15.6 Compton Scattering

Compton scattering is when a photon collides with a free electron, loses energy, and comes away with a longer wavelength. The size of the wavelength shift depends only on the scattering angle, and you can predict it with the equation $\Delta \lambda = \frac{h}{m_e c}(1 - \cos \theta)$.

Why This Matters for the AP Physics 2 Exam

Compton scattering shows up in the modern physics part of AP Physics 2, where you treat light as a stream of photons instead of just a wave. The skills here connect to how you handle collisions: you apply conservation of energy and conservation of momentum, with momentum resolved into two dimensions because the photon and electron go off in different directions.

This topic is good practice for questions that ask you to calculate a quantity, justify a claim with physics principles, and explain your reasoning clearly. The free-response section can ask you to build and use a model, then write a coherent explanation that cites the right principles, and a photon-electron collision is a clean setup for that kind of reasoning.

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Key Takeaways

• In Compton scattering, a photon hits a free electron and comes out with lower energy, lower frequency, and longer wavelength, while the electron recoils with kinetic energy.
• The wavelength shift follows $\Delta \lambda = \frac{h}{m_e c}(1 - \cos \theta)$, so the change depends only on the scattering angle, not on the photon's original wavelength.
• No shift happens at $\theta = 0$ (forward scattering), and the largest shift happens at $\theta = 180^\circ$ (backscatter), where $(1 - \cos \theta) = 2$.
• The quantity $\frac{h}{m_e c}$ is the Compton wavelength of the electron, about $2.43 \times 10^{-12}$ m.
• You explain the whole process by treating the photon as a particle and applying conservation of energy and conservation of momentum in two dimensions.
• Compton scattering is direct evidence that light is made of discrete photons that carry energy ($E = hf$) and momentum ($\lambda = \frac{h}{p}$).

The Photon-Electron Collision

When a photon hits a free electron, it transfers some of its energy and momentum to the electron. After the collision:

• The photon has lower energy than before.
• The scattered photon has a longer wavelength.
• The electron recoils and gains kinetic energy.
• Both the photon and electron change direction.

The amount of change depends on the scattering angle. When the photon scatters at a larger angle from its original path, it gives up more energy to the electron, so the photon's wavelength increases more.

You model this in AP Physics 2 by treating light as discrete particles called photons and applying conservation of energy and conservation of momentum to the photon-electron collision. Because the scattered photon and recoiling electron can move in different directions, you resolve momentum into components and treat it in two dimensions.

Why This Is Evidence for Photons

Compton scattering supports the idea that light is made of discrete, quantized energy packets called photons rather than only continuous waves. The key reasons:

• The interaction follows conservation of energy and momentum when you treat photons as particles with specific energy and momentum.
• The measured changes in the photon's properties match the predictions of a particle-particle collision.
• The wavelength shift depends on the scattering angle exactly the way a photon-electron collision predicts.

During a Compton scattering event, the photon's energy decreases, so its frequency decreases and its wavelength increases, while its momentum changes direction and the electron picks up kinetic energy.

These photon properties are connected by:

$E = hf \qquad \lambda = \frac{h}{p}$

Since the photon hands off energy and momentum to the electron, the photon's energy and frequency drop while its wavelength goes up.

Wavelength Change vs Scattering Angle

The change in wavelength is set by the scattering angle through the Compton equation:

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

Where:

• $\Delta \lambda$ is the change in wavelength
• $h$ is Planck's constant
• $m_{e}$ is the mass of the electron
• $c$ is the speed of light
• $\theta$ is the scattering angle of the photon relative to its original direction

What this equation tells you:

• When the photon keeps going straight ahead ($\theta = 0^\circ$), there is no wavelength change, since $\cos(0^\circ) = 1$ makes $(1 - \cos \theta) = 0$.
• As the scattering angle grows, the wavelength change grows too.
• The term $\frac{h}{m_e c}$ is the Compton wavelength of the electron, about $2.43 \times 10^{-12}$ m.
• The biggest wavelength change happens at a full backscatter ($\theta = 180^\circ$), where $(1 - \cos \theta) = 2$.

Notice that the original wavelength of the photon does not appear in this equation. The shift depends only on the angle, which is a common detail that questions test.

How to Use This on the AP Physics 2 Exam

Problem Solving

• Identify what you are given: the initial wavelength or energy, and the scattering angle.
• Use $\Delta \lambda = \frac{h}{m_e c}(1 - \cos \theta)$ to find the wavelength shift.
• Add the shift to the initial wavelength to get the scattered wavelength: $\lambda_f = \lambda_i + \Delta \lambda$.
• Convert between energy and wavelength with $E = \frac{hc}{\lambda}$ when a problem gives or asks for energy.
• Find the electron's kinetic energy using conservation of energy: $K_e = E_i - E_f$.

