The Thomson Problem

During the course of my PhD at the University of Manchester, under the supervision of Prof. Mike Moore, I worked on the Thomson problem, which is widely regarded as one of the most important unsolved packing problem (it is number 7 on Smale’s list of problems for the 21st century). In essence the Thomson problem is concerned with finding the minimal energy ground state of a cluster of charges. In some cases the charges are confined to a surface, such as a sphere, or it may be that the charges are trapped by some sort of potential or boundary, such as a hard wall boundary (see below). In such systems although the bulk of the charges have six nearest neighbours (which is the optimal packing arrangement for spheres confined to the surface of an infinite 2D plane) the lattice also contains topological defects such as disclinations and dislocations. Disclinations are charges with five or seven nearest neighbours, while a dislocations is a tightly bound pair of five-seven disclinations. The total number of disclinations in the system depends on the Euler characteristic of the surface to which the particles are confined.

The Thomson problem is important for two reasons. Firstly because it is of central importance to the field of strongly correlated Coulomb systems and finds applications in areas such as quantum dots, dusty plasmas, and colloidal crystals. Secondly because research into the Thomson problem has yielded fundamental insights into the role played by geometry and topology in ordered systems.

Thomson problem on a sphere

The central challenge of the Thomson problem is: given N charges confined to the surface of a sphere what is the arrangement of charges which minimizes the  total electrostatic energy? This is an example of an NP hard problem and so progress in this area has only been possibly by the use of computational heuristic techniques such as simulated annealing or the method of conjugate gradients (see the following link for a fantastic applet
which can be used to solve the Thomson problem on a sphere/torus).

This image shows the (near) ground state arrangement of 1000 charges confined to the surface of a sphere and was produced using a conjugate gradient routine. It can be seen that although the charges locally form a triangular lattice, in which most lattice sites have 6 nearest neighbours, the cluster also includes at least 12 pentagonal regions or sites with 5 nearest neighbors, similar to the arrangement of hexagons and pentagons on a football. In addition the lattice may also include dislocations - which are comprised of a tightly bound pair of 5 and 7 coordinated disclinations (coloured red and green respectively). Together with disclinations, the dislocations form an intricate pattern of scars on the surface of the sphere consisting of alternating 5 and 7 coordinated points.

Thomson Problem on a disk

This image shows the (near) ground state arrangement of 5000 charges confined to the surface of a disk which is bounded by a hard wall boundary. This system has a non-uniform density. With my PhD supervisor, we showed that the system is able to have a non-uniform density and locally maintain a crystalline structure by including more 7 coordinated disclinations than 5 coordinated disclinations in the lattice interior. A consequence of the interior containing an excess of seven coordinated disclinations is that lattice acquires curvature - this curvature is evident from the remarkable arched like structure that can be seen towards the edge of the system. In addition we were  able to successfully compute the charge density of the system in the continuum limit. This allowed us to compute the density of disclinations in the lattice interior and the strength of the local curvature. In this case the ground state configuration of the charges on the disk was found using a combination of simulated annealing and conjugate gradient methods.

The above images were produced using the packages Qhull and Geomview.