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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