Topic 15.6 Notes – Compton Scattering

1. What Compton Scattering Is

In Compton scattering, a photon hits an electron that is initially at rest (or nearly so). After the collision:

• The photon changes direction
• The photon’s energy decreases
• Its wavelength increases
• The electron recoils with kinetic energy

The scattered photon always has lower energy and longer wavelength than the incoming photon.

This only makes sense if light behaves like particles called photons.

For photons:

$$E = hf = pc$$

So:

• $E$ = energy
• $p = \frac{E}{c}$ = momentum
• $f$ = frequency
• $c$ = speed of light

That equation $E = pc$ is huge here. It lets you treat the photon like a particle in a collision.

On tests, if they describe a photon losing energy and an electron gaining kinetic energy after a collision, you should immediately think Compton scattering.

2. The Photon-Electron Collision Model

This is a two‑particle collision in two dimensions.

You apply:

• Conservation of energy
• Conservation of momentum in both x and y directions

Here’s the geometry you should picture for Compton scattering:

Compton scattering: momentum diagram

An incoming photon strikes an electron at rest. After the collision, the photon scatters at an angle $\theta$, and the electron recoils at an angle $\phi$. The bottom triangle in the diagram represents the vector addition required by momentum conservation.

What changes during the collision

• Photon energy decreases → frequency decreases
• Wavelength increases since $c = f\lambda$
• Photon momentum changes magnitude and direction
• Electron gains kinetic energy

Conservation equations

Energy conservation

$$E_{i} = E_{f} + K_{e}$$

The photon’s lost energy becomes the electron’s kinetic energy.

Momentum conservation (2D)

• x-direction: initial photon momentum equals sum of x-components after
• y-direction: total y-momentum must remain zero since it started at zero

Because $p = E/c$, you can substitute photon energy directly into momentum equations.

AP Physics 2 expects you to be comfortable setting up both component equations, even if you don’t fully derive the final formula.

3. The Compton Wavelength Equation

When you combine energy and momentum conservation, you get:

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

Where:

• $\Delta \lambda = \lambda_{f} - \lambda_{i}$
• $m_{e}$ = electron mass
• $\theta$ = photon scattering angle

The constant

$$\frac{h}{m_{e} c} = 2.43 \times 10^{-12} \text{ m}$$

is called the Compton wavelength of the electron.

What the equation tells you

• If $\theta = 0^\circ$:
  $\cos 0 = 1$ → $\Delta \lambda = 0$
  No energy transfer.
• As $\theta$ increases:
  $1 - \cos\theta$ increases → bigger wavelength shift → more energy to the electron.
• At $180^\circ$ (backscatter):
  $1 - \cos 180^\circ = 2$
  Maximum wavelength increase.

Important idea: The wavelength shift depends only on the scattering angle, not on the initial wavelength. That surprises students every year.

4. Energy and Wavelength Changes

Since

$$E = \frac{hc}{\lambda}$$

a larger wavelength means smaller energy.

A typical calculation might look like this:

1. Use $\Delta \lambda = \frac{h}{m_{e} c}(1 - \cos\theta)$.
2. Find $\lambda_{f} = \lambda_{i} + \Delta \lambda$.
3. Use $E = hc/\lambda$ to find $E_{f}$.
4. Compute electron kinetic energy:
   $K_{e} = E_{i} - E_{f}$.

Notice how energy, wavelength, frequency, and momentum are all linked. If one changes, they all change.

On free response, you may be asked to explain in words why the wavelength increases. A strong answer mentions:

• Photon transfers energy to electron
• Lower photon energy means lower frequency
• Since $c = f\lambda$, lower frequency means longer wavelength

5. Why Compton Scattering Matters

A pure wave model of light cannot explain:

• Angle‑dependent wavelength shift
• Momentum transfer like a particle collision

Compton scattering matches predictions only if photons:

• Carry discrete energy $E = hf$
• Carry momentum $p = E/c$
• Obey conservation laws like particles

This is one of the clearest experimental proofs that electromagnetic radiation consists of quantized photons.