Compton Scattering Calculator (wavelength shift and scattered photon energy)

Understanding the Compton scattering calculator — concept and principle

When an X-ray or gamma-ray photon bounces off an electron, it loses energy and its wavelength becomes longer. This calculator applies the Compton formula to find the scattered photon's wavelength and energy and the energy given to the electron.

The wavelength grows by 2.4263 pm, from 1.873 to 4.299 pm, so the photon keeps only 288 keV.

How it is calculated

Wavelength shift Δλ = λc (1 − cos θ), with the electron's Compton wavelength λc = h ÷ (mec) ≈ 2.426 pm.
Scattered wavelength λ′ = λ + Δλ.
Photon energy E = hc ÷ λ, with hc ≈ 1,239.84 keV·pm.
Electron recoil energy = E − E′.
Maximum shift = 2λc at θ = 180° (back-scatter).

The principle

Arthur Compton showed in 1923 that X-rays scattered from electrons behave like particles colliding: both energy and momentum are conserved. Solving the collision for a photon and an electron at rest gives the shift formula. The shift depends only on the angle, not on the incoming wavelength, so it is significant only for short wavelengths. This experiment was key evidence that light carries momentum in discrete quanta.

Worked example

Default inputs: incident energy 662 keV (the gamma line of caesium-137), angle 90°.

Incident wavelength = 1,239.84 ÷ 662 ≈ 1.8729 pm.
Δλ = 2.4263 × (1 − cos 90°) = 2.4263 pm.
Scattered wavelength = 1.8729 + 2.4263 ≈ 4.2992 pm.
Scattered energy = 1,239.84 ÷ 4.2992 ≈ 288.39 keV; electron gets 373.61 keV (56.44% of the photon energy).
Maximum shift at 180° = 2 × 2.4263 = 4.8526 pm.

Each point is the calculator re-run with only “Scattering angle θ (°)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Scattering angle θ (°)	Scattered photon energy (keV)
22.5	602.577
45	479.904
67.5	367.832
90	288.39
112	237.168
135	206.13
180	184.35

How to read the result

High-energy photons lose a large share of their energy in Compton scattering, which is why gamma spectra show a broad "Compton continuum" below the photopeak. For visible light, the shift is negligible compared with the wavelength. Radiation shielding and medical imaging must account for scattered photons.

Common mistakes

Using degrees where radians are needed in a hand calculation.
Adding the shift to the energy instead of the wavelength.
Expecting large relative shifts for long-wavelength light.
Forgetting that the formula assumes a free electron at rest.

Keep in mind: Radioactive sources are hazardous. Experiments must follow radiation safety rules and licensing requirements.

Frequently asked questions

What is the Compton shift at 90°?

Exactly one Compton wavelength, about 2.426 pm.

Why does the shift not depend on wavelength?

Conservation laws give Δλ as a function of the angle and the electron mass only.

Why isn't Compton scattering noticeable with visible light?

Visible wavelengths are hundreds of thousands of times larger than the shift.

Sources: Compton, A. H. (1923). A Quantum Theory of the Scattering of X-rays by Light Elements. Physical Review, 21(5), 483–502. · CODATA 2018 recommended values (electron Compton wavelength), NIST

Checked 9 Nov 2026. OpenStax, University Physics Volume 3 (2016), §6.3 “The Compton Effect” and Chapter 6 Key Equations: Δλ = λ_c(1 − cos θ), λ_c = h/(m₀c) = 0.00243 nm, E_f = hc/λ. h, c and e exact (SI 2019); electron mass: standard value (CODATA 2018).

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