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.