Phonon dispersion and instability in linear-chain crystals

This paper deals extensively with the coupling of phonons to an electron gas with a Fermi surface consisting of two parallel planes and its effect on the stability of the lattice and on the phonon dispersion around $2p_F$. The phonon dispersion and the phonon spectral density distributions are calculated at zero temperature for the equidistant and the deformed lattice, and at temperatures above the transition temperature to the deformed lattice. Two excitation branches are found under certain conditions. The calculation is performed in the random-phase approximation, but the effect of vertex renormalization by summing all ladder diagrams is discussed. The phonon dispersion in the distorted lattice is also treated within the framework of the Sawada model.