Compton Scattering

In 1923 Arthur Compton shone X-rays onto a graphite target (loosely bound, effectively free electrons) and measured the scattered X-rays at various angles $\theta$. Classical wave theory predicted that scattered X-rays should have the same wavelength as the incident ones (the electron should re-radiate at the driving frequency). Instead, Compton found the scattered wavelength was always longer by an amount depending only on $\theta$ — not on the material or incident wavelength.

This was definitive proof that electromagnetic radiation carries momentum and behaves as discrete particles (photons) in collisions. Treating the interaction as a relativistic billiard-ball collision between a photon ($E = hc/\lambda$, $p = h/\lambda$) and a stationary electron (mass $m_e$), conservation of energy and momentum yields the Compton formula exactly.

The recoiling electron carries kinetic energy equal to the photon's energy loss, directly observable as ionisation in the detector.

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

The quantity $\lambda_C = h/(m_e c) = 2.426\,\text{pm}$ is the Compton wavelength of the electron.

• At $\theta = 0°$: $\Delta\lambda = 0$ (no scattering, no shift).
• At $\theta = 90°$: $\Delta\lambda = \lambda_C \approx 2.43\,\text{pm}$.
• At $\theta = 180°$ (backscatter): $\Delta\lambda = 2\lambda_C \approx 4.85\,\text{pm}$ (maximum shift).

Incident X-rays have $\lambda = 0.0711\,\text{nm}$. Find the scattered wavelength at $\theta = 90°$.

$\Delta\lambda = \frac{h}{m_e c}(1 - \cos 90°) = 2.426\,\text{pm} \times (1 - 0) = 2.426\,\text{pm}$

$\lambda' = 0.0711\,\text{nm} + 0.00243\,\text{nm} = 0.0735\,\text{nm}$

Energy of incident photon: $E = hc/\lambda = (6.626\times10^{-34})(3\times10^8)/(7.11\times10^{-11}) \approx 2.80\,\text{keV}$.

Energy loss $\approx E - hc/\lambda' \approx 0.09\,\text{keV}$, which goes to the recoil electron.

Compton scattering is the dominant X-ray interaction in human tissue for diagnostic energies (30–150 keV), scattering photons and creating secondary electrons that ionise cells — this is why radiation is biologically damaging. PET scanners detect pairs of 511 keV photons (from electron-positron annihilation); Compton scatter in tissue is a major source of image noise. Astrophysicists use inverse Compton scattering (energetic electrons boosting photon energy) to explain cosmic gamma-ray sources.

1. Calculate the Compton shift $\Delta\lambda$ for scattering at $\theta = 60°$. Use $\lambda_C = 2.426\,\text{pm}$.

2. Why does Compton scattering prove the particle nature of light, when the photoelectric effect alone was not considered fully conclusive evidence?