Free-Energy Difference Between Normal and Superconducting States

The Eliashberg expression for the free-energy difference between superconducting and normal states for an electron-phonon interaction model is evaluated so as to estimate the errors involved in expressions based on the weak-coupling limit. It is shown that the major correction comes from the difference in self-energy terms ${\Sigma }_{1s}$ and ${\Sigma }_{1n}$ and is relatively of order ${\left[\right(\frac{\Delta }{{\omega }_0}\left)\ln \right(\frac{\Delta }{{\omega }_0}\left)\right]}^2$, where ${\omega }_0$ is an average phonon energy. The correction may be appreciable for strong-coupling superconductors such as lead.