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Consider a collision between a photon with frequency moving in the x direction, with an electron of rest mass initially at rest [ see Figure 2.3 ].
Figure 2.3: Diagram illustrating Compton scattering.
Before the collision, the four- momenta of the photon and electron are
After the collision we have
where is the photon's frequency after the collision, v is the speed of the electron and , are shown in Figure 2.3.
Conservation of four- momentum gives
Rearranging and squaring we get
Using and we obtain
which gives the Compton scattering formula