel-78 nucleus are infinitesimal. And yet, calculating that nucleus’s structure entailed an enormous computing effort: 5 million CPU hours on the most powerful supercomputer in the United States. The results could offer key insights into the potential existence of the long-sought “island of stability”: ultraheavy but unusually stable nuclei.

The most powerful supercomputer in the United States, Titan at Oak Ridge National Laboratory, needed 5 million CPU hours to calculate the nickel nucleus's structure. Image courtesy of the Oak Ridge National Laboratory and US Department of Energy.

In October, three scientists at Oak Ridge National Laboratory and the University of Tennessee in Knoxville published the results of this nickel-78 calculation in Physical Review Letters (1). The researchers demonstrated that the nucleus’s combination of protons and neutrons formed a very stable structure, what physicists call a doubly magic nucleus. It was no small feat: the finding involved solving a quantum mechanical system, including the strong nuclear force acting between 28 protons and 50 neutrons. “The immensity of these calculations are way beyond anything you can think about doing on a laptop or desktop or small cluster,” says Gaute Hagen of Oak Ridge, the study’s first author.

It was the latest example of a supercomputing trend that is shaking up the field of nuclear physics. For decades, scientists have sought to find the “magic numbers” of protons and neutrons that form these especially stable nuclei. But although experimentalists have forged heavier and heavier such magic nuclei in the beams of their accelerators, theorists have lacked the computational power to keep up. Although theoretical calculations of the structure and stability of nuclei with small numbers of nucleons have been tractable, there has simply been no way to perform the calculations necessary to accurately predict and understand what can already be measured in the laboratory.

But now, advances in computational techniques and faster hardware are finally allowing theorists to work in parallel to experimentalists for the first time as they push up the nuclear chart to higher numbers of nucleons. “To make predictions that can be verified [experimentally] in parallel to the theoretical developments, it’s fair to say, have revolutionized the field,” says Achim Schwenk of the Technical University of Munich.

Mostly, physicists pursuing these magic nuclei are enthralled by the technical challenges inherent in exploring these fundamental forces and structures, although some early research was driven by the possibility of applications that use heavy stable nuclei to shrink nuclear bombs, power sources, and technologies.

In Search of Magic Magic nuclei derive their stability from their simple structures: when they have just the right number of nucleons in their outermost layer, the forces between them achieve a sort of equilibrium; they are able to form a complete shell, making them more strongly bound. The most commonly accepted of these “magic numbers” are 2, 8, 20, 28, 50, 82, and 126. These numbers don’t necessarily correspond to specific elements, which are defined only by their proton number, but to any isotope with the right proton or neutron number. Most of these magic numbers were calculated from simple theoretical models over 60 years ago. The next wave of models had their limits, too. Instead of treating nucleons as individual particles, these models only computed densities of nucleons—a smeared-out version of reality—with experimentally justified but theoretically mysterious terms added on. “For computing masses and binding energies, for well-known nuclei, they do remarkably well,” says Hagen. But when extrapolated to higher levels of nucleons unexplored by experiment, the models gave wildly different answers. “And since it’s not based on the fundamental theory, it’s very difficult to say which one has more reliable predictions,” says Hagen. This limited their use in making predictions that guided experimentalists in finding stable nuclei. To improve these models, theorists sought to perform calculations based on first principles—so-called ab initio models—without the fudge factors of earlier efforts. But those are computationally prohibitive. In their most basic form, the computing power required to solve the equations scales exponentially as particles are added. Only 5 years ago, the heaviest element computed was carbon-12: six protons and six neutrons. “This was a tremendous effort,” says Schwenk. But in the last few years a bevy of techniques have come into fashion: in particular, the coupled-cluster method, which divvies up the nucleons into clusters, calculating the forces between only two or three at a time. Long popular in quantum chemistry where it was “Our definition of magicity has to be redefined, because these simple, regular patterns don't happen anymore.” —Ronald Fernando Garcia Ruiz applied to pairs of electrons, the method is now undergoing a renaissance as nuclear physicists apply it to nucleons. The computational costs increase with nucleon number much more manageably than for past methods. At the same time, supercomputing power continues to follow the exponential growth of Moore’s law, which has allowed theorists to finally catch up to the experimentalists, working with isotopes approaching 100 nucleons. It’s now possible to solve immensely complicated quantum-mechanical many-body problems. “That for me is just amazing,” says Hagen. Another advantage of using ab initio models is that because they are built from first principles, it’s possible to quantify meaningful uncertainties, and even compare between different types of ab initio techniques, making the predictions much more useful. This is a need in nuclear theory that “was not that present even 5 years ago,” says Witold Nazarewicz, a theorist at Michigan State University. “If the output is just one number, it is much less useful than if this prediction was accompanied by uncertainties.”

