Compton Scattering Calculator

Find the scattered wavelength, the shift and the energy lost by a photon when it bounces off an electron.

What it does

Compton scattering is what happens when an X-ray or gamma ray hits an electron and bounces off with a slightly longer wavelength. It was the first experiment that showed light behaves like particles, and it is the reason a photon is treated as a packet of energy with momentum rather than as a smooth wave.

OpenStax University Physics explains the measurement: when X-rays scatter off graphite, the scattered radiation shows a peak at the incident wavelength and a second peak at a longer wavelength, and the separation between the two depends on the angle of the outgoing beam. Compton's explanation was that the valence electrons in graphite behave as free electrons, that the incident radiation is a stream of photons, and that each photon hands its energy to one electron.

The formula that falls out of conservation of energy and momentum is short. The wavelength shift is the Compton wavelength of the electron times one minus the cosine of the scattering angle, and the Compton wavelength is just the Planck constant over the electron mass times the speed of light. Because that constant is a property of the electron and not of the light, the shift in any given direction is the same for every photon, which is exactly what the experimental data showed.

This page runs that formula. Give it the incident wavelength and the scattering angle, and it returns the scattered wavelength, the size of the shift, the energy of the outgoing photon and the kinetic energy the electron picks up. It also shows the Compton wavelength itself, so you can see how much of the shift comes from that constant.

The model assumes the electron is free and initially at rest, which is a good description for the outer electrons of a light atom such as carbon but not for tightly bound electrons in heavy atoms. Those give a different, much smaller shift, and the experimental spectra show a second, sharper peak for them. The page also assumes a single scattering event, since a photon that scatters twice has two shifts added together.

How to use

1. Choose the unit for the wavelength. Picometres suit X-ray work, nanometres suit visible light and gamma rays are often quoted in angstroms.
2. Enter the wavelength of the photon that hits the electron.
3. Enter the scattering angle, which is the angle between the incoming beam and the direction you detect the scattered beam.
4. Read the scattered wavelength and the shift, then the energy the photon keeps.
5. Read the energy the electron takes, which is the difference between the incoming and outgoing photon energy.
6. Compare the shift with the Compton wavelength: at 90 degrees they are equal, and at 180 degrees the shift is twice that value.

Formula

lambda

Incident photon wavelength.

lambda'

Wavelength of the scattered photon.

theta

Scattering angle between the incident and the outgoing photon.

h

Planck's constant, 6.62607015e-34 J s.

m

Electron rest mass, 9.1093837015e-31 kg.

The constant in the shift is fixed by the electron, which is why a 0.1 nm photon and a 0.001 nm photon shift by the same amount at the same angle, even though the first carries far more energy. The shift can never exceed two Compton wavelengths, which happens for a backscattered photon at 180 degrees. The constants used here are the CODATA values from NIST.

Worked example

Reference table

Limits

• The model treats the electron as free and at rest, so it does not describe the sharper peak from tightly bound electrons in heavy atoms.
• Only one scattering event is included. A photon that scatters twice has the shift applied twice.
• The electron is treated as non-relativistic in the recoil, which is fine for the energy transfers involved but not for very high energy photons.
• Pair production and other photon processes are outside the scope of this calculation.

Common mistakes

• Expecting the shift to scale with the incident wavelength. It does not, and that is the point of the experiment.
• Using the Compton shift at 0 degrees. A photon that goes straight through is not scattered, so the shift is zero there.
• Forgetting that the photon loses energy rather than gaining it. The shift in wavelength goes up and the energy goes down.
• Mixing up the angle of the electron with the angle of the photon. The photon angle sets the shift, and the electron carries the rest.
• Applying the formula to a photon scattering off a nucleus, where the mass is far larger and the shift is negligible.

Try it yourself

Quick checks. Answers sit tucked under each question.

Questions

Sources

• OpenStax University Physics Volume 3, 6.3: the Compton effect and the Compton shift
• University of York: Introduction to Quantum Physics, section on Compton scattering
• NIST: CODATA values of the fundamental constants

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