Compton Scattering Calculator – Wavelength Shift & Energy
Calculate Compton scattering wavelength shift, scattered photon energy, and electron recoil energy from photon energy and scattering angle.
Enter the incident photon energy and scattering angle. Select the energy unit (keV or MeV). The calculator computes the wavelength shift, scattered photon energy, and energy transferred to the electron.
Compton Scattering Calculator – Wavelength Shift & Energy
Calculate Compton scattering wavelength shift, scattered photon energy, and electron recoil energy from photon energy and scattering angle.
About the Compton Scattering Calculator
Compton scattering is a fundamental quantum mechanical phenomenon in which a photon — typically an X-ray or gamma ray — collides with a free or loosely bound electron and transfers part of its energy to the electron. As a result, the scattered photon emerges with a longer wavelength (lower energy) than the incident photon, while the electron recoils with the transferred kinetic energy. The discovery of this effect by American physicist Arthur H. Compton in 1923 provided crucial experimental evidence for the particle-like nature of light and earned him the Nobel Prize in Physics in 1927.
The wavelength shift produced by Compton scattering is given by the Compton formula: Δλ = λ_c(1 − cosθ), where Δλ is the change in wavelength, λ_c = h/(m_e c) = 2.42631 pm is the Compton wavelength of the electron, and θ is the scattering angle between the incident and scattered photon directions. The scattered photon energy is: E′ = E₀ / [1 + (E₀/m_e c²)(1 − cosθ)], where E₀ is the incident photon energy and m_e c² = 511 keV is the electron rest energy. The kinetic energy transferred to the electron equals E₀ − E′.
The scattering angle governs how much energy is transferred. At θ = 0° (forward scattering), no energy transfer occurs and the photon passes through unaffected. At θ = 90°, partial energy transfer occurs and the wavelength shift equals exactly one Compton wavelength (2.426 pm). At θ = 180° (backscattering), the maximum possible energy transfer occurs and the wavelength shift equals 2λ_c = 4.853 pm.
Compton scattering has extensive applications across science and medicine. In diagnostic radiology and computed tomography (CT), Compton scattering is the dominant interaction mechanism for X-ray photons in the diagnostic energy range (30–150 keV), contributing to image noise and scatter artefacts. In nuclear medicine and positron emission tomography (PET), understanding Compton interactions is essential for accurate image reconstruction. Compton cameras exploit the scattering geometry to determine the direction of incoming gamma rays without physical collimators, enabling new approaches to gamma-ray imaging.
In radiation physics and shielding design, Compton scattering dominates over photoelectric absorption and pair production in the intermediate energy range (around 100 keV to 10 MeV for common materials). Astrophysicists study Compton scattering in cosmic X-ray sources, and inverse Compton scattering — where energetic electrons boost photon energies — is responsible for some of the highest-energy radiation observed in the universe.
Compton Scattering Examples
Typical photon energies and scattering angles showing wavelength shift and energy transfer.
| Photon Energy & Angle | Wavelength Shift / Scattered Energy | Application |
|---|---|---|
| E₀ = 100 keV, θ = 90° | Δλ = 2.426 pm, E′ ≈ 83.6 keV | Typical diagnostic X-ray energy; about 16 keV transferred to the recoil electron. |
| E₀ = 662 keV, θ = 180° | Δλ = 4.853 pm, E′ ≈ 184 keV | Cs-137 gamma ray backscatter — maximum energy transfer, electron receives ~478 keV. |
| E₀ = 1.17 MeV, θ = 90° | Δλ = 2.426 pm, E′ ≈ 0.356 MeV | Co-60 gamma ray; large energy transfer (~0.814 MeV to electron) because photon energy >> electron rest energy (0.511 MeV). |
| E₀ = 511 keV, θ = 90° | Δλ = 2.426 pm, E′ ≈ 255.5 keV | Positron annihilation photon; exactly half the energy transferred to the electron at 90°. |
How to Use the Compton Scattering Calculator
- Select the energy unit — keV (kiloelectron volts) for X-rays and lower-energy gamma rays, or MeV (megaelectron volts) for high-energy gamma radiation.
- Enter the incident photon energy. Typical X-ray energies are 30–150 keV; typical gamma ray energies are 100 keV to several MeV.
- Enter the scattering angle θ in degrees (0° = forward scattering, 90° = perpendicular, 180° = backscattering).
- Click Calculate. The tool computes the wavelength shift Δλ = λ_c(1 − cosθ), scattered photon energy E′, and energy transferred to the recoil electron.
- Use the example buttons to load common scenarios: medical X-ray at 90°, Cs-137 backscatter, or Co-60 gamma at 90°.
Compton Scattering FAQ
What is Compton scattering?
Compton scattering is the inelastic scattering of a photon by a free or loosely bound electron. The photon transfers part of its energy and momentum to the electron, emerging with a longer wavelength. This quantum effect demonstrates the particle nature of light and is described by the Compton formula: Δλ = (h/m_e c)(1 − cosθ). It was discovered by Arthur Compton in 1923 and is a cornerstone of quantum mechanics.
What is the Compton wavelength?
The Compton wavelength of the electron (λ_c) is the fundamental length scale of Compton scattering: λ_c = h/(m_e c) = 2.42631 × 10⁻¹² m = 2.42631 pm, where h is Planck's constant, m_e is the electron mass, and c is the speed of light. It sets the maximum possible wavelength shift per interaction: the largest shift is 2λ_c = 4.853 pm at 180° backscattering. At this length scale, quantum mechanical effects become dominant over classical wave optics.
At what angle is the wavelength shift maximum?
The wavelength shift is maximum at θ = 180° (backscattering), where Δλ = 2λ_c = 4.853 pm and energy transfer to the electron is greatest. At θ = 0° (forward scattering), Δλ = 0 and no energy is transferred. At θ = 90°, Δλ = λ_c = 2.426 pm, which is an important benchmark value. The formula Δλ = λ_c(1 − cosθ) makes these relationships explicit.
How does Compton scattering differ from the photoelectric effect?
In the photoelectric effect, a photon is completely absorbed by an atom, ejecting a bound electron with kinetic energy equal to hν − φ (where φ is the work function). In Compton scattering, the photon is not absorbed but redirected, losing only part of its energy to a recoil electron. The photoelectric effect dominates at low photon energies (below ~100 keV), Compton scattering dominates at intermediate energies (~100 keV to ~10 MeV), and pair production dominates above ~1.02 MeV.
What is inverse Compton scattering?
Inverse Compton scattering occurs when a high-energy electron collides with a low-energy photon and boosts the photon to much higher energy. This is the time-reverse of normal Compton scattering. In astrophysics, relativistic electrons in cosmic sources boost microwave background photons up to X-ray or gamma-ray energies. The Sunyaev-Zeldovich effect in galaxy clusters is a well-known example, and inverse Compton cooling of electron populations is important in many high-energy astrophysical environments.
Why is Compton scattering important in radiation therapy?
In radiation therapy, megavoltage X-ray beams (4–25 MeV) interact with tissue predominantly through Compton scattering. This energy range was deliberately chosen because Compton interactions do not depend on atomic number, meaning bone and soft tissue receive similar doses per unit mass. Treatment planning systems must accurately model Compton scattering to calculate dose distributions and protect healthy tissue surrounding the treatment volume.