THE INFLUENCE OF THE ELECTRON–ION INTERACTION ON THE PHONON FREQUENCIES OF SIMPLE METALS: Na, Al, AND Pb

The phonon frequencies of simple metals are calculated from an effective interaction between ions arising from the direct Coulomb interaction between ions and the conduction-electron response to the ion motion. The response of the electrons is treated by the many-body perturbation theory. The central quantity in the theory is the scattering amplitude for Bloch electrons from the displaced ions (electron–ion matrix element). The screening effects produced by the electron–electron interaction are taken over from the free-electron theory. The electron–ion matrix element naturally splits into two parts: the potential due to the conduction electrons within the Wigner–Seitz cell and the potential which enters the one-particle Schroedinger equation for the electrons in a periodic lattice. The former is the major term for small momentum transfer, Formula: see text, and characterizes the magnitude of the screening contribution to the frequencies, while the latter dominates for large momentum transfer. This leads to a fundamental distinction between f c.c. and b.c.c. metals. To make numerical estimates, the electron wave functions are approximated by single orthogonalized plane waves. The importance of the k-dependence of the matrix element is investigated and found to be small for Na and Al, but substantial for Pb. The calculated phonon frequencies agree reasonably well with the experimental results. In addition the magnitude of the observed Kohn anomalies in Pb is explained.