Recently Friedberg, Lee and Ren have pointed out that at low density the ideal charged boson system turns out not to be a superconductor, but becomes a type II superconductor at high density. This conclusion differs from the well-known Schafroth solution of superconductivity at any density for the same problem. Schafroth's analysis is found to contain a mistake due to the neglect of the electrostatic exchange energy E ex . Based on the Schafroth solution, E ex is shown to be +∞ in the normal phase, but 0 in the condensed phase (at T = 0). Of course, the correct solution has to give a finite E ex . This research was supported in part by the U.S. Department of Energy. SCHAFROTH'S SOLUTION Schafroth's superconductivity solution of an ideal charged boson system published 35 years ago has always been considered to be the definitive work, comparable in depth to the analysis made by Landau on the diamagnetism of an ideal charged fermion system. However, recently it was found that the Schafroth solution contains a serious mistake due to the neglect of the electrostatic exchange energy E ex . It turns out that based on the Schafroth solution, E ex is +∞ in the normal phase, but 0 in the condensed phase (at T = 0). Of course, the correct solution has to give a finite E ex . For ideal charged particles, bosons or fermions, there is only the electromagnetic interaction.
It is shown that an ideal gas of charged bosons exhibits the essential equilibrium features of a superconductor. The onset of Bose-Einstein condensation marks the transition temperature
Tc. Below
Tc a Meissner-Ochsenfeld effect is exhibited which is described in a very good approximation by London's equation.The singular nature of the condensed ideal Bose gas exhibits itself in a space dependence of the London constant $\ensuremath{\lambda}$, determined by the boundary conditions on the wave function. It is shown that the electrostatic repulsion between the bosons compensates this effect and leads to a spatially constant $\ensuremath{\lambda}$, independently of the boundary conditions.The critical field
Hc(T) is determined and found to be related to the penetration depth
d(T) by ${H}_{c}=\frac{\ensuremath{\hbar}c}{2e{d}^{2}}$ (
e being the boson charge).The
B(H) law is different from the one usually assumed for actual superconductors. Corresponding changes occur in the thermodynamical relation.A comparison with superconducting metals is made. The main conclusion is that if superconductivity in metals is due to the concurrence of bosons, then the number of these bosons must be strongly temperaturedependent below
Tc.