 Solution

An unbound conduction Electron has no local restoring force, so set ${\omega }_0=0$ while retaining inertia and momentum relaxation time $\tau =1/\gamma$. This is the Drude model:Separating the response givesFor a good metal, $\gamma \ll {\omega }_p$. The real part is strongly negative at low frequency, crosses zero at $\sqrt{{\omega }_p^2-{\gamma }^2}\simeq {\omega }_p$, and approaches one from below. The imaginary part diverges as ${\omega }_p^2/(\gamma \omega )$ at low frequency, reflecting the finite dc electrical conductivity ${\sigma }_0={ϵ}_0{\omega }_p^2/\gamma$, then decays as ${\omega }^{-3}$. The low-frequency imaginary response describes conduction loss, not a finite-frequency bound-electron absorption line. Core-electron and interband responses can add a background in a real metal.