About

Compton scattering describes the inelastic collision between a photon and a free electron, where the photon loses energy and shifts to a longer wavelength. The wavelength shift Δλ depends solely on the scattering angle θ and the Compton wavelength of the electron hm_ec ≈ 2.426 pm. Miscalculating the scattered photon energy leads to incorrect calibration of gamma-ray detectors, flawed medical imaging dosimetry, and erroneous material analysis in X-ray fluorescence spectroscopy. This calculator implements the exact relativistic Compton formula alongside the Klein-Nishina differential cross-section from quantum electrodynamics. It assumes a free electron at rest. Binding energy corrections matter below 10 keV for high-Z materials.

The Klein-Nishina formula replaces the classical Thomson cross-section at photon energies approaching m_ec² ≈ 511 keV. At low energies it reduces to Thomson scattering. At high energies the forward-scattering peak becomes dominant. Pro tip: the Compton edge (backscatter at θ = 180°) defines the maximum electron recoil energy, critical for interpreting gamma-ray spectra in scintillation detectors.

Formulas

The Compton wavelength shift relates the change in photon wavelength to the scattering angle:

Δλ = λ′ − λ = hm_ec(1 − cos θ)

The scattered photon energy in terms of incident energy:

E′ = E1 + Em_ec²(1 − cos θ)

The recoil electron kinetic energy:

K_e = E − E′

The Klein-Nishina differential cross-section per unit solid angle:

dσdΩ = r₀²2 (E′E)² (EE′ + E′E − sin² θ)

Where E = incident photon energy, E′ = scattered photon energy, θ = scattering angle, m_ec² = electron rest mass energy (510.999 keV), r₀ = classical electron radius (2.818 × 10⁻¹⁵ m), h = Planck constant, K_e = electron recoil kinetic energy, Δλ = wavelength shift, λ_C = Compton wavelength of the electron.

Reference Data

Constant / Parameter	Symbol	Value	Unit
Electron rest mass energy	m_ec²	510.999	keV
Compton wavelength of electron	λ_C	2.42631 × 10⁻¹²	m
Planck constant	h	6.62607 × 10⁻³⁴	J⋅s
Speed of light	c	2.99792 × 10⁸	m/s
Classical electron radius	r₀	2.81794 × 10⁻¹⁵	m
Thomson cross-section	σ_T	6.65246 × 10⁻²⁹	m²
Common X-ray: Mo Kα	-	17.48	keV
Common X-ray: Cu Kα	-	8.04	keV
Annihilation photon	-	511	keV
Cs-137 gamma	-	661.7	keV
Co-60 gamma (line 1)	-	1173.2	keV
Co-60 gamma (line 2)	-	1332.5	keV
Na-22 gamma	-	1274.5	keV
Compton edge (Cs-137)	K_e,max	477.3	keV
Compton edge (Co-60, 1332 keV)	K_e,max	1118.1	keV
Backscatter energy (Cs-137)	E′	184.3	keV
Energy ratio threshold	γ = 1	511	keV
Pair production threshold	-	1022	keV
Max wavelength shift	2λ_C	4.853 × 10⁻¹²	m

Frequently Asked Questions

The shift Δλ = (h / mₑc)(1 − cos θ) is purely geometric. It depends only on the scattering angle θ and fundamental constants. However, the fractional energy loss E − E' does depend on incident energy. High-energy photons lose a larger fraction of their energy at the same angle because the dimensionless ratio γ = E / mₑc² determines how relativistic the interaction is.

The Thomson cross-section is the low-energy limit (γ → 0). Deviations exceed 10% when the photon energy surpasses roughly 50 keV (γ ≈ 0.1). At 511 keV (γ = 1), the total Klein-Nishina cross-section is about 60% of the Thomson value. At 10 MeV, it drops to roughly 10%. For energies below 10 keV, Thomson scattering is an adequate approximation.

This calculator assumes a free electron at rest. For tightly bound inner-shell electrons in high-Z materials, the binding energy can be a significant fraction of low-energy incident photons (below ~10 keV). In such cases, the impulse approximation or full atomic form-factor corrections are needed. For photon energies above 100 keV interacting with low-Z materials, the free-electron assumption is excellent.

The Compton edge is the maximum kinetic energy an electron can receive, occurring at θ = 180° (full backscatter). For Cs-137 at 661.7 keV, the Compton edge is at 477.3 keV. In a scintillation detector spectrum, this appears as a sharp cutoff in the continuous Compton distribution below the photopeak. Misidentifying it leads to incorrect isotope identification.

Yes, but the wavelength shift scales inversely with particle mass: Δλ = (h / mc)(1 − cos θ). For a proton, the Compton wavelength is 1836 times smaller than for an electron (≈ 1.321 fm). This makes the shift negligible at typical photon energies. The effect becomes measurable only with very high-energy gamma rays (hundreds of MeV).

At low energies (γ ≪ 1), the Klein-Nishina formula reduces to the Thomson result, which is symmetric about 90° with equal forward and backward scattering. As photon energy increases, the distribution becomes increasingly forward-peaked. At 1 MeV, forward scattering (θ < 30°) dominates. This has practical consequences for radiation shielding design, where backscatter assumptions from Thomson theory underestimate forward transmission.