A special set of numbers has formed the backbone of nuclear physics research for decades, and now we finally know how it arises from the quantum mix of nuclear particles and forces.

Nearly 80 years ago, physicist Maria Goeppert Mayer showed that when the nucleus of an atom contains certain numbers of protons and neutrons, such as 50 or 82, it becomes exceptionally stable. In the years since, researchers amassed evidence of more such “magic numbers”, which are found in the most stable, and therefore most abundant, elements in our universe.

Goeppert Mayer and her contemporaries explained these numbers by proposing that protons and neutrons occupy discrete energy levels, or shells. This model, which is still used to interpret many nuclear physics experiments, treats each particle in the nucleus as independent, but our best quantum theories assert that particles within nuclei actually interact strongly.

Jiangming Yao at Sun Yat-sen University in China and his colleagues have now resolved this contradiction and, in the process, elucidated how magic numbers emerge from these interactions.

Yao says the shell model relies on input from experiments and doesn’t encode details of interactions between each particle. Instead, he and his team started their calculations from first principles, which means they mathematically described how particles interact with each other, how they stick together and how much energy is needed to move them apart in more detail.

The two descriptions are analogous to images taken at low and high resolution, respectively, says Yao. “Before, people directly modelled the system at low resolution, or they tried to understand nuclear structure at high resolution. We used modern methods to build the bridge between these descriptions.”

He and his colleagues started with the high-resolution description, gradually made it blurrier in each step of the calculation and tracked how the structure the particles formed changed.

As they moved across their mathematical bridge, the researchers saw the symmetry of the particle’s quantum states change – drawing a graph based on the equations for these states would produce shapes with different symmetries at different resolutions. This change resulted in a nuclear structure that was at its most stable when particles grouped themselves in magic numbers.

Jean-Paul Ebran at the French Alternative Energies and Atomic Energy Commission says this work offers a theoretical probe – a mathematical microscope of sorts – that mirrors how experiments work. “Nature looks different depending on the resolution at which you observe. This [study] really maps onto what we do experimentally.”

The change in symmetry that the researchers identified is related to effects described by Albert Einstein’s theory of special relativity, thus painting an even fuller picture of how magic numbers marry together different facets of nuclear theory, says Ebran.

So far, the researchers have tested their theoretical work on a type of tin that is doubly magic because its nuclei each contain 50 protons and 82 neutrons, as well as on several additional nuclei. Going forward, they want to extend their analysis to heavier atomic nuclei, which are typically unstable, and study processes by which heavy nuclei are created in exploding stars or merging neutron stars, says Yao.