Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section B / B1 / c / Solution

A plasmon is a quantum of a collective longitudinal oscillation of electron density against the positive background. A small charge displacement creates an electric restoring field; Newton's law then gives the plasma angular frequency. Assuming approximately one conduction electron per atom and spacing $d=3\text{A˚}$ gives $n\sim d^{-3}=3.7\times 10^{28}{\mathrm{m}}^{-3}$. HenceThe ordinary frequency is about $1.7\times 10^{15}\mathrm{Hz}$, with energy $\hslash {\omega }_p\sim 7\mathrm{eV}$. A different conduction-electron count per atom changes this estimate by its square root; dielectric and band-structure corrections also modify the simplest free-electron result.