Compton Scattering Calculator - Photon Shift & Energy

A compton scattering calculator is the quickest way to find how much an X-ray or gamma-ray photon shifts in wavelength and loses in energy after it scatters off a free electron at any angle. It returns Δλ, scattered photon energy, scattered wavelength, electron recoil energy, and percent energy loss for any incident energy between 1 eV and 1 MeV.

• • Modern physics homework: Solve the standard problem set for a 20 keV X-ray at 80 degrees, where Δλ is about 2.005 pm and the photon loses about 3 percent of its energy.
• • X-ray spectroscopy check: Match a measured scattered Cu K-α line to the theoretical Compton shift to confirm that a spectrum contains inelastic scattering on top of elastic Rayleigh peaks.
• • Gamma-ray detector calibration: Estimate how much energy a back-scattered gamma photon drops into the recoil electron for sodium-iodide and germanium detector response models.
• • Medical physics context: See how Compton scattering dominates X-ray and gamma interactions in soft tissue between about 30 keV and 30 MeV.

The Compton effect describes what happens when a high-energy photon collides with a free or weakly bound electron. The photon gives up energy and momentum, so its wavelength grows and its direction changes.

This calculator covers the X-ray and gamma-ray range where Compton scattering dominates. Below about 30 keV the photoelectric effect takes over, and above about 1.022 MeV pair production becomes possible.

The interface fits the typical modern physics problem: given an incident photon energy and a scattering angle, find the wavelength shift, scattered photon energy, and recoil electron energy.

For the same kind of photon-electron problem in atomic physics, Bohr Model Calculator returns orbital radius, orbital energy, and the photon wavelength for any hydrogen-like transition.

The calculator combines the Compton wavelength shift equation with the relativistic photon energy formula, both derived from conservation of energy and momentum in a photon-electron elastic collision. NIST CODATA sets the numerical scale of every output.

• photonEnergyKeV: Incident photon energy E in kiloelectronvolts, set by the user.
• scatteringAngleDeg: Scattering angle θ in degrees. The calculator converts to radians internally.
• Wavelength shift Δλ: Output in picometers, equal to the electron Compton wavelength (2.4263 pm) times (1 − cos θ).
• Scattered photon energy E': Output in keV, equal to E divided by 1 + (E / 510.99895 keV) times (1 − cos θ).
• Compton wavelength λ_C: Reference constant h / (m_e c) = 2.4263 pm, surfaced on every result.

Every output is a direct evaluation of the Compton formula with the same NIST constants a textbook would cite, and the picometer and kiloelectronvolt values agree via E = hc / λ.

At 0 degrees the calculator reports Δλ = 0 and the photon energy unchanged. At 180 degrees the wavelength shift equals twice the Compton wavelength (about 4.8526 pm).

According to NIST CODATA 2018 - Electron Compton wavelength, the electron Compton wavelength λ_C = h / (m_e c) = 2.42631023867 × 10^-12 m

According to Omni Calculator - Compton scattering, an 80 degree scatter on a free electron shifts the photon wavelength by about 2.005 pm

When the scattered photon energy needs to be converted from electronvolts into joules per photon or per mole, Work-Energy-Power Calculator handles the energy and power side of the same intro-physics toolkit.

Four ideas drive every output: a fixed Compton wavelength scale, angle-dependent shift, conservation of energy and momentum, and the X-ray band where the compton scattering calculator effect dominates.

The 1 − cos θ factor captures the geometry, the Compton wavelength sets the scale, the conservation laws justify the result, and the energy band explains where the effect shows up.

Because Compton scattering is taught in the same modern-physics unit as relativity, Gravitational Time Dilation Calculator covers the static gravity side of the same unit so the relativistic kinematics piece sits next to the photon-electron collision.

The same R_∞ constant that sets the Compton wavelength scale also appears in the Rydberg formula for hydrogen spectral lines, which is why the Rydberg Equation Calculator is the natural next step after reviewing photon-electron scattering.

