Kramers formula

This is a placeholder entry. The Kramers formula is the classical thick-target bremsstrahlung X-ray spectrum derived by the Dutch theoretician Hendrik Anthony Kramers in 1923. For an electron beam fired into a target thick enough that each electron slows to rest inside the target (radiating many photons on the way), integrating the single-scattering bremsstrahlung cross-section over the electron's energy-loss cascade gives the remarkably simple shape dN/dE ∝ (E_max − E)/E, where E is the photon energy and E_max = eU is the Duane-Hunt cutoff set by the accelerating voltage. The continuum climbs hyperbolically from near-zero energies, peaks at low E, and falls linearly to zero at E_max.

The formula is an essential workhorse of medical and industrial X-ray physics: it predicts the continuum-to-characteristic-line ratio in tube spectra, the efficiency of bremsstrahlung production as a function of tube voltage (which scales as Z·U²), and the optimal target material and filtration for a given imaging application. The quantum-mechanical refinement (the Bethe-Heitler cross-section, 1934) reproduces the Kramers shape at low energies and modifies it at relativistic tube voltages. The full treatment of the Kramers formula and its quantum corrections belongs to an X-ray physics or quantum-mechanics chapter in a later branch.