Unit 15 · Topic 15.6

15.6 Compton Scattering

In Compton scattering, a photon, usually an X-ray, bounces off a free electron and comes out with less energy and a longer wavelength. Treating the photon as a particle with momentum $p = \frac{h}{\lambda}$ and applying conservation of energy and momentum explains the result. The bigger the angle the photon is deflected, the bigger its wavelength shift, $\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)$ .

Key terms

Compton scattering
photon momentum
scattering angle
wavelength shift
conservation of momentum

The experiment

In 1923, Arthur Compton aimed X-rays at a target with loosely held electrons and measured the X-rays that scattered off at various angles. The scattered X-rays had longer wavelengths than the incoming ones, and the shift grew with the scattering angle.

The wave model can't explain this. A light wave shaking an electron would make it re-emit light at the same frequency, so the wavelength shouldn't change.

The photon explanation

Treat the X-ray as a photon, a particle with energy $E = \frac{hc}{\lambda}$ and momentum $p = \frac{h}{\lambda}$ , and the scattering as a collision with an electron at rest, like billiard balls.

Energy is conserved: the electron recoils with some kinetic energy, so the scattered photon has less energy than before. Less energy means a lower frequency and a longer wavelength.

Momentum is conserved in two dimensions: the incoming photon's momentum equals the vector sum of the scattered photon's momentum and the electron's momentum. Split each into components along the original direction and perpendicular to it; each component total is conserved separately.

Because the photon's energy, momentum, frequency and wavelength all change in exactly the way a particle collision predicts, Compton scattering is strong evidence that light comes in photons.

The size of the shift

$\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)$

Here θ is the angle between the photon's original and new directions. The combination $\frac{h}{m_e c}$ = 2.43 × 10⁻¹² m is called the Compton wavelength of the electron.

θ = 0° (photon barely deflected): cos θ = 1, so no shift.
θ = 90°: Δλ = 2.43 × 10⁻¹² m.
θ = 180° (photon bounces straight back): cos θ = −1, so the shift is the largest possible, 2 × 2.43 × 10⁻¹² = 4.85 × 10⁻¹² m. This is when the electron gets the biggest kick.

Why X-rays?

The shift is at most about 5 picometers, whatever the starting wavelength. For visible light (around 500 nm) that's a change of about one part in 100,000, which is almost impossible to notice. For X-rays with wavelengths of tens of picometers, it's a change of several percent, easy to measure.

The shift doesn't depend on the starting wavelength. It depends only on the angle.

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
Scattering at 90°
X-rays with λ = 70.0 pm scatter off free electrons at 90°. Find the scattered wavelength, the energy of the photon before and after, and the kinetic energy given to the electron.
Show the solutionHide the solution
1. Step 1: $\Delta\lambda = \dfrac{h}{m_e c}(1 - \cos 90^\circ) = (2.43\text{ pm})(1 - 0) = 2.43$ pm, so λ′ = 70.0 + 2.43 = 72.4 pm.
2. Step 2: Before: $E = \dfrac{1240\text{ eV·nm}}{0.0700\text{ nm}} = 17{,}700$ eV = 17.7 keV.
3. Step 3: After: $E' = \dfrac{1240}{0.07243} = 17{,}100$ eV = 17.1 keV.
4. Step 4: By conservation of energy, the electron gains the difference: about 0.59 keV (590 eV).
Answer: λ′ ≈ 72.4 pm; 17.7 keV before, 17.1 keV after; electron gets about 0.59 keV.
Example 2Calculator allowed
Momentum in two dimensions
In the 90° scattering above, the incoming photon moves in the +x direction and the scattered photon moves in the +y direction. Find the direction of the recoiling electron's momentum.
Show the solutionHide the solution
1. Step 1: Before the collision, the only momentum is the photon's: $\frac{h}{\lambda}$ in +x, with λ = 70.0 pm. The y-total is zero.
2. Step 2: After: the photon has $\frac{h}{\lambda'}$ in +y, with λ′ = 72.4 pm, and nothing in x.
3. Step 3: x: the electron must carry all of the x-momentum, $+\frac{h}{\lambda}$ . y: to keep the y-total at zero, the electron must have $-\frac{h}{\lambda'}$ .
4. Step 4: Angle below the +x axis: $\tan\phi = \dfrac{h/\lambda'}{h/\lambda} = \dfrac{\lambda}{\lambda'} = \dfrac{70.0}{72.4}$ , so φ = 44.0°.
Answer: The electron moves at 44.0° below the +x axis (on the opposite side from the scattered photon).

Common mistakes

Thinking the scattered photon is slower. All photons move at c; the scattered photon has less energy and a longer wavelength.
Saying the wavelength gets shorter after scattering. The photon gives energy away, so its wavelength gets longer.
Thinking the shift depends on the incoming wavelength. It depends only on the scattering angle, which is why it's noticeable only for short wavelengths.
Forgetting that momentum is a vector. Conserve the x- and y-components separately.

On the exam

Expect to explain why Compton scattering supports the photon model. Point to the wavelength change and the use of energy and momentum conservation, treating the photon as a particle.
Questions may compare shifts at two angles (largest at 180°, zero at 0°) or ask for an electron's direction using momentum components.

Connected topics

Videos

AP Physics 2 - Unit 15 - Lesson 7 - Compton Scattering
Allen Tsao The STEM CoachWatch on YouTube (opens in a new tab)
Compton Effect and Compton Wavelength
The Organic Chemistry TutorWatch on YouTube (opens in a new tab)
What is Compton Scattering?
Physics ExplainedWatch on YouTube (opens in a new tab)
Physics - Modern Physics (7 of 26) Compton Scattering
Michel van BiezenWatch on YouTube (opens in a new tab)
Compton Scattering at Washington University in St. Louis | The Compton Effect | Doc Physics
Doc SchusterWatch on YouTube (opens in a new tab)

Check yourself

4 questions on 15.6 Compton Scattering. Pick an answer to see if you got it, and why.

Question 1 of 4Calculator allowed

An X-ray photon has a wavelength of 0.0100 nm. What is its momentum?

Question 2 of 4Calculator allowed

An X-ray photon scatters off a free electron that was at rest. Compared with the incoming photon, the scattered photon has

Question 3 of 4Calculator allowed

X-ray photons scatter off free electrons at different angles. For which scattering angle is the wavelength of the scattered photon greatest?

Question 4 of 4Calculator allowed

A 20.0 keV X-ray photon (wavelength 0.0620 nm) scatters from a free electron at rest and leaves with a wavelength of 0.0644 nm. About how much kinetic energy does the electron gain? (Use hc = 1240 eV·nm.)

0 of 4 answered