High-Frequency Damping in a Degenerate Electron Gas

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, $\epsilon (\overrightarrow{\mathrm{k}},\omega )$, valid in the limit $\omega \gg E_0$, $k<\mathrm{}{k}_0$, where $E_0$ is the Fermi energy and $k_0$ the Fermi wave number. For $\omega >2\mathrm{}{E}_0$ this expression gives values of $\mathrm{Im}\epsilon (\overrightarrow{\mathrm{k}},\omega )$ 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 $\mathrm{Im}\epsilon (\overrightarrow{\mathrm{k}},\omega )$ for smaller values of $\omega (\mathrm{but} \omega >\frac{k{k}_0}{m})$, 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.