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Derivation of the scattering formula

Fig. 3: Energies of a photon at 500 keV and an electron after Compton scattering.

Plot of photon energies calculated for a given element (atomic number Z) at which the cross section value for the process on the right becomes larger than the cross section for the process on the left. For calcium (Z=20), Compton scattering starts to dominate at hυ=0.08 MeV and ceases at 12 MeV.

Consider a photon γ with wavelength λ colliding with an electron e in an atom, which is treated as being at rest. The collision causes the electron to recoil, and a new photon γ' with wavelength λ' emerges at angle θ from the photon's incoming path. Let e' denote the electron after the collision. Compton allowed for the possibility that the interaction would sometimes accelerate the electron to speeds sufficiently close to the velocity of light as to require the application of Einstein's special relativity theory to properly describe its energy and momentum.