When identical charges repel each other but are confined to the surface of a sphere, they seek the arrangement with the lowest possible energy. The idea is simple, but the problem gets harder to solve with every charge added. The so-called Thomson problem dates back to the "plum-pudding" model of the atom, which Nobel laureate Joseph J. Thomson proposed in 1904. The atomic model has long been superseded, but the mathematical problem has endured. In 1998, mathematician Stephen Smale listed it as Problem No. 7 among his 18 major mathematical challenges for the 21st century. Exact solutions still exist only for a few small systems. The question also matters well beyond mathematics, for example for the structure of viruses, fullerenes and colloidal shells.
Gold particles as markers
At the Department of Ion Physics and Applied Physics at the University of Innsbruck, Paul Scheier's team has now brought this abstract optimization problem into the lab. The researchers produce nanodroplets of superfluid helium, just a few tenths of a degree above absolute zero, and charge them multiple times with an electron beam. The charges move to the droplet surface and arrange themselves there almost undisturbed. The droplets then pass through gold vapor. Gold atoms attach to the charge centers and grow into singly charged nanoparticles of about 700 atoms each. "Superfluid helium nanodroplets give us an exceptionally clean and cold environment in which up to several thousand charges can organize on a curved surface with very little external disturbance," says Paul Scheier. When the droplets land on a transmission electron microscopy grid, the helium evaporates. The gold particles remain as an imprint of the original charge arrangement. "By tagging the charge centers with gold nanoparticles, we can preserve and image these arrangements after the helium has evaporated," Scheier explains.
Agreement with theory
The Innsbruck team analyzed 25 imprints with 5 to 36 charges per droplet. The observed patterns closely match the lowest-energy configurations predicted by theoretical calculations. Larger droplets, with radii of up to about one micrometer, show a different picture: the gold particles form a hexagonal lattice interrupted by isolated fivefold defects. These defects are necessary because a flat hexagonal pattern cannot cover a curved surface without gaps, much like the pentagons on a soccer ball. "This opens a new experimental route to studying how charge ordering evolves from small, well-defined configurations to much larger systems on curved surfaces," says Scheier. "What excites me most is that this reverses the usual relationship between theory and experiment. Instead of mathematics simply predicting what an experiment should see, we can now let nature generate solutions and ask mathematics to explain what we observe," adds collaborator Shengfu Yang from the University of Leicester.
The researchers now plan to refine the method further. Slower droplets should make the imprints distortion-free even at large charge numbers. Because rotating helium droplets can take on non-spherical shapes, the approach could also give access to charge ordering on more complex curved surfaces.
Besides the Innsbruck team around first author Thomas Pohl, researchers from the University of Leicester, Durham University and the University of Cambridge contributed to the work. The research was funded in part by the Austrian Science Fund FWF and the State of Tyrol.
Publication: Real-Space Imaging of Thomson Charge Configurations in Superfluid Helium Nanodroplets. Thomas Pohl, Jan Mayerhofer, Hanqing Liu, Arin Mizouri, Hao Sha, Masoomeh Mahmoodi Darian, Fabio Zappa, Andrew M. Ellis, David J. Wales, Shengfu Yang, and Paul Scheier. Phys Rev Lett 2026. DOI: 10.1103/qby4-bm6q
