Transition temperature of strong-coupled superconductors reanalyzed

A through analysis is made of the dependence of the superconducting transition temperature T₂ on material properties (, ^*, phonon spectrum) as contained in Eliashberg theory. The most striking new feature of the analysis is in the asymptotic regime of very large where T₂ is found to equal 0. 15 (〈{^2〉) }^1{2} (assuming ^*=0. 1). This result implies the surprising conclusion that within Eliashberg theory T₂ is not limited by the phonon frequencies, and also shows that McMillan's "=2 limit" is spurious. The McMillan equation (with a prefactor altered from {₃}1. 45 to {₋₎₆}1. 2) is found to be highly accurate for all known materials with <1. 5 but in error for large values of. Correction factors to McMillan's equation are found in terms of, ^*, and one additional parameter, (〈{{^2〉) }^1{2}}{₋₎₆}. The frequency ₋₎₆ is defined as exp 〈ln〉 where the averages 〈ln〉 and 〈^2〉 are defined using (2) ^2F () as a weight factor. These conclusions are based on a combination of analytic and numerical solutions of the Eliashberg equations, and are supported by a comparison with tunneling data. Especially strong support comes from a new experimental result for amorphous Pb₀. ₄₅Bi₀. ₅₅ reported herein. This material has parameters =2. 59 and T₂{₋₎₆}=0. 284, in serious disagreement with McMillan's formula but in good agreement when the correction factors are included. The McMillan-Hopfield parameter or N (0) 〈I^2〉 is extracted from tunneling measurements or from a combination of empirical values of and neutron-scattering measurements of phonon dispersion. It is proposed that (which is now known not to be accurately constant) is the most significant single parameter in understanding the origin of high T₂ and the limitation of T₂ by colvalent instabilities.