Bethe-Heitler formula
The 1934 quantum-mechanical bremsstrahlung cross-section derived by Hans Bethe and Walter Heitler, extending Kramers's classical formula to relativistic electrons and thin targets. Full treatment in a later QED branch.
Definition
This is a placeholder entry. The Bethe-Heitler formula is the quantum-mechanical differential cross-section for bremsstrahlung emission by a relativistic electron scattering in the Coulomb field of a screened nucleus, derived by Hans Bethe and Walter Heitler in 1934. It was among the first systematic applications of Paul Dirac's electron theory and the newly developed perturbative formalism for quantum electrodynamics to a specific scattering process, and it successfully reproduced the observed X-ray continuum in thin targets where Kramers's classical thick-target formula breaks down.
The formula gives the photon-energy spectrum dσ/dk ∝ Z²α r_e² · [function of (k, E_e, E_e − k, Z)] where α is the fine-structure constant, r_e is the classical electron radius, Z is the target nuclear charge, E_e is the incoming electron energy, and k is the emitted photon energy. At relativistic energies it includes a logarithmic factor from the nuclear-electric-field screening by atomic electrons, and it predicts the pair-production cross-section as a time-reversed cousin by crossing symmetry. The full quantitative derivation via Feynman diagrams and the modern radiative-correction refinements belong to a later quantum-electrodynamics branch.