Free Response

• When asked to justify why Compton scattering supports the photon model, point to conservation of energy and conservation of momentum applied to a particle-particle collision.
• If the photon and electron leave at different angles, state that momentum is conserved in two dimensions and that you would resolve it into components.
• Keep your written explanation organized: name the principle, connect it to the equation, and tie it back to the observed wavelength shift.

Common Trap

• Do not assume the original wavelength affects the size of the shift. The shift depends only on the angle.

Practice Problem 1: Wavelength Change Calculation

Solution

Use the Compton scattering equation:

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

First, calculate the Compton wavelength of the electron: $\frac{h}{m_e c} = \frac{6.63 \times 10^{-34} \text{ J} \cdot \text{s}}{(9.11 \times 10^{-31} \text{ kg})(3.00 \times 10^8 \text{ m/s})} = 2.43 \times 10^{-12} \text{ m} = 0.00000243 \text{ nm}$

Now find the wavelength change when the photon scatters at 60°: $\Delta \lambda = (0.00000243 \text{ nm})(1-\cos 60°)$ $\Delta \lambda = (0.00000243 \text{ nm})(1-0.5)$ $\Delta \lambda = (0.00000243 \text{ nm})(0.5)$ $\Delta \lambda = 0.000001215 \text{ nm}$

The final wavelength is the initial wavelength plus the change: $\lambda_f = \lambda_i + \Delta \lambda = 0.005 \text{ nm} + 0.000001215 \text{ nm} = 0.005001215 \text{ nm}$

Practice Problem 2: Energy Transfer in Compton Scattering

Solution

Model the interaction as a collision between a photon and an electron and apply conservation of energy and momentum. The wavelength shift for a photon scattered through angle θ is

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

At θ = 90°, cos 90° = 0, so

$\Delta \lambda = \frac{h}{m_e c} = 2.43 \times 10^{-12} \text{ m}$

Next, find the initial wavelength from $E = \frac{hc}{\lambda}$:

$\lambda_i = \frac{hc}{E_i} = \frac{(6.63 \times 10^{-34} \text{ J·s})(3.00 \times 10^8 \text{ m/s})}{(0.511 \times 10^6 \text{ eV})(1.602 \times 10^{-19} \text{ J/eV})} = 2.43 \times 10^{-12} \text{ m}$

Then

$\lambda_f = \lambda_i + \Delta \lambda = 2.43 \times 10^{-12} \text{ m} + 2.43 \times 10^{-12} \text{ m} = 4.86 \times 10^{-12} \text{ m}$

So the scattered photon energy is

$E_f = \frac{hc}{\lambda_f} = \frac{(6.63 \times 10^{-34} \text{ J·s})(3.00 \times 10^8 \text{ m/s})}{4.86 \times 10^{-12} \text{ m}} = 4.09 \times 10^{-14} \text{ J} = 0.2555 \text{ MeV}$

By conservation of energy, the electron's kinetic energy equals the photon's lost energy: $K_e = E_i - E_f = 0.511 \text{ MeV} - 0.2555 \text{ MeV} = 0.2555 \text{ MeV}$

This result works together with conservation of momentum in two dimensions, since the recoiling electron also carries momentum and moves in a different direction from the scattered photon.

Common Misconceptions

• The wavelength shift depends on the photon's starting wavelength. It does not. The shift $\Delta \lambda$ depends only on the scattering angle. A high-energy and a low-energy photon scattered at the same angle get the same shift in wavelength.
• The photon gets absorbed in Compton scattering. The photon is not absorbed. It scatters off the electron and continues with less energy and a longer wavelength, similar to a particle bouncing off another particle.
• Energy is lost in the collision. Total energy is conserved. The photon loses energy, but that energy goes to the electron as kinetic energy, so nothing disappears.
• Compton scattering and the photoelectric effect are the same thing. They both show light acting like particles, but they are different. The photoelectric effect ejects bound electrons from a material, while Compton scattering is a collision with a free electron that leaves a scattered photon behind.
• You can ignore direction when applying momentum conservation. Because the photon and electron leave at different angles, momentum must be conserved in two dimensions, so you resolve it into components.
• A larger scattering angle always means the photon loses all its energy. Even at full backscatter, the photon keeps some energy. The shift is largest there, but the photon does not vanish.

Related AP Physics 2 Guides

• 15.1 Quantum Theory and Wave-Particle Duality
• 15.2 The Bohr Model of Atomic Structure
• 15.3 Emission and Absorption Spectra
• 15.7 Fission, Fusion, and Nuclear Decay
• 15.8 Types of Radioactive Decay
• 15.4 Blackbody Radiation