Enter the incident photon energy and the scattering angle, then read the seven outputs in the result panel. The page recalculates as you type.

1. 1 Pick the photon energy: Type the incident X-ray or gamma-ray energy in keV. Soft X-rays sit around 0.5 to 10 keV, hard X-rays around 10 to 100 keV, and gamma rays above 100 keV.
2. 2 Set the scattering angle: Enter the angle between the scattered and incident photon directions in degrees. 0 is no collision, 90 is the textbook Compton wavelength shift, and 180 is direct back scatter.
3. 3 Read the wavelength shift: The first output is Δλ in picometers. For a 90 degree scatter it will always read about 2.4263 pm.
4. 4 Check the scattered photon energy: The second output is E', the photon energy after scattering. For soft X-rays it will be very close to E, and for hard X-rays or gamma rays the difference becomes meaningful.
5. 5 Use the recoil and percent outputs: The electron recoil energy ΔE and the percent energy loss show how much energy is transferred in a single collision. Compare these to the photon energy to judge whether a Compton event dominates the photon transport.
6. 6 Verify against the reference constant: The Compton wavelength constant is shown on the right side of the result panel so you can sanity-check any Δλ value against the textbook λ_C = 2.4263 pm.

For a standard 20 keV X-ray problem, leave the photon energy at 20 and the angle at 80, then read Δλ ≈ 2.0050 pm and scattered photon energy ≈ 19.3734 keV. Flip the angle to 90 to see the canonical Compton wavelength shift of 2.4263 pm at the same energy.

When the photon-electron collision is replaced by a classic mechanics problem involving mass, force, and acceleration, Forces & Newton's Laws Calculator handles the F = ma view of the same intro-physics course.

The calculator is most useful when an X-ray, gamma-ray, or modern physics problem asks for a Compton shift and you want the numerical answer in seconds.

• • Direct Compton formulas: Uses the standard wavelength shift and the relativistic photon energy formula with NIST CODATA constants, so the answer matches the textbook value to four or more significant digits.
• • Seven outputs in one place: Returns Δλ, scattered photon energy, initial wavelength, scattered wavelength, electron recoil energy, percent energy loss, and the Compton wavelength reference constant from a single form submission.
• • Wide energy range: Handles soft X-rays from about 1 keV up to gamma rays near 1 MeV, which is the band where Compton scattering is the dominant photon-matter interaction in light elements.
• • Limits built in: 0 degrees returns Δλ = 0 and 180 degrees returns the maximum 2λ_C shift, so the canonical Compton edge cases are visible without separate calculations.
• • Cross-validation friendly: Defaults are set to the Omni Calculator 80 degree worked example (20 keV, θ = 80), so the result panel can be checked against the reference value of Δλ = 2.005 pm in seconds.

For photon-electron collisions the Compton numbers are the right ones to quote. For photoelectric or pair-production events, pair the result with a cross-section table.

Two input factors change every output, and three physical constraints decide when the Compton formula is the right model.

• • Compton scattering assumes a free or weakly bound electron. For inner-shell electrons in high-Z materials the binding energy is not negligible.
• • The free-electron Compton formula ignores Doppler broadening from the electron momentum distribution. In warm or hot targets the observed Compton peak is broadened by the initial electron momentum.
• • Above about 1.022 MeV the photon has enough energy to produce an electron-positron pair, and below about 30 keV in high-Z materials the photoelectric effect dominates. Compton scattering is the leading interaction in the band between roughly 30 keV and a few MeV.

For textbook problems at 20 to 100 keV the free-electron formula is exact to NIST CODATA precision, and the calculator agrees with a hand calculation to four or more significant digits.

According to NIST CODATA 2018 - Electron mass and energy, the electron rest energy m_e c² = 8.1871057769 × 10^-14 J ≈ 510.99895 keV

To turn the scattered photon energy into a recoil electron velocity using momentum conservation, Kinematics Motion Calculator covers displacement, velocity, and acceleration problems on the same physics worksheet.