A Study of the Cross--Over Temperature between the Adiabatic and Non--Adiabatic Contributions to the Electron--Phonon Free Energy in Na, K, Al, and Pb

We calculate the electron--phonon contribution to the free energy and entropy for four elemental metals, Na, K, Al, and Pb, using realistic phonon spectra and pseudopotentials for temperatures between $0 \leq T < 1.5 \,\, T_{melt}$. We show that the non--adiabatic contribution dominates at low temperatures whereas the adiabatic contribution dominates at high temperatures. We calculate the cross--over temperatures between the two contributions which is roughly between 0.5 and 0.8 $T_{melt}$. Where we are able to compare, we find good agreement with experiment.