Compton Wavelength Calculator
Enter a particle mass to get its Compton wavelength and reduced Compton wavelength instantly. Switch to scattering mode to add a photon energy and scattering angle, and the calculator returns the wavelength shift and the energy transferred to the particle. Reverse mode lets you enter a Compton wavelength and recover the corresponding particle mass.
What is the Compton wavelength?
The Compton wavelength of a particle is the quantum length scale at which relativistic and quantum effects both matter simultaneously. For a particle of rest mass m, it is defined as lambda_C = h / (mc), where h is the Planck constant and c is the speed of light. At lengths shorter than this, confining a particle to a smaller region would require enough energy to create a particle-antiparticle pair, so ordinary one-particle quantum mechanics breaks down and quantum field theory takes over. Arthur Compton introduced this length in his 1923 paper that explained X-ray scattering off electrons, the first experimental confirmation of the photon nature of light.
Reduced Compton wavelength and why it appears in equations
The reduced Compton wavelength lambda-bar = h-bar / (mc) = lambda_C / (2 pi), where h-bar is the reduced Planck constant. This shorter length appears naturally in the Dirac equation, the Klein-Gordon equation, and most expressions in relativistic quantum mechanics because those equations are written in terms of h-bar rather than h. For the electron, lambda-bar is about 386.16 fm (0.386 pm). It also appears in the fine-structure constant and in expressions for atomic energy levels, making it one of the most fundamental length scales in physics.
Compton scattering: the wavelength shift formula
When a high-energy photon (X-ray or gamma ray) scatters off a particle of mass m at angle theta, the photon wavelength increases by delta-lambda = lambda_C (1 - cos theta). The shift is zero at forward scattering (theta = 0), and reaches its maximum of 2 lambda_C at backscattering (theta = 180 degrees). For electron scattering this maximum shift is about 4.85 pm, which is small compared to visible light wavelengths (400-700 nm) but comparable to hard X-ray wavelengths (tens of pm), making the effect measurable. The energy lost by the photon is transferred to the recoiling particle as kinetic energy: delta-E = E_0 - E_0 / [1 + (E_0 / mc^2)(1 - cos theta)], where E_0 is the incident photon energy.
How to use this calculator
Choose a mode from the dropdown. In "Compton wavelength from mass" mode, select a particle preset (electron, proton, neutron, muon, or charged pion) or enter a custom mass in kilograms, and the calculator returns the Compton wavelength and the reduced Compton wavelength. In "Compton scattering" mode, also enter a scattering angle (0 to 180 degrees) and the energy of the incoming photon in keV: you will see the wavelength shift, the scattered photon energy, and the energy transferred to the recoiling particle, plus a chart of how all these values change across all angles. In "Reverse" mode, enter a Compton wavelength in pm and recover the corresponding particle mass.
Compton wavelengths of common particles
| Particle | Mass (kg) | Compton wavelength (pm) | Reduced wavelength (pm) |
|---|---|---|---|
| Electron | 9.1094 x 10^-31 | 2.426310 | 0.386159 |
| Muon | 1.8835 x 10^-28 | 0.011734 | 0.001868 |
| Charged pion | 2.4880 x 10^-28 | 0.008878 | 0.001413 |
| Proton | 1.6726 x 10^-27 | 0.0013214 | 0.00021031 |
| Neutron | 1.6749 x 10^-27 | 0.0013196 | 0.00021000 |
Values calculated from CODATA 2018 recommended constants. The reduced wavelength (lambda-bar) equals lambda_C divided by 2 pi.
Frequently asked questions
What is the Compton wavelength of the electron?
The electron Compton wavelength is 2.42631023867 pm (picometres), or about 2.426 x 10^-12 m. The reduced Compton wavelength is 0.38615926796 pm, or about 386.16 fm. These values are derived from the electron rest mass (9.1094 x 10^-31 kg) using lambda_C = h / (mc).
Why is the Compton wavelength important in quantum mechanics?
It sets the boundary between ordinary quantum mechanics and quantum field theory. When a particle is confined to a region smaller than its Compton wavelength, the uncertainty in momentum is so large that the kinetic energy exceeds the rest-mass energy, making particle-antiparticle pair creation energetically possible. This is why quantum field theory is needed at very short distances.
What is the difference between the Compton wavelength and the de Broglie wavelength?
The Compton wavelength lambda_C = h / (mc) depends only on the rest mass and is a fixed property of a particle species. The de Broglie wavelength lambda_dB = h / p depends on the particle momentum p and changes with speed. For a particle at rest the de Broglie wavelength is infinite; the two become equal when the particle moves at about 70% of the speed of light.
What is the maximum possible wavelength shift in Compton scattering?
The maximum shift occurs at backscattering (theta = 180 degrees), where 1 - cos(180) = 2, so delta-lambda_max = 2 lambda_C. For electron scattering this maximum is 2 x 2.426 pm = 4.852 pm. At all other angles the shift is smaller, and at forward scattering (theta = 0) the shift is exactly zero.
Does Compton scattering work for visible light?
The shift in wavelength is the same regardless of the photon energy, but the fractional shift is tiny for long-wavelength photons. Visible light has wavelengths of 400-700 nm (400,000-700,000 pm), so a shift of a few pm is less than 0.001% and is completely undetectable. The effect is only significant when the incident wavelength is comparable to the Compton wavelength of the target, which is why it was first observed with hard X-rays.
How does particle mass affect the Compton wavelength?
Compton wavelength is inversely proportional to mass: doubling the mass halves the wavelength. The proton, which is about 1836 times heavier than the electron, has a Compton wavelength about 1836 times shorter (roughly 0.00132 pm, or 1.32 fm). This means heavy particles have very short quantum length scales, and Compton scattering effects on protons require much higher photon energies to be measurable.