text: Returning to the United States, Compton was appointed Wayman Crow Professor of Physics, and head of the department of physics at Washington University in St. Louis in 1920. In 1922, he found that X-ray quanta scattered by free electrons had longer wavelengths and, in accordance with Planck's relation, less energy than the incoming X-rays, the surplus energy having been transferred to the electrons. This discovery, known as the "Compton effect" or "Compton scattering", demonstrated the particle concept of electromagnetic radiation.
  In 1923, Compton published a paper in the Physical Review that explained the X-ray shift by attributing particle-like momentum to photons, something Einstein had invoked for his 1905 Nobel Prize–winning explanation of the photo-electric effect. First postulated by Max Planck in 1900, these were conceptualized as elements of light "quantized" by containing a specific amount of energy depending only on the frequency of the light. In his paper, Compton derived the mathematical relationship between the shift in wavelength and the scattering angle of the X-rays by assuming that each scattered X-ray photon interacted with only one electron.  His paper concludes by reporting on experiments that verified his derived relation:
  where
  : is the initial wavelength,
  : is the wavelength after scattering,
  : is the Planck constant,
  : is the electron rest mass,
  : is the speed of light, and
  : is the scattering angle.
  The quantity  is known as the Compton wavelength of the electron; it is equal to . The wavelength shift  lies between zero (for ) and twice the Compton wavelength of the electron (for ). He found that some X-rays experienced no wavelength shift despite being scattered through large angles; in each of these cases the photon failed to eject an electron. Thus the magnitude of the shift is related not to the Compton wavelength of the electron, but to the Compton wavelength of the entire atom, which can be upwards of 10,000 times smaller.
  "When I presented my results at a meeting of the American Physical Society in 1923", Compton later recalled, "it initiated the most hotly contested scientific controversy that I have ever known." The wave nature of light had been well demonstrated, and the idea that it could have a dual nature was not easily accepted. It was particularly telling that diffraction in a crystal lattice could only be explained with reference to its wave nature. It earned Compton the Nobel Prize in Physics in 1927. Compton and Alfred W. Simon developed the method for observing at the same instant individual scattered X-ray photons and the recoil electrons. In Germany, Walther Bothe and Hans Geiger independently developed a similar method.