First Direct Images of Charge Patterns on Spheres Solve Thomson Puzzle

Physicists in Austria and the UK have obtained the first direct images of how identical electrical charges arrange themselves on a sphere, 122 years after J J Thomson first posed the question. Up to 36 charges, the experimental imprints match computed solutions of the Thomson problem exactly, according to Paul Scheier of the University of Innsbruck, who led the study with Shengfu Yang of the University of Leicester. The work appears in Physical Review Letters and also involved researchers at Durham University and the University of Cambridge.

"Electrical repulsion pushes [the charges] apart as far as possible and the arrangement corresponds to a minimum-energy configuration," Scheier said. The problem dates to Thomson's long-abandoned atomic model, a ball of positive charge studded with electrons. Yang noted that the model is dead but the physics is not: the same arrangement question governs curved crystals, nanoscale assemblies, the protein shells of spherical viruses and cage-like fullerenes.

Why has no one measured this before?

The Thomson problem has no general solution. Rigorous answers exist only for small cases, as Yang explained: the tetrahedron for four charges, the octahedron for six, the icosahedron for twelve. Computational searches have identified likely best arrangements for up to a few thousand charges, but proof stalls beyond that. "As the number of charges grows, there are increasingly many competing configurations, and proving that none has a lower energy becomes extremely difficult," he said.

The puzzle had been "tackled only by computation," in the words of Nadine Halberstadt of France's CNRS, who was not part of the study. The experimental obstacle is plain: in most liquid droplets, charges spread through the volume rather than staying on the surface. Superfluid helium removes that barrier. In 2024, Halberstadt and colleagues calculated that multiple charges can sit stably just below the surface of a helium droplet.

How did the team make the invisible visible?

The method turns charge positions into physical markers.

• An electron beam strikes a stream of helium droplets, leaving each with multiple positive charges; the mobile charge centres adopt Thomson configurations.
• The droplets pass through gold vapour, where atoms migrate to the nearest charge and clump into nanoparticles of roughly 700 atoms each, carrying a single charge.
• Each droplet then crashes into a thin carbon film, leaving a 2D imprint that a transmission electron microscope can read.

"Nanoparticles that contain a few hundred gold atoms are large enough that they can be seen in an electron microscope," Scheier explained. Comparing imprints against calculated projections recovers the original spherical arrangements.

The largest droplets measured about 2 microns across and carried hundreds of charges. On a flat surface, such charges would each gain six evenly spaced neighbours, but a sphere cannot be tiled that way, just as a ball cannot be stitched from hexagons alone. Some charges must settle for five. "In larger systems, we observe hexagonal ordering interrupted by fivefold defects," Yang said, adding that the same simple repulsion yields both precise small arrangements and collective crystalline order.

What are the limits and the next steps?

Caveats deserve attention. The technique performed less well for droplets carrying large numbers of charges, yielding only qualitative agreement with theory. Larger droplets also tend to distort on impact, so future runs may slow the droplets before they strike the target. Scheier's first goal is to test simulated solutions above 1,000 charges.

The interpretation carries a deeper caution. "Experiments cannot provide a mathematical proof, but they can reveal something calculations alone cannot: which configurations a real physical system actually selects," Yang said. "The mathematically lowest-energy arrangement is not necessarily the one that will form." As charge counts rise, near-optimal arrangements multiply, and a real system can get stuck in any of them.

Halberstadt framed the result as closing the loop between prediction and observation, opening an experimental platform for charge ordering on curved surfaces. She suggests spinning helium droplets, which take non-spherical shapes, as a next target that would stimulate both experimental and theoretical work.

There is also an applied angle. The gold nanoparticles produced are nearly identical in size, and Scheier believes the method could build 2D nanoparticle lattices; he is now developing a route to particles of chosen sizes. Yang's ambition reaches further, from observing self-organization toward controlling it, by predicting which structure will actually form and tuning preparation conditions to favour a chosen outcome.