Corrections to the Dielectric Constant of a Degenerate Electron Gas

Abstract

The Lindhard approximation to the frequency- and wave-number dependent dielectric constant, $\epsilon (\mathbf{k},\omega )$, provides a good description of many properties of the degenerate electron gas. However, it is known that the short-range behavior of the gas is not adequately represented by this function and it is necessary to include certain additional terms. DuBois incorporated some exchange terms into $\epsilon (\mathbf{k},\omega )$ and was able to obtain the correction to the plasmon excitation frequency. Though his final results are reasonable and have been corroborated using alternative approaches the "corrected" dielectric constant is found to violate certain a priori restrictions. In this paper a more accurate dielectric constant is derived. In order to obtain an acceptable function which does not violate the sum rule and positive definiteness restrictions on the imaginary part it is necessary to account for three types of corrections. These corrections originate in (1) the effective screening of the long-range interaction between particles; (2) the shift in single-particle energies of electrons and holes; and (3) the tendency of particles and holes to form bound states when any repulsive interparticle interaction is present. With these corrections all spurious singularities in the dielectric constant disappear. Numerical calculations of $\epsilon (\mathbf{k},\omega )$ and of moments of the imaginary part of this function have been carried out for an intermediate electron density equal to the density of conduction electrons in aluminum. The resulting dielectric constant departs by as much as 50% from the Lindhard form for low frequencies, but has similar qualitative features. The moments can be used to determine the high-frequency behavior and other properties of the electron gas.