Compton scattering is the inelastic collision of a photon with a free or loosely bound electron. The photon transfers part of its energy and momentum to the electron, emerges at an angle with a longer wavelength, and the electron recoils. The calculator above returns the wavelength shift, the scattered photon energy and the recoil energy for any incident energy and angle.
Arb Digital publishes free physics calculators that own a single mechanism properly. The photoelectric effect calculator on this site covers the process where an electron absorbs a whole photon and is ejected; that page has no scattering and no wavelength shift. This one covers the process where the photon survives with less energy, which is a different interaction that dominates at higher photon energies.
What This Compton Scattering Calculator Does
You enter an incident photon energy, a scattering angle and the rest energy of the target particle. The tool converts the energy into a wavelength, applies the Compton shift formula, and returns the scattered wavelength and the corresponding scattered photon energy. The difference between incident and scattered energy is the kinetic energy carried away by the recoiling electron, which is reported alongside.
The grid also shows the fraction of the original energy the photon retains, which is the figure that matters most in radiation shielding and imaging. A photon that keeps 90 per cent of its energy after a small-angle scatter is still very much a hazard and still very much a source of image-degrading noise; one that keeps a third after backscattering is a different problem entirely.
The target is selectable because the shift scales inversely with the target's rest mass. An electron gives the familiar picometre-scale shift. A proton, nearly two thousand times heavier, gives a shift nearly two thousand times smaller, which is why Compton scattering from nuclei is negligible in practice and why the process is understood entirely as a photon-electron interaction.
How to Use It
- Enter the incident energy in the unit you have. Diagnostic X-rays are quoted in kiloelectronvolts, gamma sources in kilo- or megaelectronvolts, and the selector handles both.
- Set the angle you actually care about. Backscatter at 180 degrees gives the maximum possible energy loss and is the case that matters for shielding behind a source.
- Leave the target as the electron unless you have a reason not to. Effectively all Compton scattering in matter is from atomic electrons.
- Read the retained fraction, not just the shift. The absolute wavelength shift is the same at a given angle regardless of energy, so it is the fractional energy change that tells you how significant the scatter is.
- Compare the recoil energy with binding energies. The free-electron treatment is valid when the transferred energy far exceeds the electron's binding energy, which is why it works well for outer-shell electrons in light elements.
The Formula: How the Compton Shift Is Calculated
The wavelength shift is Δλ = (h ÷ mec)(1 − cos θ), where h is the Planck constant, me the electron mass, c the speed of light and θ the scattering angle. The quantity h ÷ mec is the Compton wavelength of the electron, and it sets the entire scale of the effect. The NIST CODATA value for the Compton wavelength is 2.426 310 235 38 × 10−12 m, which is the constant this tool uses.
The result is remarkable for what it leaves out. The shift depends on the scattering angle and on the target's mass, and on nothing else at all. A radio photon and a gamma photon scattered through the same angle by the same electron suffer the same absolute wavelength change. HyperPhysics' page on Compton scattering derives this from conservation of relativistic energy and momentum, and notes that the wavelength change depends only on the angle for a given target particle.
In energy terms the same relation becomes E′ = E ÷ [1 + (E ÷ mec²)(1 − cos θ)], where mec² is 511 keV for an electron. The electron's recoil kinetic energy is simply E − E′, since the photon has nowhere else to put the energy it lost.
Work the defaults. An incident photon of 511 keV has exactly the electron rest energy, so E ÷ mec² = 1. At 90 degrees, cos θ = 0, so the bracket is 1 + 1 = 2 and E′ = 255.5 keV. The photon loses exactly half its energy and the electron recoils with the other half. Checking through wavelength: a 511 keV photon has a wavelength of 2.4263 pm, which is the Compton wavelength itself; adding a shift of one Compton wavelength doubles it to 4.8526 pm, and halving the wavelength doubles the energy, so the scattered photon is at half of 511 keV. The two routes agree exactly, as they must.
Why the Shift Is Independent of Wavelength
This is the result that made Compton's 1923 experiment decisive, and it is worth being clear about why it matters. Classical wave theory predicted that scattered radiation would have the same wavelength as the incident radiation, because an oscillating electron driven by a wave re-radiates at the driving frequency. Compton measured a wavelength change, and one that depended on angle rather than on intensity or wavelength.
The only way to get that result is to treat light as a particle with momentum h ÷ λ and apply conservation of energy and momentum to a two-body collision. Doing so produces exactly the observed angular dependence. The experiment is therefore direct evidence that photons carry momentum, and it is the reason the photon picture was accepted as physical rather than as a calculational device.