The Calcium “Battleground” One of those major pushes has been on the calcium chain, a fertile nucleus for study. “Calcium has been the battleground of many concepts of nuclear physics,” says Ronald Fernando Garcia Ruiz of the University of Leuven in Belgium. Its most common form, calcium-40, is doubly magic: each of its proton and neutron counts is magic in its own right, 20 of each in this case. So is calcium-48, with 20 protons and 28 neutrons. Furthermore, it falls in a range of masses that can be modeled with multiple approaches and compared: it’s light enough to apply ab initio methods, but also heavy enough to bear scrutiny from the older, cruder techniques. In 2012, Hagen was the lead author on a study in Physical Review Letters (2) that calculated the ground state and excitation energies up and down the calcium-chain. One of their predictions was that calcium-54 featured a subshell closure, not a fully magic shell closure like calcium-48. A Japanese team at RIKEN in Wakō, outside Tokyo, experimentally confirmed this the next year (3). The next big modeling problem for ab initio calculations to tackle was the charge radius, an indication of the size of the nucleus based on electron-scattering measurements. In nuclei with magic numbers of nucleons, the radius shrinks relative to closely related isotopes, such as in the doubly magic calcium-40. As you add neutrons, the radius starts to grow as they contort the nucleus outward. But when you reach eight extra neutrons, making calcium-48, the outer shell is completed, and the nucleus collapses back to nearly the same size as calcium-40. This was something difficult to capture in earlier models. In 2014, Hagen began pulling together a team of theorists to prove this for calcium-48 using ab initio calculations, a team that included specialists in topics including the electric dipole moment and astrophysicists who study neutron stars. “Gaute really spearheaded assembling the different people and he knew who to ask,” says Achim Schwenk. The result was a success: a 2015 paper showed for the first time that the charge radius for calcium-48 was the same as for calcium-40, in line with the experiment (4). Garcia Ruiz calls it “the big breakthrough” for theorists. The next push on the calcium frontline was a collaboration between theorists and experimentalists to determine the charge radius of calcium-52. Experimentalists had found that calcium-52 required a lot of energy to excite, making it a prime suspect for a doubly magic nucleus, implying that its 32 neutrons could mark a new magic number for neutrons. (Not all magic numbers hold for both protons and neutrons.) “At this point, theory was doing a great job,” says Garcia Ruiz, the first author of the resulting study (5). “We thought we were close to having a unified description of the nucleus, even if only for calcium.” But the result was surprising: both ab initio predictions and observations agreed that the charge radius was much larger than would be expected for a doubly magic nucleus (5). “This measurement, and theoretical calculations here, questions really the magicity of calcium-52,” says Hagen. For many, it throws into doubt the nature of magicity itself: the larger-than-expected charge radius could be evidence that strongly bound, otherwise “magic,” nuclei could have radically different structures. “Our definition of magicity has to be redefined, because these simple, regular patterns don’t happen anymore,” says Garcia Ruiz. “We cannot really think anymore in closed shells and the unique view of this nucleus.”