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

Abstract

We calculate the adiabatic contributions to the free energy due to the electron-phonon interaction at intermediate temperatures $0\ensuremath{\leqslant}{k}_{B}T<{ϵ}_{F}$ 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 $0.5{T}_{m}$ and $0.8{T}_{m}$, where ${T}_{m}$ is the melting temperature of the metal. The nonadiabatic contribution falls as ${T}^{\ensuremath{-}1}$ for temperatures roughly above the average phonon frequency.