Compton scattering describes elastic collision between photon and free electron. The photon transfers energy and momentum to electron; photon wavelength increases (frequency decreases). Formula: Δλ = (h/(m_e c))(1 - cos θ), where h is Planck's constant, m_e is electron rest mass, c is speed of light, and θ is scattering angle. The term h/(m_ec) = 2.426×10⁻¹² m is the Compton wavelength of electron. Key insight: rest-mass energy converts partially to kinetic energy of recoil electron. Discovered by Arthur Compton in 1923, demonstrating wave-particle duality—photons behave as particles with momentum p = E/c = hf/c. Compton effect proves light consists of discrete energy packets. Applications: X-ray scattering used in materials analysis (crystallography). Medical imaging: Compton effect primary energy loss mechanism in tissues. Astrophysics: inverse Compton scattering in accretion disks around black holes increases X-ray photon energies. Industrial gamma-ray sources use Compton scattering for detection. Energy-angle relationship: higher angle scattering (θ > 90°) produces greater wavelength shift, maximum at θ = 180° (backscatter) where Δλ = 2h/(m_ec). At θ = 0° (forward scattering, no collision), Δλ = 0 (wavelength unchanged). At moderate angles (θ = 90°), Δλ = h/(m_ec) exactly.

Quantum mechanics significance: Compton discovered first direct evidence for photon's particle nature. Classical electromagnetic theory predicted scattering independent of wavelength (Thomson scattering); Compton's data showed wavelength-dependent shift contradicting classical prediction. Einstein's photon hypothesis E=hf explained results perfectly—confirmed modern quantum mechanics foundation. Modern applications: synchrotron radiation uses Compton backscattering for precision energy measurements. Gamma-ray imaging employs Compton kinematics (3-position reconstruction from energy, timing, angles) for directional detection. Positron-electron pair production (competing process at high E) limits Compton dominance above few MeV. Klein-Nishina formula refines Compton for relativistic regimes. Recoil electron receives variable energy depending on scattering angle: E_e = E_γ(1 - 1/(1 + (E_γ/(m_ec²))(1-cosθ))). At low energies, electron barely recoils; at high energies (E_γ >> m_ec²), electron carries substantial kinetic energy. Pair production dominates above threshold (~1.022 MeV).