= 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 $d=3\ \text{\AA}$ gives $n\sim d^{-3}=3.7\times10^{28}\ \mathrm{m^{-3}}$. Hence
$$
\boxed{\omega_p=\sqrt{\frac{ne^2}{\varepsilon_0m_e}}\sim1.1\times10^{16}\ \mathrm{s^{-1}}.}
$$
The ordinary frequency is about $1.7\times10^{15}\ \mathrm{Hz}$, with <energy> $\hbar\omega_p\sim7\ \mathrm{eV}$. 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.
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