Adiabatic and Non-Adiabatic Contributions to the Free Energy from the Electron-Phonon Interaction for Na, K, Al, and Pb

TL;DR

This paper calculates the adiabatic electron-phonon free energy contributions in Na, K, Al, and Pb, showing they dominate over nonadiabatic parts above a certain temperature, with specific temperature dependencies.

Contribution

It provides a detailed analysis of adiabatic contributions and compares them with nonadiabatic effects across different elemental metals at intermediate temperatures.

Findings

Adiabatic contribution dominates above T_c between 0.5 and 0.8 T_m.

Adiabatic free energy scales as T^2 at low T and T^3 at high T.

Nonadiabatic contribution decreases as T^{-1} above average phonon frequency.

Abstract

We calculate the adiabatic contributions to the free energy due to the electron--phonon interaction at intermediate temperatures, $0 \leqslant k_{B} T < \epsilon_{F}$ 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 cross--over temperature, $T_{c}$, which is between 0.5 and 0.8 $T_{m}$, 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.