Electrostatic effect and superconductivity in tiny samples

It is our notion that the Anderson phase-number relationships (which he developed for the Josephson effect) should be applied to a superconductor in equilibrium. This adds to the BCS postulates a further one: there exists an intrinsic phase spread across a suitable domain in a sample and a corresponding uncertainty in the number of electrons in that domain. The new Hamiltonian contains a Coulomb repulsion term, and the new wave function contains unpaired occupied states even in the ground state. Minimization of the energy shows that the size of the domain within which the phase spread exists is the size of the sample. Detailed numerical solution finds that only for beads of $\mathit{100}\text{−}\mathit{500}\mathit{\AA }$ will the results differ significantly from the BCS or bulk case. Excited states and thermal properties are solved for in detail and other properties are discussed. Predictions are a smaller gap and a lower critical temperature than in bulk, a first order transition to the normal state, and an effective density of states for single particle transitions at very low temperatures which exceeds that in the normal state. It is well known that the samples in Knight shift experiments are of the size necessary to test these predictions; however, the effects predicted here are not found. We argue that those samples are not in electrical isolation, in violation of a condition critical for the detection of the electrostatic effect.