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After reading the wiki articles I know, that both Rayleigh scattering and Thomson scattering are elastic processes. But what is the essential difference between those two processes, their cross sections and energy dependence?

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Rayleigh scattering is scattering from polarizable entities. The incident light induces a dipole moment, which re-radiates. Thomson scattering is scattering from free unbound charged "unpolarizable" particles.

The cross section for Rayleigh scattering decreases with the fourth power of wavelength. That for Thomson scattering is independent of wavelength.

Comparing their relative cross sections is tricky, because Rayleigh scattering depends on the size of the particle, the wavelength of light, and the polarizability of the particle, all of which can vary significantly. Thomson scattering has none of that.

The Wikipedia pages you cite have a few examples. Air in visible light has a Rayleigh cross section on the order of $10^{-31}\, \mathrm{m}^2$, and the Thomson cross section for an electron is on the order of $10^{-28}\, \mathrm{m}^2$

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From the perspective of radiation oncology (i.e. higher energy) physics, here is the main difference, in a few words.

Rayleigh scattering is the elastic scattering of a photon off an entire atom.

Thomson scattering is the elastic scattering of a photon off a single unbound electron.

Both are more likely to happen at lower energies. In particular, Rayleigh scattering cross section depends on the inverse square of initial photon energy. Thomson scattering cross section has a more complicated energy dependence, with many terms (find it in many textbooks), generally stronger than inverse square.

FYI Thomson scattering is a particular case (the low energy case) of the Compton effect, which is non elastic: the electron is accelerated and the scattered photon has lower energy than the initial photon.

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Thomson scattering occurs by free electron and Rayleigh scattering occurs by bound electron.

Again, Thomson scattering is independent of nature of the incident beam like wavelength or frequency, but Rayleigh depends on frequency.

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