Energetic Stability of Persistent Currents in a Long Hollow Cylinder

Abstract

This paper represents an extension of the work of Schafroth in which the carriers of electricity in a superconductor are assumed to constitute an ideal Bose gas. The long-range electromagnetic interactions are fully taken into account by the method of the self-consistent field. The corresponding general equations are obtained from a variational principle applied to the total energy of the system, and the change of the total energy due to individual particle transitions is considered. Axially symmetric solutions are found for the geometry of a long hollow cylinder and for the case in which all particles are in the same lowest single-particle level compatible with the angular momentum ${l}^{\ensuremath{'}}\ensuremath{\hbar}$ around the cylinder axis. This case represents a state of the system in which ${l}^{\ensuremath{'}}$ flux quanta are trapped in the cylinder. Assuming the lattice to be at $T=0$, such a state will be stable against single-particle transitions if the corresponding energy change of the system is positive. This state need not be the state of lowest energy, but it would have a practically infinite lifetime if the stability criterion against single-particle transitions is satisfied. Indeed, its energy could only be lowered by a simultaneous transition of a number of particles comparable to their total number, which would be an extremely rare event. It is shown that stability depends on the fields ${H}_{1}$ and ${H}_{2}$ in the cylinder hole and outside the cylinder, respectively. For ${H}_{1}={H}_{2}$, stability requires that these fields be less than ${H}^{*}=4\ensuremath{\pi}n{\ensuremath{\mu}}_{0}$, where $n$ is the density of the bosons and ${\ensuremath{\mu}}_{0}$ is the Bohr magneton. For ${H}_{2}=0$, it requires that ${H}_{1}<{({H}_{c})}_{T}$ where ${({H}_{c})}_{T}$ depends upon the hole radius ${r}_{1}$ and the wall thickness $d$. For $d\ensuremath{\gg}{r}_{1}$, one finds ${({H}_{c})}_{T}={H}^{*}$, while for $d\ensuremath{\ll}{r}_{1}$, one has ${({H}_{c})}_{T}=(\frac{d}{{r}_{1}}){H}^{*}$. The difference between ${({H}_{c})}_{T}$ and ${H}^{*}$ for $d\ensuremath{\ll}{r}_{1}$ is further investigated, and it is shown that the case for which ${H}^{*}>{H}_{1}>{({H}_{c})}_{T}$ may nevertheless experimentally appear to be stable since the stability conditions are strongly modified considering that the transfer of very small but finite amounts of energy to the lattice may not occur during the time of a measurement.