Physicists image 120-year-old Thomson charge puzzle in helium droplets

For more than a century, mathematicians have wrestled with a deceptively simple question posed by Joseph J. Thomson in 1904: how do identical electrical charges arrange themselves on the surface of a sphere? A team led by ion physicist Paul Scheier at the University of Innsbruck has now imaged such charge arrangements directly for the first time, using tiny droplets of superfluid helium as spherical laboratories. The work, published in Physical Review Letters, turns a famously abstract optimization problem into something that can be photographed.

The Thomson problem arises because identical charges repel one another, yet when they are confined to the surface of a sphere they must seek the arrangement with the lowest possible energy. The idea is simple, but the problem becomes dramatically harder with every charge added. It originated in Thomson’s “plum-pudding” model of the atom, an atomic picture long since superseded, but the underlying mathematical challenge has endured.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, and the question matters well beyond mathematics, with relevance to the structure of viruses, fullerenes and colloidal shells.

The Innsbruck experiment brought this problem into the laboratory. 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 migrate to the droplet surface and arrange themselves there almost undisturbed. The droplets then pass through gold vapor, where 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, and 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.

The team analyzed 25 such imprints containing between 5 and 36 charges per droplet.

Larger droplets, with radii of up to about one micrometer, revealed a different picture. On these bigger surfaces, the gold particles formed a hexagonal lattice interrupted by isolated fivefold defects. Such defects are geometrically necessary because a flat hexagonal pattern cannot cover a curved surface without gaps, much like the pentagons on a soccer ball. The imaging therefore captured not only the small, well-defined configurations of the classic Thomson problem but also how charge ordering evolves toward much larger systems on curved surfaces.

“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, extending the technique beyond the idealized sphere at the heart of Thomson’s original question.

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. The study, titled “Real-Space Imaging of Thomson Charge Configurations in Superfluid Helium Nanodroplets,” appeared in Physical Review Letters in October 2026.

Subject of Research: Experimental imaging of Thomson charge configurations on superfluid helium nanodroplets

Article Title: A 120-year-old puzzle made visible in a helium droplet

Article References: A 120-year-old puzzle made visible in a helium droplet. Original publication (Bibliographic details unavailable.)

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Keywords: Thomson problem, superfluid helium nanodroplets, charge arrangements, gold nanoparticles, University of Innsbruck, Paul Scheier, Stephen Smale, curved surfaces, defects, Physical Review Letters, mathematical physics, electron microscopy