The role of the chemical potential in the BCS theory

We study the effect of the chemical potential on the results of the BCS theory of superconductivity. We assume that the pairing interaction is manifested between electrons of single-particle energies in an interval [mu - h omega(c), mu + h omega(c)], where mu and omega(c) are parameters of the model-mu, need not be equal to the chemical potential of the system, denoted here by mu(R). The BCS results are recovered if mu = mu(R). If mu not equal mu(R) the physical properties change significantly: the energy gap Delta is smaller than the BCS gap, a population imbalance appears, and the superconductor-normal metal phase transition is of the first order. The quasiparticle imbalance is an equilibrium property that appears due to the asymmetry with respect to mu(R) of the single-particle energy interval in which the pairing potential is manifested.