Compton Scattering: Photon Momentum and the Compton Formula

Light That Bounces Like a Ball

The photoelectric effect showed that light energy comes in packets called photons. But do photons also carry momentum, like particles? In 1923 the American physicist Arthur Compton fired X-rays at a block of graphite and found that the scattered X-rays had a longer wavelength than the incoming ones, by an amount that depended on the scattering angle. Classical wave theory could not explain this at all. Compton explained it by treating each X-ray photon as a particle that collides with an electron, like one snooker ball hitting another. Compton scattering was one of the clearest proofs that light has particle properties. This tutorial explains the experiment, photon momentum, the Compton formula and its derivation, worked examples and the effect's importance in medicine and astronomy.

Photon Energy and Momentum

A photon of frequency f and wavelength λ has energy

E = hf = hc/λ,

where h = 6.63 × 10⁻³⁴ J s is Planck's constant. Although a photon has no rest mass, relativity shows that anything carrying energy E at the speed of light carries momentum p = E/c. So a photon's momentum is

p = h/λ.

Shorter wavelengths mean more energetic and more "punchy" photons. An X-ray photon of wavelength 0.071 nm has an energy of about 17.5 keV and a momentum of about 9.3 × 10⁻²⁴ kg m/s, tiny by everyday standards but comparable to the momentum of an electron moving at a few per cent of the speed of light.

The Compton Experiment

Compton directed a beam of X-rays of a single wavelength (the molybdenum Kα line, λ = 0.0710 nm) at a graphite target and measured the wavelengths of the X-rays scattered at different angles, using an X-ray spectrometer based on Bragg diffraction.

He found two peaks at each angle:

one at the original wavelength, from photons scattered by tightly bound inner electrons (effectively by the whole atom);
one at a longer wavelength, whose shift increased with the scattering angle, from photons scattered by loosely bound outer electrons.

Classical physics predicted that the X-ray wave would make the electrons vibrate at the same frequency and re-radiate at the same wavelength. A wavelength shift that depends only on angle had no classical explanation.

The Compton Scattering Formula

Compton treated the event as an elastic collision between a photon and a free electron at rest, conserving both energy and momentum and using relativistic mechanics for the electron. The result is

Δλ = λ′ − λ = (h / mₑc)(1 − cos θ),

where λ is the incident wavelength, λ′ the scattered wavelength, θ the angle through which the photon is scattered, and mₑ the electron mass.

The constant h/(mₑc) is called the Compton wavelength of the electron:

h/(mₑc) = 6.63 × 10⁻³⁴ / (9.11 × 10⁻³¹ × 3.0 × 10⁸) ≈ 2.43 × 10⁻¹² m (0.00243 nm).

Key features of the formula:

The shift does not depend on the incident wavelength or the target material, only on the angle.
At θ = 0° (no scattering), Δλ = 0.
At θ = 90°, Δλ = h/mₑc = 0.00243 nm.
At θ = 180° (back-scattering), the shift is greatest: Δλ = 2h/mₑc = 0.00485 nm.

Where the Formula Comes From

An outline of the derivation shows how it combines particle ideas:

Momentum is conserved as a vector: the incoming photon's momentum h/λ equals the vector sum of the scattered photon's momentum h/λ′ and the electron's recoil momentum p. Using the cosine rule on this triangle gives p² = (h/λ)² + (h/λ′)² − 2(h/λ)(h/λ′) cos θ.
Energy is conserved: hc/λ + mₑc² = hc/λ′ + √(p²c² + mₑ²c⁴), using the relativistic energy of the recoiling electron.
Eliminating p between these two equations and simplifying gives λ′ − λ = (h/mₑc)(1 − cos θ).

The scattered photon has a longer wavelength simply because it has given some of its energy to the electron, which recoils.

Worked Examples on Compton Scattering

Example 1. X-rays of wavelength 0.0710 nm are scattered through 90°. Find the scattered wavelength.

Δλ = 2.43 × 10⁻¹² × (1 − cos 90°) = 2.43 × 10⁻¹² m = 0.00243 nm. λ′ = 0.0710 + 0.00243 = 0.0734 nm.

Example 2. For the same X-rays back-scattered through 180°, find the energy given to the electron.

λ′ = 0.0710 + 2 × 0.00243 = 0.0759 nm.

Incident energy = hc/λ = (6.63 × 10⁻³⁴ × 3.0 × 10⁸)/(0.0710 × 10⁻⁹) ≈ 2.80 × 10⁻¹⁵ J ≈ 17.5 keV.

Scattered energy = hc/λ′ ≈ 16.4 keV.

The electron gains the difference, about 1.1 keV of kinetic energy.

Example 3. Why can't Compton scattering be seen with visible light?

For green light (λ = 500 nm), the largest possible shift, 0.00485 nm, is less than one part in 100,000 of the wavelength, far too small to detect, and visible photons have too little energy to free electrons in this way. The effect is only noticeable when the wavelength is comparable to the Compton wavelength, that is for X-rays and gamma rays.

Example 4. A gamma photon of energy 0.662 MeV (from caesium-137) is back-scattered. Using the energy form of the formula, E′ = E / [1 + (E/mₑc²)(1 − cos θ)], with mₑc² = 0.511 MeV:

E′ = 0.662 / (1 + 2 × 0.662/0.511) ≈ 0.184 MeV. For high-energy photons, back-scattering removes most of the energy.

Why Compton Scattering Matters

Evidence for photons: together with the photoelectric effect, it established that electromagnetic radiation carries energy and momentum in quanta. Compton received the Nobel Prize in Physics in 1927.
Wave–particle duality: the wavelength of the X-rays is measured by diffraction (a wave property), yet the shift is explained by collisions (a particle property). Light shows both behaviours.
Medical imaging and radiotherapy: in the X-ray and gamma energy range used in hospitals, Compton scattering is the main way radiation interacts with soft tissue. Scattered radiation blurs X-ray images and exposes staff, which is why anti-scatter grids and lead shielding are used.
Radiation detectors: gamma-ray spectrometers show a "Compton edge" in their spectra, corresponding to the maximum energy given to electrons by back-scattering.
Astronomy: inverse Compton scattering, where fast electrons give energy to low-energy photons, produces X-rays and gamma rays around black holes and in galaxy clusters.

Summary

Photons carry momentum p = h/λ as well as energy E = hc/λ.
In Compton scattering, X-ray photons collide with electrons and emerge with a longer wavelength.
Δλ = (h/mₑc)(1 − cos θ); h/mₑc = 2.43 × 10⁻¹² m is the Compton wavelength.
The shift depends only on the scattering angle, and is greatest (2h/mₑc) at 180°.
The effect is significant only for X-rays and gamma rays, and it is strong evidence for the particle nature of light.