= Scattering cross-section of a harmonically bound charge
{title2=$\sigma(\omega)=\sigma_T\omega^4/(\omega_0^2-\omega^2)^2$}
For a charge $q$ of <mass> $m$ bound by spring constant $k=m\omega_0^2$, the undamped equation $m\ddot x+kx=qE_0e^{-i\omega t}$ has <complex amplitude> $x_0=qE_0/[m(\omega_0^2-\omega^2)]$. The induced <electric dipole moment> is $p_0=qx_0$. Averaged <electric dipole radiation> <power>, divided by incident <light intensity>, gives
$$
\sigma(\omega)=\frac{q^4\omega^4}{6\pi\epsilon_0^2c^4(k-m\omega^2)^2}
=\sigma_T\frac{\omega^4}{(\omega_0^2-\omega^2)^2}.
$$
The <Rayleigh scattering> limit $\omega\ll\omega_0$ gives $\sigma\sim\sigma_T(\omega/\omega_0)^4$, proportional to $\lambda^{-4}$. For $\omega_0\ll\omega\ll mc^2/\hbar$, the charge behaves as free and the <Thomson cross-section> is recovered. At resonance the undamped steady-state formula is invalid; a finite <damping> mechanism or <radiation reaction> must be included. The formula describes independent weakly driven scatterers, not arbitrary coherent scattering from an extended medium.
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