= Solution
Both <Thomson scattering> and <Rayleigh scattering> can be understood as radiation from a dipole driven by the incident <electric field>. Their different <frequency> dependences come from the response of a free charge compared with a <bound charge>.
For a free particle of charge $q$ and <mass> $m$, a harmonic field gives $m\ddot{\mathbf x}=q\mathbf E$, hence $\mathbf p=q\mathbf x=-q^2\mathbf E/(m\omega^2)$. The radiated field contains two time derivatives of $\mathbf p$, cancelling the response's $\omega^{-2}$. The Thomson cross section is consequently independent of <frequency> in the classical low-recoil regime:
$$
r_q=\frac{q^2}{4\pi\epsilon_0mc^2},\qquad
\boxed{\sigma_T=\frac{8\pi}{3}r_q^2}.
$$
For incident <linear polarization> with unit polarization $\boldsymbol\epsilon$, summing over outgoing polarizations gives $d\sigma/d\Omega=r_q^2[1-|\widehat{\mathbf n}\cdot\boldsymbol\epsilon|^2]$. Radiation vanishes along the acceleration and is strongest perpendicular to it. Averaging an incident <unpolarized light> beam gives
$$
\frac{d\sigma_T}{d\Omega}=\frac{r_q^2}{2}(1+\cos^2\Theta),
$$
where $\Theta$ is the <scattering angle>. Scattering through $90^\circ$ is fully linearly polarized perpendicular to the <scattering plane> for this ideal isotropic dipole model. The <electron> case dominates over equally charged heavier particles because the cross section scales as $m^{-2}$. <recoil> corrections become necessary when <photon energy> is no longer small compared with $mc^2$.
For a <bound charge> with <natural frequency> $\omega_0$ and damping rate $\gamma_d$,
$$
\mathbf p=\alpha_{\rm e}(\omega)\mathbf E,\qquad
\alpha_{\rm e}=\frac{q^2/m}{\omega_0^2-\omega^2-i\gamma_d\omega}.
$$
Far below <resonance>, $\alpha_{\rm e}$ is approximately constant, so
$$
\boxed{\sigma_R=\frac{k^4|\alpha_{\rm e}|^2}{6\pi\epsilon_0^2}
\ \propto\omega^4\propto\lambda^{-4}}.
$$
This is <Rayleigh scattering>. It requires a scatterer small compared with the <wavelength> and, for the simple expression, an isotropic linear dipole response. Atoms, molecules and small <dielectric> particles provide examples. The stronger scattering of shorter visible <wavelengths> explains blue skylight; <absorption> and the nonuniform solar spectrum prevent treating that simple scaling as a complete color calculation.<Rayleigh Scattering by Richard Fitzpatrick>[section 9.6, bound-charge response and low-frequency limit]{ref}
For isotropic electric scatterers the normalized angular and polarization patterns are the same as in <Thomson scattering>. The <frequency> scaling distinguishes their limits: a free charge has a displacement proportional to $\omega^{-2}$, whereas a tightly <bound charge> has an approximately frequency-independent displacement. Near <resonance>, far outside the small-particle limit, or with anisotropic <polarizability>, the simple Rayleigh formula must be modified. In particular, the electric-only pattern must not silently be applied to a perfectly conducting sphere whose magnetic dipole response is also retained.
A scattering measurement should subtract stray-light and detector backgrounds, control scatterer density and optical thickness, and calibrate detector response and incident <power> when comparing <wavelengths>. A polarization analyzer at $90^\circ$ tests the dipole pattern; multiple scattering or unresolved nonisotropic contributions can reduce its polarization.
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