Compton Scattering Calculator
Enter the incident photon energy and scattering angle to instantly find the scattered photon energy, wavelength shift, recoil electron kinetic energy and recoil angle, energy loss fraction, and the differential Klein-Nishina cross section. You can also solve from wavelength instead of energy. Results update in real time with a full show-your-work panel.
Formula
Worked example
A 100 keV X-ray photon scatters off a free electron at 90 degrees. The electron Compton wavelength is 2.42631 pm. Wavelength shift: 2.42631 * (1 - cos 90) = 2.42631 pm. Incident wavelength: hc/100 keV = 12.398 pm. Scattered wavelength: 12.398 + 2.426 = 14.824 pm. Scattered photon energy: hc/14.824 pm = 66.98 keV. Recoil electron energy: 100 - 66.98 = 33.02 keV (33% of the original energy transferred).
What is Compton scattering?
Compton scattering is the inelastic scattering of a photon by a charged particle, most commonly a free electron. When a high-energy photon (an X-ray or gamma ray) collides with an electron, it transfers some of its energy and momentum to the electron. The photon bounces off at a new angle with a longer wavelength and lower energy, while the electron recoils with the energy difference. Arthur Holly Compton discovered and explained this effect in 1923, earning the Nobel Prize in Physics in 1927. The experiment was pivotal because it confirmed the particle nature of light: classical wave theory predicted no wavelength change, but the observed shift matched perfectly with treating the photon as a particle obeying relativistic conservation of energy and momentum.
How to use this calculator
Select whether you want to specify the incident photon by energy (keV) or wavelength (pm), enter your value, and choose the scattering angle between 0 and 180 degrees. You can also select the scattering particle: the default is a free electron (standard Compton scattering), but you can choose a proton or enter any custom rest-mass energy to explore inverse Compton or nuclear scattering. All outputs update instantly: wavelength shift, scattered photon energy, recoil electron kinetic energy and angle, energy loss fraction, and the Klein-Nishina differential cross section. The energy chart shows how scattered and electron energies vary across all angles for your photon.
The Compton scattering formula and its derivation
The Compton wavelength shift formula is Delta-lambda = (h/mc)(1 - cos theta), where h is Planck's constant (6.626 x 10^-34 J*s), m is the rest mass of the scattering particle, c is the speed of light, and theta is the scattering angle. The factor h/(mc) is called the Compton wavelength of the particle and equals 2.42631 pm for an electron. It is derived by applying conservation of relativistic energy and momentum to the two-body photon-electron collision. The scattered photon energy follows directly: E_1 = E_0 / [1 + (E_0/mc^2)(1 - cos theta)]. The recoil electron kinetic energy is simply T_e = E_0 - E_1. The electron recoil angle phi is found from cot(phi) = (1 + E_0/mc^2) tan(theta/2).
Klein-Nishina cross section and angular distribution
The probability that a photon scatters at a given angle is described by the Klein-Nishina formula, the first quantum-relativistic scattering formula ever derived (Klein and Nishina, 1929). The differential cross section is dSigma/dOmega = (r_e^2/2) P^2 [P + 1/P - sin^2(theta)], where r_e = 2.818 fm is the classical electron radius and P = E_1/E_0 is the ratio of scattered to incident energy. At low photon energies (E_0 much less than mc^2) this reduces to the classical Thomson cross section, which is symmetric front-to-back. As photon energy increases, the distribution peaks sharply in the forward direction. This directional asymmetry is exploited in medical imaging (PET scanners) and security screening detectors.
Compton scattering in medicine and science
Compton scattering is the dominant photon interaction in soft tissue for X-ray energies between about 30 keV and 30 MeV, making it the primary attenuation mechanism in diagnostic CT and radiotherapy. In positron emission tomography (PET), the 511 keV annihilation photons scatter in the patient and detector, introducing positional errors that modern corrections algorithms handle. In astrophysics, inverse Compton scattering (where energetic electrons boost low-energy photons) is a major X-ray and gamma-ray source in cosmic jets, supernova remnants, and the cosmic microwave background spectrum. Compton telescopes such as COMPTEL and INTEGRAL use the kinematics derived here to reconstruct gamma-ray source positions from the Compton scatter geometry.
