Coulomb repulsion and Tc in BCS theory of superconductivity

Coulomb repulsion among the many electrons in a metal is in a balance, which can be toppled by even a weak electron-phonon attractive interaction. Therefore neglecting the Coulomb term from the BCS reduced Hamiltonian has little effect on Tc. This is shown by a field-theoretic argument, an analysis based on the Bogoliubov model potential and a direct numerical calculation. Detailed knowledge about electrons and phonons for various materials can be incorporated into the BCS theory through a refined treatment of the self-consistent gap equation. Consequently the universal ratio 3.5 in the BCS theory is replaced by a range of values varying from 3.51 for Ga to 4.76 for Hg. It is found that the phonon cutoff frequency is much lower than the Debye frequency. Extraordinarily high Tc could be expected if all phonons were involved in pairing electrons in a BCS superconductor.