Compton Scattering refers to the process of photons scattering from atomic electrons. Using conservation of energy and momentum, it is easy to derive the relationship between the photon's energy loss and its angle of deflection (known as the Compton Scattering Formula). The cartoon to the right shows an initial photon and electron, along with a final photon and electron. Initially the photon has all the momentum, , and the total energy is the energy of the photon plus the rest energy of the electron, . (This isn't exactly true. The atomic electron has some momentum and energy, but we are concerned with photons with energies very large compared to atomic energies, so the electron's momentum and kinetic energy are negligible.) After the photon scatters, the total energy is the energy of the scattered photon plus the energy of the electron, including whatever momentum it gained from the photon's kick. Conserving momentum in the direction of the incident photon yields , while the total momentum transverse to the incident photon must be zero, which gives . Solving for pe2 by using our favorite smash hit trig identity with the two momentum equations leaves an expression with only the photon angle. Substitute this result for pe2 into the expression for the energy after the collision, replace the wave numbers using the dispersion relation for photons , then set the before and after collision expressions for energy equal to each other. Isolate the radical, divide everything by hbar, then square both sides. Now we can write an expression for the final photon's energy in term of the initial photon's energy. Reintroduce the hbars using . The expression in terms of energy is useful to an experimentalist looking at the energy spectrum from a detector. The more famous formula is given in terms of the wavelengths. Here λe is the electron's de Broglie wavelength at rest, . The change to wavelengths was made with the substiution . The maximum energy is transferred to the electron when the photon scatters backward, returning whence it came. In this case the scattered photon's energy is . Here we have just used conservation of energy and momentum to derive the relationship between the photon's loss of energy and the angle by which it's deflected. QED allows us to calculate the probability of scattering as a function of energy. Taking into account the geometry of an experimental setup, the likelihood of a photon scattering in various materials due to their electron densities, the probability of a detector absorbing a photon of given energy, and electronics nooise an actual x ray would look something like below. This is part of a spectrum taken from the decay of 241Am; the x axis is energy in keV, the y axis is the number of counts in the energy bin; the red points are data, the green points are a simulation.