Compton Scattering Formula
Compton scattering: delta-lambda = (h/mec)(1 - cos theta).
Calculate the wavelength shift of X-rays scattered by electrons in matter.
The Formula
The Compton scattering formula describes how the wavelength of a photon increases after it collides with and transfers energy to a free electron. American physicist Arthur Holly Compton discovered this effect experimentally in 1923, providing direct proof that photons carry momentum — a cornerstone result of quantum mechanics. Before this discovery, it was believed that light scattering could not change the wavelength of the incident radiation. Compton's work earned him the Nobel Prize in Physics in 1927.
In the formula, h is Planck's constant (6.626 × 10⁻³⁴ J·s), me is the rest mass of the electron (9.109 × 10⁻³¹ kg), c is the speed of light (2.998 × 10⁸ m/s), and θ is the angle at which the photon is scattered relative to its original direction. The combination h/(mec) is called the Compton wavelength of the electron, equal to approximately 2.426 × 10⁻¹² m (2.426 pm).
The wavelength shift Δλ depends only on the scattering angle θ — not on the energy or wavelength of the incident photon. When θ = 0° (no deflection), Δλ = 0 and the photon passes through unaffected. When θ = 90°, Δλ equals the full Compton wavelength (2.426 pm). When θ = 180° (photon bounced straight back), the shift is at maximum: Δλ = 2 × 2.426 pm = 4.852 pm. This angular dependence is uniquely characteristic of particle-particle collisions, confirming the particle nature of light.
Compton scattering is most significant for high-energy photons such as X-rays and gamma rays. For visible light, the wavelength shift is negligibly small compared to the wavelength itself, which is why the effect was not observed until high-energy sources were available. The formula is used in medical imaging, radiation physics, and astrophysics.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| Δλ | Wavelength shift of the scattered photon | m (or pm) |
| h | Planck's constant = 6.626 × 10⁻³⁴ | J·s |
| me | Rest mass of the electron = 9.109 × 10⁻³¹ | kg |
| c | Speed of light = 2.998 × 10⁸ | m/s |
| θ | Scattering angle between incident and scattered photon directions | degrees or radians |
Example 1
An X-ray photon is scattered at θ = 90° by a free electron. What is the wavelength shift?
Compton wavelength = h/(mec) = 2.426 pm
Δλ = 2.426 pm × (1 − cos 90°) = 2.426 pm × (1 − 0) = 2.426 pm
Wavelength shift = 2.426 pm (the full Compton wavelength)
Example 2
A gamma ray photon scatters at θ = 180° (backscatter). What is the maximum possible wavelength shift?
Δλ = 2.426 pm × (1 − cos 180°) = 2.426 pm × (1 − (−1)) = 2.426 pm × 2
Maximum wavelength shift = 4.852 pm (occurs only in direct backscatter)
When to Use It
The Compton scattering formula applies when:
- Analyzing X-ray or gamma ray interactions with matter in radiation physics
- Calculating energy loss of photons in medical imaging and radiation therapy planning
- Studying Compton telescopes used in high-energy astrophysics to detect gamma rays
- Verifying the particle nature of electromagnetic radiation in quantum mechanics coursework
- Designing radiation shielding where Compton scattering is the dominant interaction mechanism
- Interpreting experimental X-ray diffraction data at high energies where wavelength shifts matter