When X-rays strike matter, most pass straight through, but some come out in a different direction. This is called scattering. In 1923 Compton sent the K_\alpha line of molybdenum (wavelength 0.0709 nm) onto graphite and measured, with an X-ray spectrometer, the wavelength of the X-rays scattered at 45°, 90° and 135°.
The scattered X-rays included X-rays of longer wavelength than those sent in. The larger the scattering angle \theta, the larger the increase; at 90° it was about 0.0024 nm, turning 0.0709 nm into 0.0733 nm. With other light elements in place of graphite, the increase at the same angle was the same.
The measured spectrum had two peaks, one at the incident wavelength and one at the increased wavelength. As the scattering angle grows, the two peaks move farther apart. Why the unshifted peak remains is taken up in the last scene.
Along with the scattering, one electron is knocked out. It is called the recoil electron. The energy the scattered X-ray lost has become the kinetic energy of this electron.
If light is an electromagnetic wave, scattering is explained as follows. The electric field of the incoming X-ray shakes an electron, and the shaken electron gives off an electromagnetic wave at the same frequency. This scattering, calculated by Thomson, is called Thomson scattering, and the outgoing wave has the same frequency as the incoming one.
Same frequency means same wavelength. The wave leaving in every direction has the incident wavelength, and the wavelength does not change with angle. That is why the three waves in the figure are drawn with the same wavelength.
Another explanation was also considered: the electron is pushed by the X-rays, starts to move, and the Doppler effect lengthens the wavelength. But then the increase would depend on how much the electron had been pushed, and would change with the intensity of the X-rays and how long they had been shining. In the measurements, the increase did not depend on intensity and was set by the angle alone.
Compton treated X-rays as photons and scattering as a collision between one photon and one electron. To calculate a collision, the photon needs a momentum as well as its energy E = h\nu.
That electromagnetic waves carry momentum was already known in classical electromagnetism. A wave carrying energy E carries momentum \dfrac{E}{c} and pushes whatever it strikes. This is radiation pressure, measured in the early 1900s by Lebedev and by Nichols and Hull.
If one photon has energy E = h\nu, its momentum is p = \dfrac{h\nu}{c}. Using c = \nu\lambda, p = \dfrac{h}{\lambda}: the shorter the wavelength, the larger the momentum. In 1917 Einstein, studying the motion of gas molecules in thermal equilibrium with radiation, had shown that a momentum of \dfrac{h\nu}{c} must be exchanged each time a photon is absorbed or emitted.
A photon of the molybdenum K_\alpha line has an energy of 17.5 keV and a momentum of 9.3 \times 10^{-24} kg m/s. What differs from classical radiation pressure is that this momentum is handed over in amounts of \dfrac{h}{\lambda}, to one electron in one collision.
Write the momentum of the incident photon as \dfrac{h}{\lambda}, that of the scattered photon as \dfrac{h}{\lambda'} and that of the recoil electron as p, and take the electron to be at rest before the collision. By conservation of momentum, the momentum of the incident photon equals the sum of the momenta of the scattered photon and the electron, so the three vectors form a triangle, as in the figure.
The angle between the momenta of the two photons is the scattering angle \theta, so the law of cosines gives p^2 = \left(\dfrac{h}{\lambda}\right)^2 + \left(\dfrac{h}{\lambda'}\right)^2 - \dfrac{2h^2}{\lambda\lambda'}\cos\theta.
For conservation of energy, the energy of the electron is written with the relativistic formula. An electron at rest has energy mc^2, and an electron with momentum p has energy \sqrt{p^2c^2 + m^2c^4}, so \dfrac{hc}{\lambda} + mc^2 = \dfrac{hc}{\lambda'} + \sqrt{p^2c^2 + m^2c^4}.
Writing the kinetic energy of the electron as \dfrac{p^2}{2m} gives the increase only approximately. The recoil electron moves much more slowly than light, so the numbers hardly change, but with the relativistic formula the increase follows with no approximation.
Rewrite the energy equation as \sqrt{p^2c^2 + m^2c^4} = \dfrac{hc}{\lambda} - \dfrac{hc}{\lambda'} + mc^2, square both sides, and substitute p^2 from the momentum equation. The terms \dfrac{h^2c^2}{\lambda^2}, \dfrac{h^2c^2}{\lambda'^2} and m^2c^4 appear on both sides and cancel, p disappears, and what remains involves only the wavelengths of the photons and the scattering angle.
Simplifying gives \left(\dfrac{h}{\lambda} - \dfrac{h}{\lambda'}\right)mc = \dfrac{h^2}{\lambda\lambda'}(1 - \cos\theta), and multiplying both sides by \dfrac{\lambda\lambda'}{hmc} gives \lambda' - \lambda = \dfrac{h}{mc}(1 - \cos\theta). Debye derived the same formula independently in the same year, 1923.
The result, \Delta\lambda = \dfrac{h}{mc}(1 - \cos\theta), contains neither the incident wavelength nor any property of the material. The measured fact that the increase depends only on the angle appears directly in the formula. The curve in the figure is this formula: the increase is 0 at \theta = 0 and largest, \dfrac{2h}{mc}, at \theta = 180^\circ.
\dfrac{h}{mc} has the dimension of length and is called the Compton wavelength of the electron; its value is 2.43 \times 10^{-12} m. At 90° the increase is exactly this value, and the 0.0709 nm of the molybdenum K_\alpha line becomes 0.0733 nm. The increases Compton measured agreed well with the formula.
The energy of the recoil electron is the energy the photon lost. For scattering at 180°, a 17.5 keV photon becomes 16.4 keV, and the electron receives about 1.1 keV. Visible light has a wavelength of about 500 nm, so the increase is at most about one hundred-thousandth of the wavelength and hard to measure. With X-rays the increase is a few percent of the wavelength.
The unshifted peak remains because some scattering happens on electrons bound tightly to atoms. If the electron stays with its atom and the whole atom recoils, the m in the formula becomes the mass of the whole atom. A carbon atom has about 22000 times the mass of an electron, so the increase is 22000 times smaller, and in the measurements it is the same as no increase at all.
In 1924 Bohr, Kramers and Slater put forward a theory in which energy and momentum are not conserved in each individual process, but only on average over many processes. If it were right, a scattered photon and a recoil electron would not necessarily appear together, one of each at the same moment.
In 1925 Bothe and Geiger counted scattered photons and recoil electrons with two separate counters and found that the two were counted at the same time. In the same year Compton and Simon used a cloud chamber to compare the direction in which each recoil electron flew with the direction of the scattered photon, and showed that momentum is conserved in each collision. Compton received the 1927 Nobel Prize in Physics for the discovery of this effect.