The ratio2Δ0kBTcwithin Eliashberg theory

We have calculated the ratio of the energy gap ${\Delta }_0$ to the critical temperature $T_c$ for a large number of known electron-phonon spectral densities with the use of the imaginary-axis formulation of Eliashberg theory with an analytic-continuation technique to the real axis. Agreement with experiment is within a few percent. It is found that the general trend obtained for $\frac{2{\Delta }_0}{k_B{T}_c}$ against $\frac{T_c}{{\omega }_{\ln }}$, where ${\omega }_{\ln }$ is the popular Allen-Dynes expression for the average phonon energy, can be reproduced by a two-parameter functional form of the general type first given by Geilikman and Kresin. The details, however, can only be understood with complete calculations.