7.4 Compton scattering and pair production

Quantum Electrodynamics (QED) explains how light and matter interact. Compton scattering and pair production are two key processes in QED that show how photons can transfer energy to electrons or create particle-antiparticle pairs.

These processes are crucial for understanding high-energy physics and astrophysics. They demonstrate the wave-particle duality of light and the conversion between energy and mass, fundamental concepts in quantum mechanics and relativity.

Compton Scattering

Process and Key Features

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• Inelastic scattering of a photon by a charged particle, usually an electron, resulting in a decrease in energy (increase in wavelength) of the photon
• A portion of the photon's energy is transferred to the recoiling electron, and the photon is deflected from its original path by a scattering angle
• The change in wavelength of the scattered photon, known as the Compton shift, depends only on the scattering angle and is independent of the initial photon energy
• The Compton scattering cross-section is proportional to the classical electron radius squared and depends on the photon energy and scattering angle

Klein-Nishina Formula and Applications

• The Klein-Nishina formula describes the differential cross-section for Compton scattering, taking into account relativistic effects and the spin of the electron
• Compton scattering is an important process in high-energy astrophysics, as it can modify the spectra of X-rays and gamma-rays from celestial sources (X-ray binaries, active galactic nuclei)
• Compton scattering is also used in medical imaging techniques such as Compton scatter imaging and Compton camera imaging for nuclear medicine and radiation therapy
• In material science, Compton scattering is employed to study the electronic structure and momentum distribution of electrons in materials (Compton profile analysis)

Pair Production

Phenomenon and Threshold Energy

• Creation of an elementary particle and its antiparticle from a neutral boson, typically a photon, in the presence of a strong electromagnetic field, such as that of an atomic nucleus
• In quantum electrodynamics, pair production usually refers to the creation of an electron-positron pair from a high-energy photon interacting with a nucleus
• The threshold energy for pair production is $2mc^2$, where $m$ is the rest mass of the created particle (electron mass for electron-positron pair production), and $c$ is the speed of light, corresponding to the rest mass energy of the two particles
• Pair production demonstrates the equivalence of mass and energy as described by Einstein's equation $E=mc^2$

Cross-section and Importance

• The cross-section for pair production increases with photon energy above the threshold and is proportional to the square of the atomic number of the nucleus involved
• Pair production is the dominant photon interaction process at high energies, typically above a few MeV, and plays a crucial role in the absorption of high-energy photons in matter (gamma-ray absorption in lead shielding)
• Pair production is essential in the development of electromagnetic cascades in high-energy particle detectors (calorimeters) and in the Earth's atmosphere (cosmic-ray air showers)
• The creation of electron-positron pairs is also possible in strong electric fields, a process called Schwinger pair production, which is relevant in extreme astrophysical environments (magnetars, black hole vicinity)

Cross-sections for Compton Scattering and Pair Production

Calculating Cross-sections using Feynman Rules

• Feynman rules are a set of prescriptions for constructing the mathematical expressions describing the probability amplitudes of particle interactions in quantum field theory
• To calculate the cross-section for Compton scattering using Feynman rules:
  1. Draw the Feynman diagrams representing the process, including the incoming photon and electron, the outgoing photon and electron, and the intermediate virtual electron state
  2. Assign four-momenta to each particle in the diagram, ensuring conservation of four-momentum at each vertex
  3. Write down the mathematical expression for each vertex and propagator in the diagram using the QED Feynman rules
  4. Multiply the expressions for each diagram and sum over all possible diagrams to obtain the total amplitude
  5. Square the total amplitude and sum over final spins and polarizations, then average over initial spins and polarizations to obtain the spin-averaged squared amplitude
  6. Integrate the spin-averaged squared amplitude over the phase space of the final-state particles to obtain the differential cross-section
• To calculate the cross-section for pair production using Feynman rules, follow a similar procedure:
  1. Draw the Feynman diagram representing the process, including the incoming photon, the outgoing electron-positron pair, and the intermediate virtual electron state
  2. Assign four-momenta to each particle in the diagram, ensuring conservation of four-momentum at each vertex
  3. Write down the mathematical expression for each vertex and propagator in the diagram using the QED Feynman rules
  4. Square the amplitude and sum over final spins and polarizations to obtain the squared amplitude
  5. Integrate the squared amplitude over the phase space of the final-state particles to obtain the differential cross-section

Complexity of Calculations

• The calculations of cross-sections using Feynman rules involve complex mathematical expressions and require proper handling of spinors, polarization vectors, and gamma matrices
• The evaluation of Feynman diagrams often involves performing loop integrals, which can be challenging and require the use of regularization and renormalization techniques to obtain finite results
• Automated tools and software packages (FeynCalc, MadGraph) are available to assist in the calculation of cross-sections and the manipulation of Feynman diagrams
• Perturbative expansions and approximation methods (leading order, next-to-leading order) are employed to simplify calculations and obtain results with the desired level of accuracy

Angular and Energy Distributions of Scattered Photons and Pairs

Compton-scattered Photons

• The angular distribution of Compton-scattered photons is described by the Klein-Nishina formula, which gives the differential cross-section as a function of the scattering angle and the initial photon energy
• The Klein-Nishina formula predicts that the scattering is strongly forward-peaked at high photon energies, while at lower energies, the distribution becomes more isotropic
• The energy distribution of Compton-scattered photons can be derived from the Compton scattering kinematics, relating the scattered photon energy to the initial photon energy and the scattering angle
• The energy of the scattered photon decreases with increasing scattering angle, with the maximum energy loss occurring for backscattering (180-degree scattering angle)

Electron-Positron Pairs

• The angular distribution of electron-positron pairs produced by pair production is peaked in the forward direction, with the opening angle between the electron and positron decreasing with increasing photon energy
• The energy distribution of the produced electron and positron is continuous, ranging from zero to the maximum available energy (initial photon energy minus the rest mass energy of the two particles)
• The energy sharing between the electron and positron is symmetric on average, but individual pairs can have asymmetric energy distributions

Experimental Measurements

• The angular and energy distributions of Compton-scattered photons and produced pairs can be measured experimentally using detectors such as scintillators, semiconductor detectors, or pair spectrometers
• Compton polarimeters exploit the angular distribution of Compton-scattered photons to measure the polarization of X-rays and gamma-rays (astrophysical sources, synchrotron radiation)
• Pair spectrometers, consisting of a converter foil and a magnetic spectrometer, are used to measure the energy and angular distributions of electron-positron pairs in high-energy physics experiments (particle colliders)
• The comparison of experimental measurements with theoretical predictions provides a means to study these processes and test the validity of quantum electrodynamics in various energy regimes