Electromagnetic instability of the Thomson Problem

– The classical Thomson problem of n charged particles confined to the surface of a sphere of radius a is analyzed within the Darwin approximation of electrodynamics. For n < nc(a) the ground state corresponds to a hexagonal Wigner crystal with a number of topological defects. However, if n > nc(a) the Wigner lattice is unstable with respect to small perturbations and the ground state becomes spontaneously magnetized for finite n. The Thomson problem, finding the ground state of electrons inside a sphere with a uniform neutralizing background, has a time honored position in the history of modern physics [1–7]. The original question was posed by J.J. Thomson [8] after his discovery of the electron in 1897. Thomson conjectured that the knowledge of the positions of the electrons inside the atoms is essential to understanding the regularity of the chemical elements in the periodic table. At the time, however, proton still had to wait 14 years to be discovered, so in order to keep his atom neutral, Thomson was forced to introduce a uniform neutralizing background. The model became known as the “plum pudding” atom and the question that needed to be answered was: What are the positions of the electrons inside a uniformly (positively) charged sphere? Surprisingly, after more than a century this problem still has no general solution. If the background charge is made to vanish, the electrostatic energy will be a minimum only if all the electrons are located at the surface. This is a general consequence of the Earnshaw theorem [9] which precludes existence of a stable equilibrium with purely electrostatic interactions. Curiously, the Coulomb potential is precisely on the border line where this behavior is possible. If instead of 1/r, the electrons would interact by a 1/r potential with ǫ > 0, the bulk occupation of the sphere would be energetically favorable for a sufficiently large number of electrons [10]. Unfortunately, even the restricted surface Thomson problems remains unsolved for an arbitrary number of electrons [11, 12]. (∗) E-mail: [email protected] (∗∗) E-mail: [email protected] (∗∗∗) E-mail: [email protected]