Thomson scattering means that a low-energy photon is scattered by an electron at rest.
Compton scattering is the same phenomenon with a high-energy (> 511 keV) photon.
The cross section of Thomson scattering is of the order of the electron classical size. The classical electron radius is 3 * 10^-15 m. That is also the distance where the potential energy of two close electrons is equal to 511 keV, that is, the mass of the electron.
The cross section of Compton scattering is of the order of the electron classical size divided by the energy of the photon (given in units of 511 keV).
Let us assume that a "photon" moves in a medium of coupled electron-positron dipoles. Oscillation of such a dipole spreads to the neighbor dipole through the electric force. The photon is really a phonon of this medium. We do not assume the existence of any electromagnetic waves. The oscillation is strictly in the dipoles.
Suppose then that we have a free electron in the medium. What is the cross section of its collision with a phonon?
We may model a phonon as a moving oscillation of a single dipole. The oscillation of a single dipole jumps to the neighboring dipole at (almost) the speed of light. The phonon moves fast through the medium.
If the free electron happens to be within 3 * 10^-15 meters from the positron or the electron in the oscillating dipole of the phonon, then there is very strong interaction between the free electron and the phonon. This might explain why the cross section of a photon-electron collision is of the order of that length.
The free electron robs energy and momentum from the oscillation of the dipole.
We may assume that the dipole has before the collision assumed an equilibrium position in the electric field of the electron.
When the dipole starts to oscillate, what is the effect on the free electron? If the electron is not close to the ends of the dipole, the momentum transfer is inversely proportional to the distance to the ends of the dipole, and the periodically changing field probably cancels away most of the momentum transfer to the free electron.
Why is the cross section inversely proportional to the energy of the photon in Compton scattering?
The history of the Klein-Nishina formula
In 1928, Klein and Nishina were able to derive the correct differential cross section formula for Compton scattering, based on the brand-new Dirac equation. Yuji Yazaki in the link (2017) tells about the history of the discovery.
In 1926, Dirac treated scattering as a state transition of the system electron & an oscillating electromagnetic field. The apparent "collision of a photon" is a state transition which happens at a certain probability per second. Dirac derived the correct formula for a "spinless" electron. Klein and Nishina included the magnetic field of the electron in the formula.
We need to find out what is the relationship between the Feynman approach to scattering and the Klein-Nishina approach.
https://arxiv.org/abs/1501.06838
Waller and Tamm (1930), and in unpublished notes, Ettore Majorana, modified the Klein-Nishina semiclassical approach to a quantum field theoretical framework. It turned out that the electron goes through intermediate states. Thomson scattering is produced by negative-energy, that is, positron, intermediate states.
We need to compare the ideas of Waller, Tamm, Majorana, and Feynman.
https://arxiv.org/abs/1501.06838
Waller and Tamm (1930), and in unpublished notes, Ettore Majorana, modified the Klein-Nishina semiclassical approach to a quantum field theoretical framework. It turned out that the electron goes through intermediate states. Thomson scattering is produced by negative-energy, that is, positron, intermediate states.
We need to compare the ideas of Waller, Tamm, Majorana, and Feynman.
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