1. Introduction to the Compton Effect
The Compton effect, also known as incoherent scattering or Compton scattering, occurs when a photon interacts with a loosely bound or essentially 'free and stationary' electron. In this interaction, part of the photon’s energy is transferred to the electron, resulting in a scattered photon with reduced energy (\( h\nu' \)) and an ejected recoil (Compton) electron with kinetic energy (\( E_K \)). The scattering angle of the photon is denoted by \( \theta \), and the angle of the recoil electron is denoted by \( \phi \).
2. Conservation Laws in the Compton Effect
The Compton effect is governed by the conservation of energy and momentum. The key equations for the interaction are:
\[ E_{\gamma} + E_{\text{K}} = E_{\gamma'} + E_{\text{K}}' \]
\[ p_{\gamma} \cos \theta = p_{\text{K}} \cos \phi \]
\[ p_{\gamma} \sin \theta = p_{\text{K}} \sin \phi \]
Where: - \( m_e c^2 \) is the rest energy of the electron (0.511 MeV), - \( E_K \) is the kinetic energy of the recoil (Compton) electron, - \( v \) is the velocity of the recoil electron, - \( c \) is the speed of light in a vacuum.
3. The Compton Wavelength-Shift Equation
The basic Compton equation, also called the wavelength-shift equation, is derived from the conservation of energy and momentum and is given by:
Where: - \( \lambda \) is the wavelength of the incident photon, - \( \lambda' \) is the wavelength of the scattered photon, - \( \Delta \lambda \) is the wavelength shift, - \( \lambda_C \) is the Compton wavelength of the electron (\( 0.024 \, \text{Å} \)).
4. Energy and Recoil Electron Kinetics
The energy of the scattered photon (\( h\nu' \)) and the kinetic energy of the recoil electron (\( E_K \)) depend on the incident photon energy and the scattering angle \( \theta \). These are expressed as:
Specific cases for scattering angles are given as: - For \( \theta = 0 \) (forward scattering), \( h\nu' = h\nu \). - For \( \theta = \pi \) (back scattering), \( h\nu' = \frac{h\nu}{1 + 2 \frac{h\nu}{m_e c^2}} \).
5. Compton Attenuation Coefficients
The Compton effect is important for photon attenuation. The electronic attenuation coefficient \( \sigma_C \) describes how the intensity of the photon decreases as it interacts with electrons. At low photon energies, the coefficient is \( 0.665 \times 10^{-24} \, \text{cm}^2/\text{electron} \), but it decreases as photon energy increases. The mass attenuation coefficient \( \sigma_C/\rho \) depends on the material and photon energy but is generally independent of the atomic number \( Z \) of the material.
For example, at photon energies of around 1 MeV, the mass attenuation coefficient \( \sigma_C/\rho \) is the same for both carbon (Z=6) and lead (Z=82), about \( 0.1 \, \text{cm}^2/\text{electron} \).
6. Energy Transfer in the Compton Effect
The maximum energy transfer to the recoil electron occurs when the photon is backscattered (\( \theta = \pi \)), and the maximum fractional energy transfer to the electron is:
Where \( \epsilon = \frac{h\nu}{m_e c^2} \) is the normalized incident photon energy.
The mean energy transfer \( f_{\text{C}} \) is plotted against photon energy in the Compton graph, showing that the energy transfer is low at low photon energies and increases as the photon energy increases.
Example Calculation: Energy Transfer in Compton Scattering
For an incident photon energy \( h\nu = 1 \, \text{MeV} \) and a scattering angle \( \theta = \pi/2 \), the energy of the scattered photon can be calculated using the Compton equation:
Therefore, the kinetic energy of the recoil electron is: