Skyrme-Hartree-Fock-Bogoliubov mass models on a 3D mesh: V. The N2LO extension of the Skyrme EDF

The universe is built from a vast collection of tiny, heavy cores called atomic nuclei, which sit at the heart of every atom. While we can see the effects of these cores in the stars and the matter around us, we cannot see the cores themselves directly. To understand how they hold together, how they break apart, and how they behave under the crushing pressure of a dying star, scientists rely on theoretical maps. These maps are not drawn with ink and paper, but with mathematical rules that describe the invisible forces between protons and neutrons. The most successful maps so far have been built on a framework called the Skyrme energy density functional. Think of this framework as a set of instructions that tells a computer how to calculate the energy of a nucleus based on how its particles are arranged. For decades, these instructions have been good enough to predict the weight of many known atoms, but they have struggled to explain the most extreme environments in the cosmos, such as the interiors of neutron stars, where matter is packed so tightly that a single teaspoon would weigh billions of tons.

A team of researchers from Belgium, Germany, and France has now updated these instructions with a more sophisticated set of rules, creating a new model called BSkG5. This model is the first large-scale attempt to apply a "next-to-next-to-leading order" approach to the problem. In simpler terms, while previous versions of the map used a basic set of rules to describe how particles interact, this new version adds layers of complexity that account for how the particles move and change shape in much finer detail. The researchers did not just tweak the numbers; they fundamentally changed the structure of the equations to include terms that describe interactions involving up to four different layers of movement. This allowed them to build a model that is just as accurate at predicting the weights of known atoms as the best previous versions, but with a crucial difference: it also correctly describes the behavior of matter inside neutron stars.

The challenge the team faced was that the old rules worked well for atoms on Earth but failed when applied to the dense matter of a neutron star. Previous attempts to fix this involved adding extra, somewhat awkward terms to the equations that made the math difficult to handle and sometimes led to unstable results. The new BSkG5 model solves this by using the more complex, higher-order rules to achieve the same stability and accuracy without needing those awkward extra terms. In fact, the new model is more efficient, using two fewer adjustable numbers than its immediate predecessor while still matching experimental data with high precision. When tested against the known masses of 2,457 different atomic nuclei, the model's predictions were off by a root-mean-square deviation of 0.649 MeV, a level of accuracy that rivals the best existing tools. It also correctly predicted the sizes of atomic nuclei and the energy required to split heavy atoms apart, which is vital for understanding nuclear fission.

Perhaps the most significant achievement of this work is how it bridges the gap between the tiny world of the atomic nucleus and the massive world of the neutron star. The researchers found that their new model produces a "stiff" equation of state for pure neutron matter. In physics, this means that as you squeeze neutron matter together, it resists compression with a force strong enough to support a neutron star with a mass twice that of our Sun. This is a critical requirement because astronomers have observed neutron stars with exactly this mass, and older models often predicted that such heavy stars would collapse under their own gravity. The new model achieves this without breaking the laws of physics or requiring impossible speeds for sound waves traveling through the star. It successfully reconciles the need for a heavy, stable star with the need for an accurate description of the lightest atoms, a feat that previous models could not accomplish simultaneously.

To ensure the model was robust, the team tested it in extreme scenarios beyond just calculating the weight of an atom. They simulated how a heavy nucleus like lead-208 would vibrate when hit by high-energy light, and they calculated how a rapidly spinning, distorted nucleus would behave. In both cases, the model remained stable and produced results that were physically reasonable, showing that the complex new rules do not cause the calculations to fall apart. They also used the model to simulate the creation of heavy elements during the collision of two neutron stars, a process known as the r-process. The results showed that the new model predicts the abundance of elements in the universe in a way that is very similar to the older models, suggesting that the new rules do not disrupt our understanding of how the cosmos is filled with heavy metals like gold and uranium.

The work represents a practical demonstration that these highly complex mathematical forms can be used for real-world calculations involving thousands of different atoms and extreme astrophysical conditions. While the model is not perfect—it still has some difficulty with the precise angular momentum of certain odd-numbered nuclei—it proves that the path forward for nuclear theory lies in these more detailed, higher-order descriptions. The researchers note that their model naturally leads to a description of how particles move that changes with temperature and density, a feature that was previously only found in more complicated types of models. This opens the door for future studies on how matter behaves in the hot, dense environments of exploding stars and colliding neutron stars. By refining the map of the atomic nucleus, the team has provided a tool that is both precise enough for laboratory physics and robust enough to describe the most extreme objects in the universe.