Compton Scattering Calculator
When an X-ray photon collides with a nearly-free electron, the scattered photon emerges at a longer wavelength. The wavelength increase depends only on the scattering angle theta via the Compton formula: delta_lambda = (h / m_e c)(1 - cos theta). The factor h / m_e c = 2.42631 x 10^-12 m is the Compton wavelength of the electron. This effect was first observed by Arthur Compton in 1923 and provided decisive evidence for the particle nature of photons. Unlike the photoelectric effect, which only ejects electrons at certain frequencies, Compton scattering occurs at all X-ray and gamma-ray energies and shifts the photon to a lower frequency (longer wavelength), transferring kinetic energy to the electron. Enter the initial photon wavelength and scattering angle.
Compton scattering formula
delta_lambda = (h / m_e c) * (1 - cos theta)
h / m_e c = 2.42631e-12 m = 2.42631 pm (Compton wavelength)
lambda_scattered = lambda_initial + delta_lambda
Wavelength shift is maximum at theta = 180 degrees (backscatter): delta_lambda = 2 * 2.42631 = 4.853 pm. At theta = 0 (forward scatter) there is no shift. Energy loss fraction = 1 - lambda_0 / lambda_1.
Compton scattering at key angles
- theta = 0 degrees: no shift, photon passes through without energy loss.
- theta = 90 degrees: delta_lambda = 2.426 pm, the electron Compton wavelength.
- theta = 180 degrees (backscatter): delta_lambda = 4.853 pm, maximum energy transfer to electron.
- For 71 pm X-rays at 90 degrees: scattered wavelength = 71.07 + 2.43 = 73.50 pm (about 3.4% energy loss).
Compton scattering: frequently asked questions
What is Compton scattering?
Compton scattering is the inelastic scattering of a photon by a free electron. The photon transfers some momentum and energy to the electron, emerging at a longer wavelength. The wavelength shift is delta_lambda = (h / m_e c)(1 - cos theta), where theta is the scattering angle. This confirmed the particle nature of light in 1923.
What is the Compton wavelength?
The Compton wavelength of the electron is lambda_C = h / (m_e c) = 2.42631 x 10^-12 m (about 2.43 pm). It is the natural wavelength scale for Compton scattering. The wavelength shift ranges from 0 (forward scattering, theta = 0) to 2 lambda_C (backscattering, theta = 180 degrees).
What is the maximum wavelength shift?
The maximum shift occurs at theta = 180 degrees (backscattering): delta_lambda = 2 h/(m_e c) = 4.852 pm. At theta = 90 degrees the shift is exactly one Compton wavelength: delta_lambda = 2.426 pm.
Why is Compton scattering important?
Compton scattering provided compelling evidence that electromagnetic radiation consists of discrete photon quanta carrying momentum, not just energy. It was the first direct experimental demonstration of photon momentum and confirmed Einstein's photon hypothesis. It is also exploited in gamma-ray detectors and medical imaging.
Does Compton scattering apply to other particles besides electrons?
Yes. The formula generalizes to any free particle: delta_lambda = (h / mc)(1 - cos theta) where m is the mass of the target particle. For a proton (1836 times heavier than the electron), the shift is 1,836 times smaller. Compton scattering off nuclei is negligible for common X-ray energies.
Official sources
- NIST CODATA 2018: Compton Wavelength of the Electron.
- OpenStax University Physics Vol. 3: The Compton Effect.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.