The physics of the Compton effect
Compton scattering is the inelastic scattering of a photon, typically an X-ray or gamma ray, off a charged particle, usually an electron. The photon transfers some of its energy and momentum to the electron, so the scattered photon emerges with lower energy and therefore a longer wavelength. Arthur Compton demonstrated this in 1923, and it was strong evidence that light carries momentum and behaves as a particle, supporting the quantum picture.
The wavelength change depends only on the scattering angle, not on the incoming wavelength. The shift is given by the Compton formula, in which the increase in wavelength equals the Compton wavelength of the electron, about 0.00243 nanometres, multiplied by one minus the cosine of the scattering angle.
Why the angle dependence matters
Because the shift scales with one minus the cosine of the angle, a photon scattered straight forward (zero degrees) loses essentially no energy, while one scattered straight back (180 degrees) loses the most. This angle dependence is exploited in fields from medical imaging and radiation shielding to gamma-ray astronomy, where the Compton effect dominates photon interactions in an intermediate energy range between the photoelectric effect at low energies and pair production at high energies.