The independence from wavelength has a practical consequence too. Because the shift is a fixed absolute amount, it is negligible for visible light, where the wavelength is hundreds of nanometres and the shift is picometres, and it is dominant for hard X-rays and gamma rays, where the wavelength is itself of order a picometre. That is why Compton scattering is invisible in optics and unavoidable in radiology.
Where Compton Scattering Dominates
Three processes compete when a photon meets matter: the photoelectric effect, Compton scattering and pair production. Which one wins depends on photon energy and on the atomic number of the material. The photoelectric effect dominates at low energy and rises steeply with atomic number, which is why lead is a good X-ray shield and why bone shows up against tissue. Pair production only becomes possible above 1.022 MeV, twice the electron rest energy.
Between those two, Compton scattering dominates. For soft tissue, which is mostly light elements, the Compton region stretches from roughly 30 keV to about 20 MeV. That range covers essentially all of diagnostic radiology above the lowest energies and all of radiotherapy, which is why Compton interactions are the main source of scattered dose in a clinic and the main cause of contrast loss in a radiograph.
The angular distribution is not uniform. The Klein-Nishina formula describes how likely each scattering angle is, and it becomes increasingly forward-peaked as photon energy rises. This tool computes the kinematics for whatever angle you specify; it does not compute how probable that angle is, and for shielding calculations that probability matters as much as the energy.
Where This Sits Among the Other Quantum Tools
This page handles one interaction. The photoelectric effect calculator handles complete photon absorption and electron ejection, which is the competing process at lower energies. The photon energy calculator converts between energy, frequency and wavelength for the photon itself, and the wavelength calculator does the same for waves generally. The de Broglie wavelength calculator gives the matter wavelength of the recoiling electron, and the relativistic kinetic energy calculator converts that recoil energy into a speed. For the rest-energy relation that sets the whole scale, the mass energy equivalence calculator is the relevant page, and the energy converter handles units.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Expecting the shift to depend on the incident wavelength — it does not. Only the angle and the target mass matter, which is the whole point of Compton's result.
- Entering the angle in radians — the box expects degrees, and a value of 1.57 will be read as one and a half degrees rather than a right angle.
- Confusing this with the photoelectric effect — there the photon is absorbed entirely and there is no scattered photon. Here the photon survives with reduced energy.
- Applying it to tightly bound inner electrons — the free-electron treatment needs the transferred energy to be much larger than the binding energy. Otherwise the whole atom recoils and the shift is far smaller.
- Reading the kinematics as a probability — this page tells you what happens at a given angle, not how often that angle occurs. The angular distribution comes from the Klein-Nishina cross-section.
Related Free Tools From Arb Digital
Compare the competing process with the photoelectric effect calculator, and describe the photon itself with the photon energy calculator and the wavelength calculator. Follow the recoiling electron with the de Broglie wavelength calculator and the relativistic kinetic energy calculator. The mass energy equivalence calculator gives the 511 kiloelectronvolt rest energy that sets the scale, the two photon absorption calculator covers a different multi-photon process, and the energy converter handles units. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
It is 2.42631 times ten to the minus twelve metres, or about 2.426 picometres, from the CODATA recommended values. It equals the Planck constant divided by the electron mass times the speed of light, and it is the maximum wavelength shift a photon can suffer in a single Compton scatter.
The most it can lose is at 180 degrees, a direct backscatter, where the wavelength shift is twice the Compton wavelength. Even then the photon never loses all its energy, because a photon at rest cannot exist, so some energy always continues forward as a scattered photon.
Because it falls out of conserving relativistic energy and momentum between a massless photon and a stationary electron. The algebra leaves an expression containing only the electron mass, the speed of light, the Planck constant and the scattering angle, with the incident wavelength cancelling entirely.
In the photoelectric effect the photon is absorbed completely and its whole energy goes into freeing and accelerating an electron. In Compton scattering the photon survives, continues in a new direction with less energy, and shares the difference with a recoiling electron.
Because it dominates photon interactions in soft tissue across the diagnostic and therapeutic energy range. Scattered photons reach the detector from the wrong direction, which reduces image contrast, and they carry dose away from the beam path, which is why scatter control and shielding matter.
Yes in principle, but the shift is smaller by the ratio of the masses, nearly two thousand times. The proton Compton wavelength is about 1.32 femtometres, so the effect is negligible in ordinary matter and Compton scattering is treated entirely as a photon-electron process.
Because classical wave theory predicted no wavelength change at all in scattered radiation. Measuring a shift that depended only on angle could be explained only by treating light as particles carrying momentum, which established the photon as a physical entity rather than a calculational convenience.
This tool is provided for educational and study use. It models a single scatter from a free stationary electron and does not account for electron binding, multiple scattering, Doppler broadening or the angular probability distribution, so it should not be used for radiation shielding or dosimetry work, which is done with validated transport codes by a qualified medical or health physicist.
