When light bounces off an electron
In 1923, Arthur Compton aimed X-rays at a block of graphite and found that the scattered rays had a slightly longer wavelength than the incoming ones, by an amount that depended only on the scattering angle. Classical wave theory could not explain it. Compton showed that it follows if the X-rays behave as particles, photons, each colliding with an electron like a billiard ball and giving it part of its energy and momentum.
The formulas
- 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 kinetic energy: K = E − E'
- Electron recoil angle: cot φ = (1 + E ÷ mₑc²) × tan(θ ÷ 2), with mₑc² = 511 keV
Example: Compton’s own experiment
Molybdenum Kα X-rays have a wavelength of 71.07 pm. Scattered at 90°:
- Shift: 2.426 × (1 − 0) = 2.426 pm
- Scattered wavelength: 73.50 pm
- Photon energy: 17.445 keV before, 16.869 keV after
- Electron energy: 0.576 keV, or 3.3% of the photon’s energy
- Electron recoil angle: 44.04° from the incident direction
Reading the result
- At 0°, nothing changes: the photon passes without interacting.
- At 90°, the shift equals one Compton wavelength.
- At 180°, backscatter, the shift is largest, 2λc ≈ 4.85 pm, and the electron takes the most energy.
Because the shift is fixed in absolute terms, it matters for short wavelengths. For a 662 keV gamma ray from cesium-137, with a wavelength of only 1.87 pm, a backscattered photon keeps just 184 keV. For visible light, at 500,000 pm, the shift is a negligible fraction.
Where it matters
Compton scattering dominates the interaction of X-rays and gamma rays with soft tissue between about 100 keV and 10 MeV, so it shapes dose distributions in radiotherapy and the scatter that blurs X-ray images. Gamma spectrometers show a characteristic Compton edge, and Compton cameras and telescopes use the effect to locate gamma-ray sources. The calculator assumes a free electron at rest, a good approximation when the photon energy is much larger than the electron’s binding energy.