Compton Scattering Calculator

Enter a photon's incident wavelength and scattering angle to find the Compton wavelength shift, scattered wavelength, scattered photon energy, and recoil electron energy using Δλ = (h / mₑc)(1 − cos θ).

Quick Facts

Compton wavelength (electron)
λ_C = h / (mₑc) ≈ 2.42631 pm
A fixed length scale set by fundamental constants; it does not depend on the photon's wavelength.
Shift formula
Δλ = λ_C (1 − cos θ)
The wavelength shift depends only on the scattering angle θ, never on the incident wavelength or intensity.
Maximum shift
Δλ_max = 2λ_C ≈ 4.853 pm at θ = 180°
Head-on backscattering produces the largest possible wavelength increase; θ = 0° gives zero shift.
Historical note
Discovered by Arthur Compton, 1923
The effect confirmed that photons carry momentum p = h/λ, earning Compton the 1927 Nobel Prize in Physics.

Your Results

Calculated
Compton Wavelength Shift (Δλ)
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Δλ = (h / mₑc)(1 − cos θ)
Scattered Wavelength (λ')
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λ' = λ + Δλ
Scattered Photon Energy (E')
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E' = hc / λ'
Electron Recoil Kinetic Energy
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KE = E − E' (energy conservation)

Ready

Enter an incident wavelength and scattering angle, then press Calculate.

Formula and Method for Compton Scattering

Compton scattering describes what happens when a photon collides with a free (or loosely bound) electron: the photon transfers part of its energy and momentum to the electron and scatters off at an angle θ with a longer wavelength — and therefore lower energy — than it had going in. Applying conservation of energy and momentum to this photon-electron collision gives the Compton scattering formula: Δλ = λ' − λ = (h / mₑc)(1 − cos θ), where h is Planck's constant, mₑ is the electron's rest mass, c is the speed of light, and θ is the angle between the incoming and scattered photon directions. This calculator applies that formula to find the wavelength shift, the scattered wavelength, and how the lost photon energy splits off as kinetic energy for the recoiling electron.

How the calculation works

Enter the incident photon's wavelength λ and the scattering angle θ. The calculator first evaluates the Compton wavelength of the electron, λ_C = h / (mₑc) ≈ 2.42631 pm, a fixed quantity built from fundamental constants. The wavelength shift is Δλ = λ_C(1 − cos θ), which is added to the incident wavelength to get the scattered wavelength, λ' = λ + Δλ. Photon energy follows from E = hc/λ (and E' = hc/λ' for the scattered photon); by conservation of energy, the kinetic energy transferred to the recoiling electron is KE = E − E'.

Common mistakes

  • Ignoring the scale: the Compton shift is at most a few picometers, so it's negligible next to visible light (~500 nm) but significant for X-rays and gamma rays (~10-100 pm), which is why Compton's original 1923 experiment used X-rays.
  • Mixing degrees and radians: θ in the formula is an angle, not a slope — this calculator takes θ in degrees (0° to 180°) and converts to radians internally before applying cos θ.
  • Confusing scattering mechanisms: Compton scattering is inelastic (the photon loses energy), unlike classical Thomson scattering, which is elastic and only valid when the photon energy is much smaller than the electron's rest-mass energy (511 keV).

Real-world applications

  • Compton scattering is one of the three dominant photon-matter interactions (with the photoelectric effect and pair production) and is a core input to radiation shielding design and medical dosimetry calculations.
  • Compton cameras used in gamma-ray astronomy and nuclear medicine imaging (such as SPECT) reconstruct where a gamma photon came from by measuring its scattering angle and energy loss in a detector.
  • The 1923 experiment that confirmed this effect — for which Arthur Compton won the 1927 Nobel Prize in Physics — was direct evidence that light carries momentum as discrete photons, not just as a wave.

Frequently Asked Questions

What is Compton scattering?
Compton scattering is the inelastic scattering of a photon (typically an X-ray or gamma ray) off a free or loosely bound electron. The photon loses energy and increases in wavelength while the electron recoils and gains kinetic energy. Arthur Compton first observed it in 1923, providing key evidence that light carries momentum as discrete particles (photons).
What is the Compton wavelength shift formula?
The wavelength shift is Δλ = (h / mₑc)(1 − cos θ), where h is Planck's constant (6.626×10⁻³⁴ J·s), mₑ is the electron rest mass (9.109×10⁻³¹ kg), c is the speed of light, and θ is the scattering angle measured from the photon's original direction. The shift depends only on the scattering angle, not on the incident photon's wavelength.
Why is the Compton wavelength only about 2.43 picometers?
h / (mₑc) is a fixed combination of fundamental constants called the electron's Compton wavelength, equal to about 2.42631 pm. Since the shift ranges from 0 (at θ = 0°) to 2 × 2.42631 ≈ 4.853 pm (at θ = 180°), Compton scattering is only a noticeable fraction of the wavelength for very short-wavelength photons like X-rays and gamma rays.
How much energy does the recoiling electron gain?
By conservation of energy, the electron's kinetic energy equals the energy lost by the photon: KE = E − E' = hc/λ − hc/λ', where λ' = λ + Δλ is the scattered wavelength. The scattered photon always ends up with less energy than the incident photon, and the recoiling electron carries away exactly the difference.

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