We re-examine the physics of supercritical nuclei, specially focusing on the scattering phase δϰ and its dependence on the energy ε of the diving electronic level, for which we give both exact and approximate formulas. The Coulomb potential Zα/ r is rounded to the constant Zα/ R for r< R. We confirm the resonant behavior of δϰ that we investigate in detail. In addition to solving the Dirac equation for an electron, we solve it for a positron, in the field of the same nucleus. This clarifies the interpretation of the resonances. Our results are compared with claims made in previous works.