AcademyPhotons
Academy
Compton Scattering
Level 1 - Physics topic page in Photons.
Principle
Compton scattering shows that photons carry momentum. A photon scattered by an electron loses energy, its wavelength increases, and the electron recoils.
The wavelength shift depends only on the scattering angle for a free electron initially at rest.
Notation
\(\lambda\)
incident photon wavelength
\(\mathrm{m}\)
\(\lambda'\)
scattered photon wavelength
\(\mathrm{m}\)
\(\Delta\lambda\)
Compton wavelength shift
\(\mathrm{m}\)
\(\theta\)
photon scattering angle
\(\mathrm{rad}\)
\(m_e\)
electron rest mass
\(\mathrm{kg}\)
\(\lambda_C\)
electron Compton wavelength
\(\mathrm{m}\)
Method
Derivation 1: Assign photon energy and momentum
The photon is massless but carries energy and momentum.
Photon energy
\[E=hf=\frac{hc}{\lambda}\]
Photon momentum
\[p=\frac{E}{c}=\frac{h}{\lambda}\]
Derivation 2: Apply relativistic conservation
Energy and momentum are conserved in the photon-electron collision. Eliminating the recoil electron variables gives the wavelength shift.
Compton shift
\[\Delta\lambda=\lambda'-\lambda=\frac{h}{m_ec}(1-\cos\theta)\]
Electron Compton wavelength
\[\lambda_C=\frac{h}{m_ec}\]
Derivation 3: Interpret angle limits
Forward scattering has no wavelength change. Backscattering gives the largest shift.
Forward scattering
\[\theta=0\Rightarrow\Delta\lambda=0\]
Backscattering
\[\theta=180^\circ\Rightarrow\Delta\lambda=2\lambda_C\]
Rules
Photon momentum
\[p=\frac{h}{\lambda}\]
Compton shift
\[\lambda'-\lambda=\frac{h}{m_ec}(1-\cos\theta)\]
Electron Compton wavelength
\[\lambda_C=\frac{h}{m_ec}=2.43\times10^{-12}\,\mathrm{m}\]
Photon energy
\[E=\frac{hc}{\lambda}\]
Examples
Question
An x-ray photon scatters through
\[90^\circ\]
Find the wavelength shift.Answer
\[\Delta\lambda=\lambda_C(1-\cos90^\circ)=2.43\times10^{-12}\,\mathrm{m}\]
Checks
- The scattered photon wavelength is never shorter than the incident wavelength in ordinary Compton scattering from an electron at rest.
- The shift depends on angle, not on the incident wavelength.
- Energy lost by the photon becomes electron kinetic energy.
- The effect is easier to observe for x-rays and gamma rays than for visible light.