Compton Scattering Calculator

Understanding the Compton Effect

When electromagnetic radiation strikes a free or loosely bound electron, the interaction follows the laws of elastic collision. The incident photon carries momentum and energy; the struck electron recoils, carrying away some of that energy. Conservation of momentum and energy govern the outcome.

The scattered photon emerges with a longer wavelength (lower frequency and energy) than the incident photon. The degree of wavelength shift depends on two factors:

• Scattering angle — the angle between the incident and scattered photon directions
• Particle mass — more massive particles cause smaller wavelength shifts

This effect only manifests significantly at high-energy photons (X-rays and gamma rays) interacting with electrons. At lower photon energies, Compton scattering becomes negligible compared to other interaction mechanisms like the photoelectric effect.

The Compton Shift Equation

The wavelength extension—the difference between scattered and incident photon wavelengths—is given by the Compton scattering formula:

• Δλ — Wavelength shift (change in wavelength), in meters
• h — Planck constant = 6.626 × 10⁻³⁴ J·s
• m — Rest mass of the scattering particle, in kilograms
• c — Speed of light = 2.998 × 10⁸ m/s
• θ — Scattering angle, in degrees or radians

Compton Wavelength and Physical Interpretation

The quantity h / (m·c) is known as the Compton wavelength of a particle. For an electron, this equals approximately 2.426 × 10⁻¹² m or 2.426 picometres (pm).

The wavelength shift is zero when θ = 0° (photon scatters in the same direction—no actual collision). Maximum shift occurs at θ = 180° (backscattering), where the factor (1 − cos(θ)) equals 2.

The shift is independent of the incident photon's energy or wavelength. Whether you scatter an X-ray or a gamma ray, the shift for a given particle and angle remains the same. This universality was historically crucial in confirming the particle nature of light.

Practical Example: Electron Scattering

Consider a photon scattering off a free electron at θ = 80°.

• Electron rest mass: m = 9.109 × 10⁻³¹ kg
• cos(80°) ≈ 0.174
• Factor (1 − cos(80°)) ≈ 0.826
• Compton wavelength of electron: λ_c = 2.426 pm
• Wavelength shift: Δλ = 2.426 pm × 0.826 ≈ 2.00 pm

If the incident photon had a wavelength of 0.5 pm (hard X-ray), the scattered photon would have a wavelength of approximately 2.50 pm—a substantial change that would be measured in diffraction experiments.

Key Considerations and Pitfalls

When working with Compton scattering calculations, keep these practical points in mind:

1. Scattering angle precision matters — Small errors in angle measurement cause noticeable changes in wavelength shift, especially near 90°. At 90°, the cosine is zero, making the formula particularly sensitive. Use decimal degrees or radians carefully.
2. Valid only for free or quasi-free particles — Compton scattering formulas assume the target particle is either free or weakly bound in atoms. Tightly bound electrons (inner shells) behave differently; the entire atom recoils instead, producing negligible wavelength shift.
3. Non-relativistic approximation sufficient here — The formula presented does not include relativistic corrections. For most practical cases with electrons and reasonable photon energies (MeV range), this formula is accurate. At extremely high energies, full quantum field theory becomes necessary.
4. Wavelength shift is identical across the spectrum — Remember that the shift depends only on angle and particle mass, never on photon energy. A 1 keV and a 100 keV photon scattered at the same angle off an electron produce the same wavelength change in absolute terms, though the percentage change differs.