High-Frequency Damping in a Degenerate Electron Gas

Abstract

A closed form has been derived for the dissipative part of the complex frequency- and wave-number-dependent dielectric constant of a degenerate electron gas, $\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$, valid in the limit $\ensuremath{\omega}\ensuremath{\gg}{E}_{0}$, $k<{k}_{0}$, where ${E}_{0}$ is the Fermi energy and ${k}_{0}$ the Fermi wave number. For $\ensuremath{\omega}>2{E}_{0}$ this expression gives values of $\mathrm{Im}\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$ which are in excellent agreement with the results of more detailed calculations in which the difficult integrals over phase space were performed by a Monte Carlo method. The formula also appears to give good numerical estimates of $\mathrm{Im}\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega})$ for smaller values of $\ensuremath{\omega} (\mathrm{but} \ensuremath{\omega}>\frac{k{k}_{0}}{m})$, though its accuracy is not assured in that region. For example, in aluminum at the plasmon frequency, the asymptotic form agrees with the calculations of DuBois and Kivelson. The high-frequency formula derived may, therefore, be used to circumvent difficult numerical work in estimating the importance of electron correlation effects at high frequencies.