Thomson problem

The Thomson problem is that of determining the minimum (ground state) energy configuration of N classical electrons on the surface of a sphere (or 2-sphere $S^\{2\}$). The electrons repel each other with a force given by Coulomb's law. This problem is named for J. J. Thomson, who posed it in 1904 as part of the development of the plum pudding model of the atom. In this model, the electrons formed spherical shells. This one hundred year old puzzle (Thomson wrote "I have not as yet succeeded in getting a general solution") has many important physical realizations including multi-electron bubbles and the surface ordering of liquid metal drops confined in Paul traps. The Thomson problem is solved exactly for a few particles on the sphere and the energy per particle is known for a large number of particles on the surface of the sphere. Numerical results provide the best known solutions for a wide range of particle numbers. The configurations that are found have a great variety of geometrical structure.

One can also ask for ground states of particles interacting with arbitrary potentials: this is the generalized Thomson problem.The generalized Thomson problem arises, for example, in determining the arrangements of the protein subunits which comprise the shells of spherical viruses. The "particles" in this application are clusters of protein subunits arranged on a shell. Other realizations include regular arrangements of colloid particles in "colloidosomes", proposed for encapsulation of active ingredients such as drugs, nutrients or living cells, fullerene patterns of carbon atoms, and VSEPR Theory. An example with longrange logarithmic interactions is provided by the Abrikosov vortices which would form at low temperatures ina superconducting metal shell with a large monopole at the center.

The Thomson problem is of outstanding mathematical interest not only for the asymptotics (large N behavior) of the growth of minimum energy configurations but also for the characteristics of the minimum energy configurations themselves.

Best known solutions

In the following table $N$ is the number of charges on a sphere of radius 1. $E\_1$ is the energy (any pair of charges with distance r have potential energy 1/r). The symmetry type is given in Schönflies notation (see Point groups in three dimensions). $r\_i$ are the positions of the charges. Most symmetry types require the sum (and thus the center of mass) to be at the origin.

It is customary to also consider the polyhedron formed by the convex hull of the charges. Thus $v\_i$ is the number of vertices where the given number of edges meet. $e$ is the total number of edges and $f\_3$ and $f\_4$ are the number of triangle and quadrilateral faces. $heta\_1$ is the smallest angle between any two charges.

References

*J. J. Thomson, "On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure", Philosophical Magazine Series 6, Volume 7, Number 39, pp. 237--265, March 1904.

*T. Erber and G. M. Hockney, "Complex Systems: Equilibrium Configurations of $N$ Equal Charges on a Sphere $(2leq\; Nleq\; 112)$", Advances in Chemical Physics, Volume 98, pp. 495--594, 1997.

*Mark J. Bowick, Cris Cecka and Alan A. Middleton: http://physics.syr.edu/condensedmatter/thomson/thomsonapplet.htm

*David J. Wales and Sidika Ulker: http://www-wales.ch.cam.ac.uk/~wales/CCD/Thomson/table.html