2.4263
pm
0.102426
nm
12.3984
keV
12.1047
keV
0.2937
keV
2.369
%
2.4263
pm
2.4263
pm
0.102426
nm
12.3984
keV
12.1047
keV
0.2937
keV
2.369
%
2.4263
pm
Compton scattering is the inelastic scattering of a photon by a charged particle, typically an electron, resulting in a decrease in the photon's energy and an increase in its wavelength. Discovered by Arthur Holly Compton in 1923, this effect provided definitive proof that photons carry momentum and behave as particles in collisions. The Compton Scattering Calculator computes the wavelength shift, scattered photon wavelength and energy, and the kinetic energy of the recoil electron for any incident wavelength and scattering angle. The wavelength shift depends only on the scattering angle and the Compton wavelength of the electron (λc = 2.426 pm), not on the incident photon energy. This effect is significant for X-rays and gamma rays where photon wavelengths are comparable to the Compton wavelength. Compton scattering is fundamental to medical imaging (CT scans), radiation therapy, astrophysics (inverse Compton scattering in active galaxies), and gamma-ray spectroscopy.
The Compton scattering formula relates the wavelength shift to the scattering angle:
$$\Delta\lambda = \lambda' - \lambda = \frac{h}{m_e c}(1 - \cos\theta)$$
where λ is the incident wavelength, λ' is the scattered wavelength, θ is the scattering angle, and h/(mec) = 2.426 pm is the Compton wavelength of the electron.
Key features of the formula:
$$\Delta\lambda_{\max} = \frac{2h}{m_e c} = 4.853 \text{ pm (at } \theta = 180°\text{, backscattering)}$$
$$\Delta\lambda = 0 \text{ (at } \theta = 0°\text{, forward scattering)}$$
$$\Delta\lambda = \frac{h}{m_e c} = 2.426 \text{ pm (at } \theta = 90°\text{)}$$
Energy conservation gives the recoil electron kinetic energy:
$$KE_e = E_{\gamma} - E'_{\gamma} = \frac{hc}{\lambda} - \frac{hc}{\lambda'}$$
This is derived by applying conservation of energy and momentum to the photon-electron collision, treating the photon as a particle with momentum p = h/λ.
The wavelength shift is always positive (longer wavelength = lower energy after scattering) and depends only on angle, not incident energy. At θ = 0° (forward), Δλ = 0 (no energy transfer). At θ = 90°, Δλ = 2.426 pm. At θ = 180° (backscatter), Δλ = 4.853 pm (maximum shift). The fractional energy loss is largest for high-energy photons: a 1 MeV gamma ray loses 80% of its energy in backscattering, while a 10 keV X-ray loses only 5%. The recoil electron KE equals the energy difference between incident and scattered photons. For low-energy photons (λ >> λ_c), Compton scattering becomes negligible and Thomson scattering (elastic) dominates. For very high-energy photons, pair production becomes the dominant interaction.
Inputs
Results
A 12.4 keV X-ray scattered at 90° shifts by 2.43 pm. The photon retains most energy (12.1 keV), transferring only 0.29 keV to the electron.
Inputs
Results
A 1.24 MeV gamma ray backscattered at 180° transfers 1028 keV (83%) to the electron. The wavelength shift is the maximum 4.85 pm, but the fractional energy transfer is dramatic.
Compton scattering is the inelastic collision between a photon and a free (or loosely bound) electron. The photon transfers some energy and momentum to the electron, emerging with a longer wavelength (lower energy) at an angle to its original direction.
λ_c = h/(m_e·c) = 2.426 pm is the Compton wavelength of the electron. It sets the scale of the wavelength shift. When the photon wavelength is comparable to λ_c, Compton scattering is most significant. Each particle has its own Compton wavelength inversely proportional to its mass.
The shift Δλ = (h/m_ec)(1−cosθ) is purely geometric—it depends only on the collision angle. However, the energy transfer does depend on incident energy because the same Δλ represents a larger fractional change for shorter-wavelength (higher-energy) photons.
Maximum energy transfer occurs at θ = 180° (backscattering). For a photon with energy E_γ, the maximum energy transferred to the electron is: KE_max = E_γ × 2E_γ/(m_ec² + 2E_γ), where m_ec² = 511 keV.
Classical wave theory predicted no wavelength change in scattering (Thomson scattering). Compton showed the wavelength shift matches exactly the prediction from treating the photon as a particle with momentum p = h/λ in a billiard-ball-like collision. This confirmed the particle nature of light.
When a low-energy photon collides with a high-energy electron, the photon gains energy. This is important in astrophysics: cosmic microwave background photons scattered by hot gas in galaxy clusters gain energy (Sunyaev-Zel'dovich effect).
In a gamma-ray detector, the Compton edge is the maximum energy deposited by a single Compton-scattered electron (at θ = 180°). It appears as a sharp cutoff in the measured energy spectrum below the photopeak.
In CT scans and X-ray imaging, Compton scattering is the dominant interaction for photon energies above ~25 keV in tissue. Scattered photons degrade image quality, so anti-scatter grids are used. Understanding Compton scattering is essential for accurate dose calculations in radiation therapy.
At θ = 0° (forward scattering), Δλ = 0—no energy is transferred and the photon continues undeviated. This is the limiting case of no interaction. As θ increases, more energy is transferred to the electron.
Thomson scattering is the low-energy limit of Compton scattering where the photon energy is much less than m_ec² (511 keV). In Thomson scattering, no wavelength shift occurs (elastic), and the cross-section is constant. Compton scattering shows inelastic behavior with wavelength shift and reduced cross-section at high energies.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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