Compton scattering

Calculate how a photon transfers energy and momentum to a free electron initially at rest. Compare wavelengths and move the angle probe to explore Compton scattering.

Use

Enter an incident wavelength λ from 1 to 200 pm and a photon scattering angle θ from 0 to 180°, then select Calculate. After calculation, the angle field and slider move the same probe through both curves and the momentum diagram. An invalid angle clears current values and the probe. A wavelength or preset change requires a new calculation. Reset result keeps the parameters.

The Mo Kα and Cu Kα presets are rounded examples at 71 and 154 pm; the other presets are 10, 5 and 200 pm. The exact field accepts decimal values and scientific notation; the slider changes the angle in 0.1° steps. Navigation within the app retains the result. Reloading restores only validated parameters and the selected view.

Wavelength and energy

With λC = h/(mₑc), δ = 1 − cos θ = 2 sin²(θ/2), the shift is Δλ = λCδ and λ′ = λ + Δλ. The incident photon energy is E₀ = hc/λ. The energy ratio q = E′/E₀ = λ/λ′ gives E′ = qE₀ and electron kinetic energy T = E₀Δλ/λ′. Wavelengths are in pm and energies in keV. The stable half-angle and transfer formulas retain very small nonzero values.

At 0°, Δλ and T are zero and q is one. At 180°, Δλ = 2λC. The shift depends only on θ; energies and the angular factor also depend on λ. For λ = 71 pm and θ = 30°, λ′ is approximately 71.325 pm.

Momentum diagram

All arrows start at the same origin and use equal horizontal and vertical scales in units of p₀ = E₀/c. The incident photon is (1, 0), the scattered photon is (q cos θ, q sin θ), and the electron is (1 − q cos θ, −q sin θ). Their vector sum conserves momentum. The electron x component is evaluated as δ(1 + λC/λ)/(1 + (λC/λ)δ) for numerical stability. These are momentum vectors, not spatial paths or a time sequence.

The electron angle is atan2(pₑ,y, pₑ,x), measured from the positive x axis: negative for a photon scattered above that axis. At 180° it is 0°. At 0° the electron has no recoil momentum and no defined recoil direction, so its arrow is absent.

Angular factor

For unpolarized incident light, F(θ) = ½q²(q + 1/q − sin²θ) is the relative differential Klein–Nishina cross section per solid angle, normalized to its forward value F(0) = 1. F is not a normalized probability over θ, dσ/dθ, an absolute cross section, or a measured intensity. A distribution over angle intervals would also require the solid-angle element. This tool does not generate random scattering samples.

Constants and limits

The model uses the exact SI values h = 6.62607015 × 10⁻³⁴ J s, c = 299792458 m/s and e = 1.602176634 × 10⁻¹⁹ C, with the CODATA 2022 central value mₑ = 9.1093837139 × 10⁻³¹ kg. Both λC and mₑc² are derived consistently from these values. Displayed results are rounded; no uncertainty propagation is performed.

This is a single free-electron model. Atomic binding, Doppler broadening, material response and detector count rates are outside its scope.

References

• OpenStax University Physics 3, §6.3: photon momentum, conservation laws and Compton shift.
• Geant4 Physics Reference Manual 11.4: Klein–Nishina energy ratio and differential cross section; F here is an algebraic conversion to solid angle divided by its forward value, not Geant4's sampler or atomic fit.
• NIST SP 961, CODATA 2022: physical constants and the rounded Mo/Cu examples.