Compton Scattering Wavelength Shift Calculator

Compton scattering is the collision of a photon with a nearly free electron, in which the photon transfers some of its energy and momentum to the electron and comes out with a longer wavelength (lower energy). Arthur Compton's 1923 measurement of this wavelength shift provided direct evidence that light behaves as particles (photons), not just waves.

The shift is Δλ = (h / (m_e c)) times (1 − cos θ), where h is Planck's constant, m_e is the electron mass, c is the speed of light, and θ is the scattering angle. The constant h / (m_e c) ≈ 2.4263 picometres is called the Compton wavelength of the electron.

The Compton formula shows Δλ depends only on the scattering angle θ and physical constants, not on the incident photon's wavelength or energy. What does depend on the incident wavelength is how significant that fixed shift is as a fraction of the total wavelength, which is why Compton scattering is most visible with short-wavelength X-rays.

The maximum shift occurs at θ = 180°, direct backscattering, where 1 − cos θ = 2, giving Δλ = 2 × 2.4263 pm ≈ 4.8526 pm. At θ = 0° (no scattering), Δλ = 0, since the photon passes straight through unaffected.

The Compton wavelength is λC = h / (m_e c) ≈ 2.42631 x 10^-12 metres (2.4263 picometres). It represents the length scale at which quantum and relativistic effects for the electron both become significant, and it sets the overall size of every Compton wavelength shift.

In the photoelectric effect, the photon is completely absorbed and ejects an electron, transferring all its energy. In Compton scattering, the photon survives the collision but loses some energy and changes direction, coming out at a longer wavelength. Both effects demonstrate the particle nature of light, at different photon energies.

Before 1923, many physicists still doubted that light carried discrete momentum like a particle. Compton's experiment, scattering X-rays off graphite and measuring the predicted wavelength shift, could only be explained by treating the photon as a particle with energy E = hf and momentum p = h/λ, colliding elastically with an electron. It won Compton the 1927 Nobel Prize in Physics.

It happens in principle at any wavelength, but the fixed 2.4 picometre shift is negligible compared to a visible light wavelength of 400 to 700 nanometres, roughly one part in 200,000. The effect only becomes measurable for X-rays and gamma rays, whose wavelengths are comparable to or shorter than the Compton wavelength itself.

In Rayleigh scattering, the photon bounces off a bound, whole atom with no change in wavelength, since the atom is far too massive to recoil noticeably. In Compton scattering, the photon interacts with a single, effectively free electron, which does recoil and absorb momentum, producing the measurable wavelength increase described by the Compton formula.

The electron, initially at rest or loosely bound, recoils and carries away the energy and momentum the photon lost. Its recoil angle and kinetic energy are related to the photon's scattering angle by conservation of energy and momentum, the same two conservation laws used to derive the Compton wavelength shift formula.