AP Physics 2-15.6 Compton Scattering- Study Notes- New Syllabus
AP Physics 2- 15.6 Compton Scattering – Study Notes
AP Physics 2- 15.6 Compton Scattering – Study Notes – per latest Syllabus.
Key Concepts:
- Compton Scattering
Compton Scattering
Compton scattering is the phenomenon in which an X-ray or gamma-ray photon collides with a free or loosely bound electron, resulting in a photon of longer wavelength (lower energy) and a recoiling electron. This demonstrates the particle nature of light and the transfer of momentum between photons and electrons.
Physical Concept
A photon with initial wavelength \(\mathrm{\lambda}\) collides with a stationary electron. The photon is scattered at an angle \(\mathrm{\theta}\) relative to its original direction. The electron recoils with kinetic energy \(\mathrm{K_e}\).
The change in wavelength (\(\mathrm{\Delta \lambda}\)) of the photon is given by the
Compton formula:
\(\mathrm{\Delta \lambda = \lambda’ – \lambda = \dfrac{h}{m_e c} (1 – \cos \theta)} \)
where
- \(\mathrm{h}\) is Planck’s constant,
- \(\mathrm{m_e}\) is the electron mass,
- \(\mathrm{c}\) is the speed of light, and
- \(\mathrm{\theta}\) is the scattering angle.
Compton Wavelength
The quantity \(\mathrm{\lambda_C = \dfrac{h}{m_e c} \approx 2.43 \times 10^{-12} \, m}\) is called the Compton wavelength of the electron.
- Maximum wavelength shift occurs when the photon is scattered backward (\(\mathrm{\theta = 180^\circ}\)).
- This effect confirms that photons carry momentum \(\mathrm{p = \dfrac{h}{\lambda}}\) and behave as particles in collisions.
Key Features
- Photon loses energy and increases wavelength after scattering; the electron gains kinetic energy.
- Demonstrates particle-like behavior of light and momentum transfer.
- Depends on the scattering angle; no change in wavelength occurs for \(\mathrm{\theta = 0^\circ}\).
- Supports the quantum theory of electromagnetic radiation.
Example :
An X-ray photon of wavelength \(\mathrm{\lambda = 0.05 \, nm}\) is scattered at an angle \(\mathrm{\theta = 90^\circ}\). Find the wavelength shift (\(\mathrm{\Delta \lambda}\)).
▶️ Answer/Explanation
Step 1: Compton formula: \(\mathrm{\Delta \lambda = \lambda_C (1 – \cos \theta) = 2.43 \times 10^{-12} (1 – \cos 90^\circ)}\)
Step 2: Calculate: \(\mathrm{\cos 90^\circ = 0 \Rightarrow \Delta \lambda = 2.43 \times 10^{-12} \, m}\)
Step 3: Convert to nm: \(\mathrm{\Delta \lambda \approx 2.43 \times 10^{-3} \, nm}\)
Example :
Find the wavelength of the scattered photon if the incident photon has \(\mathrm{\lambda = 0.05 \, nm}\) and scattering angle \(\mathrm{\theta = 180^\circ}\).
▶️ Answer/Explanation
Step 1: Maximum wavelength shift: \(\mathrm{\Delta \lambda = \lambda_C (1 – \cos 180^\circ) = 2.43 \times 10^{-12} (1 – (-1)) = 4.86 \times 10^{-12} \, m}\)
Step 2: Scattered photon wavelength: \(\mathrm{\lambda’ = \lambda + \Delta \lambda = 0.05 \, nm + 0.00486 \, nm \approx 0.05486 \, nm}\)
Step 3: Interpretation: The photon is red-shifted due to energy transfer to the electron.