The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of N electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904 after proposing an atomic model, later called the plum pudding model, based on his knowledge of the existence of negatively charged electrons within neutrally-charged atoms. Related problems include the study of the geometry of the minimum energy configuration and the study of the large N behavior of the minimum energy.

Mathematical statement

The electrostatic interaction energy occurring between each pair of electrons of equal charges (, with e the elementary charge of an electron) is given by Coulomb's law, where \epsilon_0 is the electric constant and is the distance between each pair of electrons located at points on the sphere defined by vectors and, respectively. Simplified units of e=1 and (the Coulomb constant) are used without loss of generality. Then, The total electrostatic potential energy of each N-electron configuration may then be expressed as the sum of all pair-wise interaction energies The global minimization of U(N) over all possible configurations of N distinct points is typically found by numerical minimization algorithms. Thomson's problem is related to the 7th of the eighteen unsolved mathematics problems proposed by the mathematician Steve Smale — "Distribution of points on the 2-sphere". The main difference is that in Smale's problem the function to minimise is not the electrostatic potential but a logarithmic potential given by A second difference is that Smale's question is about the asymptotic behaviour of the total potential when the number N of points goes to infinity, not for concrete values of N.

Example

The solution of the Thomson problem for two electrons is obtained when both electrons are as far apart as possible on opposite sides of the origin,, or

Known exact solutions

Mathematically exact minimum energy configurations have been rigorously identified in only a handful of cases. Geometric solutions of the Thomson problem for N = 4, 6, and 12 electrons are Platonic solids whose faces are all congruent equilateral triangles. Numerical solutions for N = 8 and 20 are not the regular convex polyhedral configurations of the remaining two Platonic solids, the cube and dodecahedron respectively.

Generalizations

One can also ask for ground states of particles interacting with arbitrary potentials. To be mathematically precise, let f be a decreasing real-valued function, and define the energy functional Traditionally, one considers also known as Riesz \alpha-kernels. For integrable Riesz kernels see the 1972 work of Landkof. For non-integrable Riesz kernels, the Poppy-seed bagel theorem holds, see the 2004 work of Hardin and Saff. Notable cases include: One may also consider configurations of N points on a sphere of higher dimension. See spherical design.

Solution algorithms

Several algorithms have been applied to this problem. The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance: While the objective is to minimize the global electrostatic potential energy of each N-electron case, several algorithmic starting cases are of interest.

Continuous spherical shell charge

The energy of a continuous spherical shell of charge distributed across its surface is given by and is, in general, greater than the energy of every Thomson problem solution. Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. For example, a spherical shell of N=1 represents the uniform distribution of a single electron's charge, -e, across the entire shell.

Randomly distributed point charges

The expected global energy of a system of electrons distributed in a purely random manner across the surface of the sphere is given by and is, in general, greater than the energy of every Thomson problem solution. Here, N is a discrete variable that counts the number of electrons in the system. As well,.

Charge-centered distribution

For every Nth solution of the Thomson problem there is an (N+1)th configuration that includes an electron at the origin of the sphere whose energy is simply the addition of N to the energy of the Nth solution. That is, Thus, if is known exactly, then U_0(N+1) is known exactly. In general, U_0(N+1) is greater than, but is remarkably closer to each (N+1)th Thomson solution than and. Therefore, the charge-centered distribution represents a smaller "energy gap" to cross to arrive at a solution of each Thomson problem than algorithms that begin with the other two charge configurations.

Relations to other scientific problems

The Thomson problem is a natural consequence of J. J. Thomson's plum pudding model in the absence of its uniform positive background charge. Though experimental evidence led to the abandonment of Thomson's plum pudding model as a complete atomic model, irregularities observed in numerical energy solutions of the Thomson problem have been found to correspond with electron shell-filling in naturally occurring atoms throughout the periodic table of elements. The Thomson problem also plays a role in the study of other physical models including multi-electron bubbles and the surface ordering of liquid metal drops confined in Paul traps. The generalized Thomson problem arises, for example, in determining arrangements of protein subunits that 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 long-range logarithmic interactions is provided by Abrikosov vortices that form at low temperatures in a superconducting metal shell with a large monopole at its center.

Configurations of smallest known energy

In the following table N is the number of points (charges) in a configuration, is the energy, the symmetry type is given in Schönflies notation (see Point groups in three dimensions), and r_i are the positions of the charges. Most symmetry types require the vector sum of the positions (and thus the electric dipole moment) to be zero. It is customary to also consider the polyhedron formed by the convex hull of the points. Thus, v_i is the number of vertices where the given number of edges meet, e is the total number of edges, f_3 is the number of triangular faces, f_4 is the number of quadrilateral faces, and \theta_1 is the smallest angle subtended by vectors associated with the nearest charge pair. Note that the edge lengths are generally not equal. Thus, except in the cases N = 2, 3, 4, 6, 12, and the geodesic polyhedra, the convex hull is only topologically equivalent to the figure listed in the last column. According to a conjecture, if P is the polyhedron formed by the convex hull of the solution configuation to the Thomson Problem for m electrons and q is the number of quadrilateral faces of P, then P has edges.