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?
[Physics] the difference between Rayleigh scattering and Thomson scattering
electromagnetismquantum mechanicsscattering
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Raman scattering is inelastic scattering from molecules. The photon interacts with the molecule and changes the molecules vibrational, rotational or electron energy.
Rayleigh scattering is in the main elastic scattering from small particles whose size is less than that of the wavelength of the photon. The scattering can occur of atoms or molecules and for molecules the scattering can be inelastic with a change of rotational energy of the molecule.
Compton scattering is inelastic scattering of a photon from a free charged particle. If the charged particle is a bound electron then the energy of the photon must be much greater than the binding energy of the electron.
Side note: Rayleigh scattering is a particular case of Mie scattering. This theory explains in particular the white colour of objects which are made of particles of size greater than the typical wavelength : milk, clouds, chemical powders...
To add to the answer there is Thomson (no "p") scattering which is the elastic scattering of electromagnetic radiation by a free charged particle, as explained by classical electromagnetism.
It is just the low-energy limit of Compton scattering when the particle kinetic energy and photon frequency do not change as a result of the scattering.
This limit is valid as long as the photon energy is much smaller than the mass energy of the particle.
In the literature, the two terms are often mixed up and used differently. In my answer, I will use the most distinct and most common interpretation of the two.
In simple terms, elastic scattering is about energy. Specifically, "the kinetic energy of the scattering particle is conserved in the center-of-mass frame" (see wikipedia). Inelastic scattering is then a process where the scattering particle loses energy (see wikipedia).
Coherent vs. incoherent scattering is not about energy, it is about the phase or fluctuations of a the wave or scattering particle. There does not seem to be a wikipedia article on coherent scattering, but there is one for incoherent scattering. Coherent scattering is then the case where the scattered particle or wave has a fixed phase relation relative to the initial wave, such that you can observe interference between the two. This coherence can be destroyed by fluctutations of the scattering medium or by quantum effects such as inversion of level systems.
Importantly, all combinations of the two categories are allowed. That is coherent elastic scattering, incoherent elastic scattering, coherent inelastic scattering and incoherent inelastic scattering are all different physical processes.
In mathematical terms, the processes can be categorized by properties of the difference between the initial and final state of the scattering particle in the scattering process $\psi_i \rightarrow \psi_f$. For elastic scattering, $\psi_i$ and $\psi_f$ have the same energy, for inelastic scattering they do not. To investigate incoherent scattering, one usually leaves the framework of wavefunctions and introduces density matrices or statistical ensembles. A common example of coherent and incoherent scattering is the famous Mollow triplet in resonance fluorescence, where both processes occur simultaneously.
Best Answer
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 $10^{-28}\, \mathrm{m}^2$