Compton Scattering
When a photon collides with an electron, it bounces off like a billiard ball — losing some energy and gaining wavelength. The bigger the bounce angle, the more its wavelength stretches, and that wavelength shift depends only on the scattering angle, not on the photon's original energy or anything else about the collision. This was one of the clearest proofs that light travels in particle-like packets that carry momentum.
The formula
Δλ = (h / mₑc) · (1 − cos θ)
- Δλ — wavelength shift (m): how much longer the photon's wavelength becomes after scattering
- h / mₑc — Compton wavelength (m): a fixed constant equal to 2.43 picometers, set by the electron's mass
- θ — scattering angle (°): the angle between the photon's incoming and outgoing directions
Worked example
A photon scatters straight off an electron at a 90 degree angle. How much does its wavelength shift?
- θ = 90°
- h / mₑc = 2.43 pm
- Δλ = (h / mₑc) · (1 − cos θ)
- Δλ = 2.43 pm · (1 − cos 90°) = 2.43 pm · 1
Δλ = 2.43 pm
Test yourself
Now the photon bounces straight backward, at a 180 degree angle. What is the wavelength shift?
- Correct answer: 4.86 pm
- 2.43 pm
- 1.22 pm
Correct! At 180°, cos θ = −1, so (1 − cos θ) = 2, giving Δλ = 2.43 pm × 2 = 4.86 pm.
At which scattering angle does the photon's wavelength shift become zero?
- 180°
- 90°
- Correct answer: 0°
Right! At 0° the photon isn't deflected at all, cos θ = 1, so (1 − cos θ) = 0 and there's no shift.
Where you see this
In an X-ray image, Compton scatter is the fuzz an X-ray photon leaves when it bounces off an electron in your body and arrives at the detector from the wrong direction. Arthur Compton's 1923 measurement of the bounced photon's stretched wavelength was the experiment that made physicists finally accept the photon as a real particle.
Common mistakes
The key fact is that the wavelength shift depends ONLY on the bounce angle — not on the photon's original energy, not on anything else: Δλ = 2.43 pm · (1 − cos θ). At 0 degrees nothing changes (cos θ = 1 makes the shift zero — the photon barely scattered), while a straight-back bounce at 180 degrees gives the maximum, cos θ = −1, doubling the shift to 4.86 pm.
How it connects
The photoelectric effect showed photons carry energy; Compton showed they carry momentum too — a billiard ball with a wavelength, recoiling off electrons by the same collision rules as the mechanics module. The natural question this raises — if waves act like particles, do particles act like waves? — is exactly the next lesson.