Adiabatic and nonadiabatic contributions to the free energy from the electron-phonon interaction for Na, K, Al, and Pb

We calculate the adiabatic contributions to the free energy due to the electron-phonon interaction at intermediate temperatures 0 <= k(B)T <epsilon(F) for the elemental metals Na, K, Al, and Pb. Using our previously published results for the nonadiabatic contributions we show that the adiabatic contribution, which is proportional to T-2 at low temperatures and goes as T-3 at high temperatures, dominates the nonadiabatic contribution for temperatures above a crossover temperature, T-c, which is between 0.5T(m) and 0.8T(m), where T-m is the melting temperature of the metal. The nonadiabatic contribution falls as T-1 for temperatures roughly above the average phonon frequency.