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The Compton Effect in the Scattering of Photons by Electrons

In the early 1920's Arthur Compton was engaged in experiments involving the scattering of high powered radiation,
Xrays and gamma rays, from crystals and other materials. The perception of the time was that the frequency/wave length
of the radiation was not changed by the scattering process. It was Arthur Compton's contribution to science to
demonstrate experimentally and explain theoretically that the scattered radiation underwent a diminishment in energy;
i.e., a decrease in frequency and an increase in wavelength. The crucial experiment involved the scattering of
radiation by free electrons.

Theoretically it was not obvious how to establish the results of the interaction of electromagnetic waves
and material particles. If the interaction involved only particles then the conservation of momentum and energy
would determine the end result. But it was not immediately clear until the development of quantum mechanics
what the energy and momentum of a wave is.

Albert Einstein in his explanation of the photoelectric effect established that the energy of a photon is
hν, where h is Planck's constant and ν is the frequency of the radiation. Others, including Louis de Broglie,
established that the momentum of a photon is hν/c.

Let ν_{0} be the frequency of the incident radiation and ν_{θ} that of the scattered
radiation, where θ is the angle of the scattered radiation.

The electron recoils from the interaction with some velocity v. The relativistic formula for the momentum
of a particle of rest mass m_{0} is

m_{0}βc/(1−β²)^{½}

where c is the speed of light.

Suppose the interaction of a photon and an electron are as follows:

The momentum vectors can be rearranged into a triangle as shown below.