How do identical electrical charges arrange themselves when confined to the surface of a sphere? Although the question appears straightforward, finding the lowest-energy arrangement becomes increasingly difficult as the number of charges increases. Researchers at the University of Innsbruck have now brought this longstanding mathematical challenge into the laboratory using superfluid helium droplets and gold nanoparticles.
Known as the Thomson problem, the question originated from the atomic model proposed by Nobel laureate J. J. Thomson in 1904. His now-outdated “plum-pudding” model described an atom as a positively charged sphere containing negatively charged electrons. Although the atomic theory has since been superseded, the mathematical problem of determining how repelling charges distribute themselves across a spherical surface remains relevant.
In 1998, mathematician Stephen Smale included the problem as Problem No. 7 in his list of 18 major mathematical challenges for the 21st century. Exact solutions remain available for only a limited number of small systems. Beyond mathematics, the problem has implications for understanding the structures of viruses, fullerenes and colloidal shells.
Gold Nanoparticles Reveal Electrical Charge Arrangements
Paul Scheier’s research team at the Department of Ion Physics and Applied Physics at the University of Innsbruck developed an experimental method to examine these arrangements.
The researchers generate nanodroplets of superfluid helium at temperatures just a few tenths of a degree above absolute zero. They then introduce multiple electrical charges using an electron beam. Because the charges repel one another, they move towards the droplet’s surface and organise themselves into configurations with minimal disturbance from their surroundings.
The charged droplets subsequently pass through gold vapour. Gold atoms attach to the locations of the electrical charges and develop into singly charged nanoparticles containing approximately 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.
The droplets are deposited onto a transmission electron microscopy grid, where the helium evaporates. The gold nanoparticles remain behind, preserving a visible 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.
Experimental Results Closely Match Theoretical Predictions
The Innsbruck team examined 25 imprints containing between five and 36 charges per droplet. The resulting patterns closely corresponded to the lowest-energy configurations predicted by theoretical calculations.
The researchers also investigated larger droplets with radii reaching approximately one micrometre. In these systems, the gold nanoparticles formed hexagonal lattices interrupted by isolated fivefold defects.
These defects emerge because a flat hexagonal arrangement cannot cover a curved surface perfectly without structural adjustments. A familiar example is a football, whose spherical surface incorporates pentagonal panels among hexagons.
“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.
Collaborator Shengfu Yang from the University of Leicester highlighted how the findings could change the relationship between mathematical theory and experimental observation.
“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 Yang.
The team intends to improve the experimental technique by slowing the helium droplets, which could help prevent distortions in the resulting imprints, particularly when larger numbers of charges are involved. Rotating helium droplets can also develop non-spherical shapes. This could allow researchers to investigate how electrical charges organise themselves across more complex curved surfaces.
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