Compton Scattering Calculator
Calculates the scattered photon energy after a Compton scattering event given the incident photon energy, scattering angle, and target particle mass. Use it when analyzing X-ray or gamma-ray interactions with matter.
About this calculator
Compton scattering occurs when a high-energy photon collides with a nearly free particle (usually an electron) and transfers part of its energy, emerging at a longer wavelength. The scattered photon energy is E' = E₀ / [1 + (E₀/mc²)(1 − cos θ)], where E₀ is the initial photon energy in joules, m is the target particle rest mass in kg, c = 2.998×10⁸ m/s, and θ is the scattering angle in degrees. The wavelength shift is described by the Compton formula Δλ = (h/mc)(1 − cos θ), where h/m_e c ≈ 2.426×10⁻¹² m is the electron Compton wavelength. Maximum energy transfer occurs at θ = 180° (backscattering), while at θ = 0° the photon passes unaffected. The recoil kinetic energy of the target particle equals E₀ − E', conserving total energy and momentum.
How to use
Suppose a 100 keV X-ray photon (E₀ = 1.602×10⁻¹⁴ J) scatters off a free electron (m = 9.109×10⁻³¹ kg) at θ = 90°. Step 1: Compute mc² = 9.109×10⁻³¹ × (2.998×10⁸)² = 8.187×10⁻¹⁴ J. Step 2: E₀/mc² = 1.602×10⁻¹⁴ / 8.187×10⁻¹⁴ ≈ 0.1956. Step 3: (1 − cos 90°) = 1. Step 4: E' = 1.602×10⁻¹⁴ / (1 + 0.1956 × 1) = 1.602×10⁻¹⁴ / 1.1956 ≈ 1.340×10⁻¹⁴ J ≈ 83.6 keV. The electron recoils with 100 − 83.6 = 16.4 keV.
Frequently asked questions
What is the physical significance of the Compton wavelength shift?
The Compton wavelength shift Δλ = (h/mc)(1 − cos θ) shows that the change in photon wavelength depends only on the scattering angle and the rest mass of the target, not on the initial photon energy. This was revolutionary evidence that photons carry quantized momentum, confirming the particle nature of light. A larger shift means more energy transferred to the recoiling particle. For electrons the maximum shift (at 180°) is 2 × 2.426 pm = 4.852 pm, which is detectable in X-ray diffraction experiments.
How does the scattering angle affect the energy of the scattered photon?
At θ = 0° (forward scattering) the photon loses no energy because cos(0°) = 1, making the denominator equal to 1. As θ increases toward 90° the photon loses a moderate fraction of its energy, and at θ = 180° (backscattering) the energy loss is maximized. For a 100 keV photon scattering off an electron, the backscattered photon retains only about 20 keV. This angular dependence is exploited in Compton telescopes used in gamma-ray astronomy to reconstruct the direction of incoming photons.
Why does Compton scattering require a high-energy photon to be observable?
Compton scattering is significant only when the photon energy E₀ is comparable to or greater than the rest-mass energy mc² of the target particle (511 keV for electrons). At low photon energies (visible light, soft UV) the ratio E₀/mc² ≪ 1, so the wavelength shift is negligibly small and Thomson scattering (elastic, classical) dominates instead. X-rays and gamma rays have energies in the keV–MeV range where the shift becomes measurable. This is why Compton scattering is important in medical imaging (CT scatter artifacts), radiation shielding design, and high-energy astrophysics.