Thermodynamics of Dirty Superconductors

We use a method due to Tsekhmistrenko to eliminate from the Fr\"ohlich Hamiltonian the electron-phonon interaction term. We then evaluate the thermodynamic properties of a superconductor described by this Hamiltonian, using a formalism developed by Zubarev and Tserkovnikov which is based on a paper by Bloch and De Dominicis. We introduce an extra term in the Hamiltonian to take the impurity scattering into account and study the effect of this extra term on the transition temperature. For the product of the mean free path and the relative change in the transition temperature we find values of 7×{}10^-6, 9×{}10^-6, and 8×{}10^-6 cm for Sn, In, and Al.