What this calculator does
When a high-energy photon scatters off an electron, it comes away with a longer wavelength, having given up some energy. The shift depends only on the scattering angle, not on the photon's original wavelength, which is the striking feature of the effect.
Compton's demonstration of this in 1923 was decisive evidence that light behaves as particles. A wave theory predicts no wavelength change at all, so the observed shift could not be explained any other way.
The formula
The shift equals the Compton wavelength of the electron, 2.426 picometres, multiplied by one minus the cosine of the scattering angle. Adding that to the incident wavelength gives the scattered wavelength.
| Term | Meaning |
|---|---|
| Compton wavelength | h divided by the electron mass times c, equal to 2.426 pm. It sets the scale of the entire effect. |
| Scattering angle | The angle between the photon's original and final directions. Zero means undeviated; 180 degrees means straight back. |
| Backscatter | Scattering through 180 degrees, which produces the maximum possible wavelength shift of twice the Compton wavelength. |
The inputs explained
| Field | What to enter |
|---|---|
| Incident wavelength (pm) | The incident photon wavelength in picometres. X-rays are typically tens to hundreds of picometres. |
| Scattering angle (°) | The scattering angle in degrees, from 0 to 180. |
When to use it
Analysing an X-ray scattering experiment
The measured wavelength shift identifies the scattering angle, and vice versa.
Understanding evidence for photons
The angle dependence and the independence from incident wavelength are both impossible to explain with waves alone.
Estimating energy loss
The wavelength increase corresponds directly to energy transferred to the recoiling electron.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the shift depend on scattering angle?
The same incident photon scattered through a range of angles.
| Scattering angle | Wavelength shift Δλ | New wavelength λ′ |
|---|---|---|
| 0° | 0 pm | 71.000 pm |
| 45° | 0.7106 pm | 71.711 pm |
| 90° | 2.426 pm | 73.426 pm |
| 180° | 4.853 pm | 75.853 pm |
Questions
Why does the shift not depend on the incident wavelength?
Because it comes out of momentum conservation between photon and electron, and the electron's mass is what sets the scale. The absolute shift is the same whatever the photon started with, though the fractional change is far larger for short wavelengths.
Why was this important for quantum theory?
Because classical wave theory predicts scattered light at the same wavelength as the incident light. The observed shift, matching the particle prediction exactly, made the photon picture very hard to argue with.
Why is the effect invisible with visible light?
Because a shift of a few picometres against a wavelength of 500,000 pm is undetectable. The effect only becomes significant when the incident wavelength is itself comparable to a few picometres, which means X-rays and gamma rays.
What happens to the electron?
It recoils, carrying away exactly the energy and momentum the photon lost. Detecting that recoil electron alongside the scattered photon was part of confirming the picture.
For the energy of the photons involved, see the photon energy calculator. For the matter-wave relationship, see the de Broglie wavelength calculator.