In the Thomson scattering,
the classical picture of the photon-free-electron scattering,
there would be no change of energy of the electromagnetic field.
In other word, the frequency/wavelength of light should maintain
same before and after the scattering. In 1924, Compton observed
scattering between X-ray and electrons where wavelength of the
scattered light becomes larger than its initial value by
This apparent inconsistency turned out to be understood in the Quantum picture where the radiation field is equivalent to a photon. In the system illustrated in FIG. 4, The energy momentum conservation leads to the following equations:
Finally we get the recoiling photon energy as
It should be remarked that the recoiling energy given by Eq. 22 can be easily obtained by the 4-momentum treatment in the scheme of the relativistic kinematics. The photon 4-momentum vector and the electron 4-momentum vector follows the conservation law in the collision such as
The cross section of the Compton Scattering is described by the Kline-Nishina Formula:
implies the valuables described in the electron rest system.