What is the Compton effect?
In 1923 the American physicist Arthur Compton shone X-rays on a block of graphite. He measured the X-rays that came out sideways. He found two wavelengths: the original one and a longer one. The bigger the angle of scattering, the bigger the change.
This is the Compton effect: when a high-energy photon scatters from an electron, the scattered photon has a longer wavelength (less energy) than the incoming photon.
Old wave theory said the wavelength must stay the same. Only the idea that light comes in photons, each with energy and momentum, explains the result.
The photon as a particle with momentum
A photon of wavelength λ has energy E = hf = hc/λ. It has no mass, but it still carries momentum: p = E/c = h/λ. (h = 6.63 × 10⁻³⁴ J·s is Planck's constant.)
A shorter wavelength means a more energetic photon with more momentum. X-rays and gamma rays have wavelengths of a few picometres (1 pm = 10⁻¹² m), so their photons are strong enough to knock an electron about.
In the lesson on the photoelectric effect the photon is swallowed whole. Here it is scattered: it comes out again, with less energy.
The collision: conservation of energy and momentum
Treat the crash like two balls. The electron is at rest before the hit. Two rules hold:
- Energy is conserved: photon energy before = photon energy after + kinetic energy of the electron.
- Momentum is conserved along the original direction and across it. The electron must move off at some angle below the axis while the photon goes above it, so that the sideways momenta cancel.
Because the photon lost energy, its energy hc/λ′ is smaller, so λ′ is bigger. The electron's kinetic energy is K = hc/λ − hc/λ′. The electron moves so fast that you must use relativistic energy for it; this is part of the full derivation.
The Compton shift formula
Solving the two conservation equations (eliminate the electron's speed and angle) gives:
Δλ = λ′ − λ = (h / m c) (1 − cos θ)
The quantity h/(mc) = 2.43 pm is called the Compton wavelength of the electron. The shift depends only on the angle θ, not on the original wavelength.
- θ = 0°: no shift (the photon goes straight on).
- θ = 90°: Δλ = 2.43 pm.
- θ = 180°: Δλ = 4.86 pm, the largest possible shift.
Why do we only see it with X-rays and gamma rays?
The shift is at most about 4.9 pm. For visible light (wavelength about 500 nm = 500 000 pm) this is about one part in 100 000, far too small to notice. For X-rays of about 10 pm it is a big fraction of the wavelength, so it is easy to measure.
Also, electrons in an atom are held with energies of a few eV. An X-ray photon has tens of thousands of eV, so the electron behaves as if it were free and at rest. When a photon hits an electron held tightly (or the whole atom), the recoiling mass is huge, so the shift is almost zero. That is why Compton's graph shows both an unshifted and a shifted peak.
Compton effect and the photoelectric effect
Both prove that light acts as particles. The difference:
- Photoelectric effect: low-energy photons (visible or UV) are absorbed by a bound electron. The photon disappears.
- Compton effect: high-energy photons scatter from an almost free electron. A photon comes out with a longer wavelength.
Together they strongly support the photon model and lead on to matter waves.
Try it: predict, then check
In the 3D scene: set θ to 0°, then 90°, then 180°. Before you look at the numbers, guess the shift each time. Then check the readout. Notice that the electron gets the most energy when the photon bounces straight back.
At home: roll one marble into another marble that is resting on a table. The first marble slows down and goes off at an angle; the second one moves. Compare a glancing hit with a head-on hit. A glancing hit gives away little energy, a head-on hit gives away a lot, just like small and large θ.
Key formulas and definitions
- Photon energy: E = hf = hc/λ
- Photon momentum: p = h/λ
- Compton shift: Δλ = λ′ − λ = (h/mc)(1 − cos θ)
- Compton wavelength of the electron: λC = h/(mc) = 2.43 × 10⁻¹² m = 2.43 pm
- Electron kinetic energy: K = hc/λ − hc/λ′
- Useful: hc = 1240 eV·nm = 1.24 keV·nm = 1240 keV·pm
Worked examples
1. Calculate the Compton wavelength of the electron. (h = 6.626 × 10⁻³⁴ J·s, m = 9.109 × 10⁻³¹ kg, c = 3.00 × 10⁸ m/s)
λC = h/(mc) = 6.626 × 10⁻³⁴ / (9.109 × 10⁻³¹ × 3.00 × 10⁸) = 6.626 × 10⁻³⁴ / 2.733 × 10⁻²² ≈ 2.42 × 10⁻¹² m, or about 2.43 pm.
2. X-rays scatter through 90°. By how much does the wavelength change?
cos 90° = 0, so Δλ = 2.43 × (1 − 0) = 2.43 pm.
3. What is the largest possible Compton shift, and at what angle?
The shift is largest when cos θ = −1, which is θ = 180° (the photon bounces back). Δλ = 2.43 × 2 = 4.86 pm.
4. X-rays of wavelength 71.0 pm are scattered through 90°. Find the wavelength of the scattered X-rays.
Δλ = 2.43 pm. λ′ = 71.0 + 2.43 = 73.4 pm.
5. A photon of wavelength 10 pm hits an electron and scatters at 90°. Find the kinetic energy given to the electron.
λ′ = 10 + 2.43 = 12.43 pm. Photon energy before = 1240/10 = 124.0 keV. After = 1240/12.43 = 99.8 keV. Electron kinetic energy = 124.0 − 99.8 ≈ 24.2 keV.
6. At what scattering angle is the Compton shift 1.2 pm?
1.2 = 2.43 (1 − cos θ), so 1 − cos θ = 0.494 and cos θ = 0.506. θ ≈ 59.6°, about 60°.
7. Green light of wavelength 500 nm is scattered through 90° by an electron. Find the fractional change in wavelength and say why we do not see it.
Δλ = 2.43 pm = 2.43 × 10⁻¹² m. Fractional change = 2.43 × 10⁻¹² / 5 × 10⁻⁷ ≈ 4.9 × 10⁻⁶, which is about 0.0005%. No instrument or eye can see such a tiny change.
Common mistakes
- Thinking the shift depends on the incoming wavelength. It depends only on the angle θ.
- Mixing up the photoelectric effect (photon absorbed) with the Compton effect (photon scattered and still exists).
- Using degrees wrongly: with a calculator in radian mode cos 90° will not be 0. Check the mode.
- Saying the scattered photon has a shorter wavelength. It lost energy, so λ′ is always longer (or equal when θ = 0).