Compton Scattering Calculator
Compute the scattered photon's wavelength after Compton scattering off a free electron, given the initial wavelength and scattering angle. The wavelength shift demonstrates the photon's particle-like momentum exchange.
Last updated: May 2026
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About this calculator
The Compton scattering formula is λ' = λ + λ_C(1 − cos θ), where λ' is the scattered wavelength, λ is the incident wavelength, λ_C = h/(m_e·c) ≈ 2.426 × 10⁻¹² m is the Compton wavelength of the electron, m_e = 9.109 × 10⁻³¹ kg is the electron rest mass, and θ is the scattering angle (the angle between incident and scattered photon directions). The shift Δλ = λ_C(1 − cos θ) is at most 2λ_C ≈ 4.85 × 10⁻¹² m, occurring at backscattering (θ = 180°), and zero at forward scattering (θ = 0°). The formula assumes the electron is initially at rest and is treated as free (binding energy negligible compared to photon energy). Compton's 1923 discovery that X-ray photons lose energy after scattering off electrons, by exactly this formula, was a decisive demonstration that electromagnetic radiation behaves as particles with momentum p = h/λ — Einstein had hypothesised photons in 1905 but their corpuscular momentum was confirmed by Compton, who won the 1927 Nobel Prize. Edge cases: at θ = 90°, Δλ = λ_C (the canonical reference case); at θ = 180° (backscattering), Δλ = 2λ_C (maximum). For low-energy photons (λ ≫ λ_C, e.g. visible light), the shift is negligible relative to λ; for high-energy X-rays and γ-rays the shift dominates. The formula must be modified for bound electrons (atomic Compton profile) or relativistic electron motion (inverse Compton scattering).
How to use
Example 1 — X-ray photon scattered at 90°. λ = 1 × 10⁻¹² m (1 pm, an X-ray photon of energy ~1.24 MeV/c). θ = 90°. Step 1: cos(90°) = 0. Step 2: 1 − cos θ = 1. Step 3: Δλ = 2.426e−12 × 1 = 2.426e−12 m. Step 4: λ' = 1e−12 + 2.426e−12 = 3.426e−12 m. Verify: the scattered wavelength is 3.426 pm — Δλ = λ_C as expected for 90° scattering ✓. The corresponding scattered photon energy is hc/λ' ≈ 5.80e−14 J ≈ 362 keV, down from the incident ~1.24 MeV — the missing energy went into electron kinetic energy. Example 2 — γ-ray backscattering. λ = 0.5 × 10⁻¹² m, θ = 180° (fully backscattered). Step 1: cos(180°) = −1. Step 2: 1 − cos θ = 2. Step 3: Δλ = 2.426e−12 × 2 = 4.852e−12 m. Step 4: λ' = 0.5e−12 + 4.852e−12 = 5.352e−12 m. Verify: this is the maximum Compton shift (2·λ_C ≈ 4.85 pm). The incoming photon at 0.5 pm corresponds to E ≈ 2.48 MeV; after backscattering it has energy hc/5.352e−12 ≈ 3.71e−14 J ≈ 232 keV — losing nearly 91% of its energy to a single electron in one collision. This dramatic loss is why thick lead shielding is needed for γ-rays in medical and industrial settings ✓.
Frequently asked questions
Why does the Compton effect prove that photons have momentum?
Classical electromagnetism predicts Thomson scattering: a wave scattering elastically off a free electron, with no wavelength change. Compton's 1923 experiments showed X-rays scattered off graphite electrons did shift wavelength, by an angle-dependent amount that classical theory could not explain. Compton derived the observed shift by treating each X-ray as a particle (photon) with energy E = hν and momentum p = E/c = h/λ, then applying conservation of energy and momentum in a single photon-electron collision. The result λ' − λ = (h/m_e c)(1 − cos θ) matched experiment precisely. This was decisive evidence that photons carry momentum — Einstein had argued this in 1905 for the photoelectric effect, but Compton's experiment showed the momentum explicitly through scattering kinematics. The 1927 Nobel Prize cemented the photon's status as a real particle with both energy and momentum, paving the way for quantum electrodynamics.
