Understanding the Compton wavelength shift
Compton scattering increases a photon’s wavelength when it scatters from an initially stationary, effectively free electron. The shift depends on scattering angle rather than incident wavelength.
This calculator uses the electron Compton wavelength from the 2022 CODATA adjustment. It applies the ideal free-electron kinematic model and does not calculate intensity, cross-section, recoil direction, or binding effects.
How to use the Compton scattering calculator
- Enter incident wavelength: Provide a positive photon wavelength in nanometers.
- Enter scattering angle: Use an angle from 0° through 180°.
- Calculate: Review the wavelength shift in meters and scattered wavelength in nanometers.
- Apply model limits: Use a more complete treatment when electron binding, material response, or scattering probability matters.
Formula and variables
The maximum shift is twice the electron Compton wavelength at 180°, while forward scattering at 0° produces zero shift.
Δλ = (h/mₑc)(1 − cos θ); λ′ = λ + Δλ- λ — Incident wavelength
- Photon wavelength before scattering (nm)
- λ′ — Scattered wavelength
- Photon wavelength after scattering (nm)
- θ — Scattering angle
- Angle between incident and scattered photon directions (degrees)
- h/mₑc — Electron Compton wavelength
- CODATA electron Compton wavelength (m)
Ninety-degree scattering
A 0.071 nm photon scatters through 90° from a free electron initially at rest.
- Incident wavelength
- 0.071 nm
- Angle
- 90°
- Δλ = 2.42631023538 × 10⁻¹²(1 − cos 90°) m
- λ′ = 0.071 nm + 0.00242631023538 nm
Result: The scattered wavelength is approximately 0.07342631 nm.
The photon loses energy as its wavelength increases and the electron recoils.
Understanding your results
Relate angle to the shift
The ideal wavelength shift grows monotonically from forward to backward scattering.
- 0° gives no shift.
- 90° gives one electron Compton wavelength of shift.
- 180° gives the maximum shift of two electron Compton wavelengths.
Assumptions
- The target is an initially stationary, free electron.
- Relativistic energy and momentum conservation apply.
- The entered angle is the photon scattering angle.
Limitations
- Does not model electron binding, Doppler broadening, multiple scattering, cross-sections, polarization, or detector resolution.
- Does not calculate scattered-photon energy or electron recoil quantities.
- The free-electron model may not describe low-energy scattering in bound systems.
Common mistakes
- Confusing the electron Compton wavelength with the reduced Compton wavelength.
- Entering radians instead of degrees.
- Adding the shift in meters directly to a wavelength in nanometers.
- Using the electron recoil angle as the photon scattering angle.
Practical use cases
Modern physics education
Check wavelength-shift exercises and explore angular dependence.
X-ray and gamma-ray screening
Estimate the ideal kinematic shift before applying material and detector models.
Frequently asked questions
Why does incident wavelength not appear in the shift formula?
For scattering from a free stationary electron, the wavelength difference depends only on angle and the electron Compton wavelength.
What is the maximum Compton shift?
At 180°, it is twice the electron Compton wavelength, approximately 4.8526 × 10⁻¹² m.
Does this calculate scattering probability?
No. A differential cross-section model such as Klein–Nishina is required for probability or intensity.
Sources and review
- 2022 CODATA Recommended Values of the Fundamental Physical Constants — National Institute of Standards and Technology. Accessed 2026-07-13.
Reviewed 2026-07-13.