Past exam of the quantum condensed matter course of the University of Cambridge / 2022 / Section B / 4 / b / Solution

Substitute harmonic amplitudes with the specified $e^{-i\omega t}$ convention:One electron contributes a dipole moment $\mathbf{p}=q{\mathbf{u}}_0$, so the electric polarization of a density $n$ isUsing $\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P}={ϵ}_0{ϵ}_r\mathbf{E}$ givesThus the paper's dimensionless ${ϵ}_{\omega }$ is the relative permittivity, rather than the absolute permittivity ${ϵ}_0{ϵ}_r$. The electron's negative charge disappears from the electric susceptibility because both the force and dipole moment contain $q$.