Compton scattering at key angles (100 keV photon on electron)
| Angle (deg) | Scattered energy (keV) | Wavelength shift (pm) | Electron energy (keV) |
|---|---|---|---|
| 0 | 100.000 | 0.00000 | 0.000 |
| 30 | 95.110 | 0.32515 | 4.890 |
| 45 | 90.678 | 0.71129 | 9.322 |
| 60 | 83.600 | 1.21314 | 16.400 |
| 90 | 66.978 | 2.42631 | 33.022 |
| 120 | 52.480 | 3.63947 | 47.520 |
| 150 | 42.780 | 4.55468 | 57.220 |
| 180 | 39.789 | 4.85262 | 60.211 |
Scattered photon energy and wavelength shift for a 100 keV incident photon scattering off a free electron at standard angles.
Frequently asked questions
What is the Compton wavelength?
The Compton wavelength of a particle is h/(mc), where h is Planck's constant, m the particle rest mass, and c the speed of light. For an electron it is approximately 2.42631 pm (picometres). It sets the natural length scale of Compton scattering: the maximum wavelength shift (at 180 degrees backscattering) is exactly 2 times the Compton wavelength.
At what scattering angle is the energy transfer maximised?
The maximum energy transfer from photon to electron occurs at a scattering angle of 180 degrees (direct backscattering). At this angle (1 - cos 180) = 2, so Delta-lambda = 2 * lambda_C, which is the largest possible wavelength shift. The scattered photon has its minimum energy and the electron has its maximum kinetic energy.
Does Compton scattering apply to visible light?
Technically yes, but the effect is negligibly small. For visible light at about 2 eV, the wavelength is roughly 600 nm (600,000 pm), and the maximum wavelength shift from electron scattering is only about 4.85 pm - less than 0.001% of the incident wavelength. The shift is only appreciable for X-rays and gamma rays, whose wavelengths (0.001 to 10 pm) are comparable to the electron Compton wavelength.
What is the difference between Compton scattering and the photoelectric effect?
In the photoelectric effect the photon is completely absorbed by an atom, ejecting a bound electron with kinetic energy equal to the photon energy minus the binding energy. In Compton scattering the photon is scattered rather than absorbed: it survives with reduced energy and a longer wavelength. The photoelectric effect dominates at lower photon energies (below about 30 keV in soft tissue), while Compton scattering dominates from roughly 30 keV to 30 MeV. Above that, pair production takes over.
What is inverse Compton scattering?
In inverse Compton scattering a high-energy electron transfers energy to a low-energy photon, increasing the photon energy. It is the time-reversal of ordinary Compton scattering and is important in astrophysics: relativistic electrons in cosmic jets, galaxy clusters, and the early universe scatter microwave or infrared photons up to X-ray and gamma-ray energies. The Sunyaev-Zel'dovich effect, used to map galaxy clusters, is an inverse Compton process involving the cosmic microwave background.
How accurate is this calculator?
The calculator uses CODATA 2018 values for Planck's constant, the speed of light, and the electron rest mass. It applies the exact relativistic Compton formula with no small-angle approximations. Numerical errors are below 1 part per million for all valid inputs. The Klein-Nishina cross section assumes a free, stationary electron, which is a good approximation for energies well above the electron binding energy of the target material.
Sources
- Compton, A.H. (1923). A Quantum Theory of the Scattering of X-rays by Light Elements. Physical Review, 21(5), 483-502.
- Klein, O. and Nishina, Y. (1929). Uber die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac. Zeitschrift fur Physik, 52(11-12), 853-868.