Circulating electrons, superconductivity, and the Darwin-Breit interaction

The importance of the Darwin-Breit interaction between electrons in solids at low temperatures is investigated. The model problem of particles on a circle is used and applied to mesoscopic metal rings in their normal state. The London moment formula for a rotating superconducting sphere is used to calculate the number, $N$, of superconducting electrons on the sphere. This number is found to be three times the radius, $R$, of the sphere divided by the classical electron radius, i.e.\ $N=3R/r_{\rm e}$. The Darwin-Breit interaction gives a natural explanation for this relation from first principles. It also is capable of electron pairing. Collective effects of this interaction require a minimum of two dimensions but electron pairing is enhanced in one-dimensional systems.