Compton Scattering Calculator — Wavelength Shift & Energy Transfer
Compton scattering occurs when a photon (X-ray or gamma ray) collides with a loosely bound electron and transfers some of its energy to the electron. Enter the incident wavelength and scattering angle to find the wavelength shift, scattered photon wavelength, and energy transferred to the electron.
pm
°
Increase in photon wavelength after scattering off a free electron
- 1
Compton wavelength (λ_C)
h ÷ (m_e × c) = 2.42631 pmA fixed constant ≈ 2.42631 pm; the maximum possible shift occurs at 180°. - 2
Cosine term (1 − cos θ)
1 − cos(90°) = 1Equals 0 at forward scatter (θ = 0°) and 2 at back-scatter (θ = 180°). - 3
Wavelength shift (Δλ)
2.42631 × 1 = 2.4263 pm
How does this calculator work?
The Compton wavelength shift Δλ = (h/m_e c) × (1 − cos θ) ≈ 2.426 × (1 − cos θ) pm depends only on the scattering angle θ. The scattered photon has wavelength λ′ = λ + Δλ and lower energy E′ = hc/λ′; the energy difference goes to the recoiling electron. Maximum shift of ≈ 4.85 pm occurs at 180° backscatter.
Formula
How this is calculated
In 1923 Arthur Compton observed that X-rays scattered from electrons have a longer wavelength than the incident beam — a result that confirmed the particle nature of light. The wavelength shift Δλ = λ_C × (1 − cos θ) depends only on the scattering angle θ (the angle between the incident and scattered photon), not on the incident wavelength. Here λ_C = h/(m_e c) ≈ 2.42631 pm is the Compton wavelength of the electron, built from Planck's constant h, the electron rest mass m_e, and the speed of light c.
At θ = 0° (forward scattering) there is no wavelength change — the photon passes straight through. At θ = 90° the shift equals one Compton wavelength (≈2.43 pm). At θ = 180° (backscatter) the shift reaches its maximum of 2λ_C ≈ 4.85 pm. The energy lost by the photon (E − E′ = hc/λ − hc/λ′) is transferred to the recoiling electron as kinetic energy.
The Compton effect is significant only when the photon wavelength is comparable to λ_C — hard X-rays and gamma rays. For visible light (λ ≈ 500 nm ≫ λ_C) the fractional shift is negligible. The derivation uses special relativity and conservation of four-momentum, treating the photon as a particle with momentum p = h/λ.
Frequently asked questions
The Compton wavelength λ_C = h/(m_e c) ≈ 2.42631 pm is the characteristic length scale of Compton scattering. It equals the wavelength shift when the scattering angle is 90°, and twice the shift at 180° (maximum backscatter). It is a fundamental constant depending only on the electron mass.
Compton scattering becomes significant when the photon energy is comparable to or exceeds the electron rest mass energy (511 keV). For soft X-rays (wavelengths much longer than λ_C), photoelectric absorption dominates. For very high-energy gamma rays, pair production dominates. In medical CT scanning, Compton scattering in tissue is the dominant interaction for photon energies of 30 keV–25 MeV.
Rayleigh (coherent) scattering occurs when the photon scatters elastically off the entire atom — the wavelength does not change. Compton (incoherent) scattering involves an interaction with a single, approximately free electron, with measurable energy transfer to the electron and a wavelength increase. Rayleigh scattering dominates at low energies; Compton scattering dominates at higher photon energies.
Also known as
TG we-Calculate Editorial Team. (2026). Compton Scattering Calculator — Wavelength Shift & Energy Transfer [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/compton-scattering-calculator
TG we-Calculate Editorial Team. "Compton Scattering Calculator — Wavelength Shift & Energy Transfer." TG we-Calculate. 2026. https://we-calculate.com/calculator/compton-scattering-calculator.
TG we-Calculate Editorial Team, "Compton Scattering Calculator — Wavelength Shift & Energy Transfer," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/compton-scattering-calculator
@misc{wecalculate_compton_scattering_calculator, title = {Compton Scattering Calculator — Wavelength Shift & Energy Transfer}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/compton-scattering-calculator}}, year = {2026}, note = {TG we-Calculate} }
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