What's special about the electron's Compton wavelength of 2.426 pm?
λ_C = h/(m_e·c) ≈ 2.4263 × 10⁻¹² m is a fundamental length scale associated with the electron, set by its rest mass and Planck's constant. Physically, it represents the wavelength at which a photon's energy equals the electron's rest-mass energy m_e c² ≈ 511 keV. Below this wavelength (higher energy), pair production becomes possible — γ-rays with E > 2 × m_e c² ≈ 1.022 MeV can spontaneously produce electron-positron pairs in the field of a nucleus. The reduced Compton wavelength ℏ/(m_e c) = λ_C/(2π) ≈ 386 fm appears in the Dirac equation and represents the characteristic scale over which an electron's wavefunction varies relativistically — Zitterbewegung. Each fundamental particle has its own Compton wavelength: proton ~1.32 fm; neutron similar; muon ~11.7 fm. These set the natural relativistic length scales in particle physics.
How does Compton scattering compare to photoelectric and pair production?
All three are mechanisms by which photons interact with matter, dominant at different energies. The photoelectric effect dominates at low photon energies (< 100 keV for most materials), where photons are fully absorbed by tightly bound electrons in atoms; the entire photon energy goes into ejecting the electron. Compton scattering dominates at intermediate energies (100 keV–10 MeV), where photons scatter off weakly bound or 'effectively free' electrons; the photon survives but is shifted in wavelength and direction. Pair production dominates at high energies (> 1.022 MeV, the threshold to create an electron-positron pair), where the photon converts to matter in the field of a nucleus. The cross-section for each varies by photon energy and atomic number Z: photoelectric ∝ Z⁴–Z⁵, Compton ∝ Z (approximately), pair production ∝ Z². In medical imaging X-rays, contrast comes from the Z-dependence of photoelectric absorption; in PET scans, pair production reversed (positron annihilation into two 511-keV γ-rays) is the source of the signal.
What are the common mistakes when applying the Compton formula?
The biggest mistake is using degrees in cos(θ) without converting to radians — Math.cos in most programming languages expects radians, and θ in degrees gives wrong results unless multiplied by π/180. The calculator handles this internally but pen-and-paper applications must convert manually. The second is treating bound electrons as free at low photon energies — when photon energy is comparable to the electron's binding energy (e.g., K-shell binding in heavy elements is tens of keV), the simple Compton formula breaks down and you need the bound-electron Compton profile. The third is forgetting the formula assumes the electron is initially at rest; if the electron has significant initial kinetic energy (e.g., in a plasma or relativistic beam) you need the full relativistic kinematics including inverse Compton scattering where high-energy electrons boost low-energy photons. People also mix up λ and λ', or apply the formula to particles other than electrons without substituting m_e with the right rest mass. Finally, the formula gives wavelength shift but doesn't directly tell you the recoil electron's kinetic energy or angle — those need momentum and energy conservation separately.
When should I not use this calculator?
Do not use it for low-energy (visible, UV) photons scattering off bound electrons, where Rayleigh or Mie scattering dominates and the photon-electron collision approximation fails. It is not appropriate for bound electrons when photon energy is comparable to binding energy (sub-keV X-rays in heavy elements); use bound-electron Compton profiles or full atomic calculations instead. Do not use it for inverse Compton scattering (high-energy electrons boosting low-energy photons, important in astrophysics and laser-Compton sources); that requires the relativistic generalisation. It is unsuitable for non-electron targets — the 2.426 pm Compton wavelength is electron-specific; substitute the target particle's m·c for other targets. Avoid the formula for highly bound K-shell electrons in heavy elements where the impulse approximation breaks down. For accurate astrophysical Compton-shift calculations (e.g., CMB photons through galaxy clusters via the Sunyaev-Zel'dovich effect), use the full relativistic treatment with electron temperature and statistical distributions. Finally, for high-Z material radiation shielding, total cross-sections integrate Compton, photoelectric, and pair production — a single Compton shift formula tells you nothing about overall attenuation.