Bremsstrahlung

Bremsstrahlung — German for "braking radiation" — is the continuous electromagnetic radiation emitted when a charged particle decelerates in the electrostatic field of another charged particle, typically an atomic nucleus. The physics is a direct application of the Larmor formula: any acceleration produces radiation, and deceleration during a Coulomb scattering event is no exception. An electron passing through a tungsten target in an X-ray tube suffers many small-angle Rutherford deflections; at each close encounter with a nucleus its transverse acceleration peaks, producing a brief pulse of radiation whose Fourier transform contributes to the continuous X-ray spectrum.

The thick-target spectrum (where the electron slows to rest inside the target, radiating many photons on the way) was worked out classically by Hendrik Kramers in 1923 and has the remarkably simple form dN/dE ∝ (E_max − E)/E — a hyperbolic falloff in photon number from near-zero energies up to a sharp cutoff at E_max = eU, the Duane-Hunt limit, where U is the accelerating voltage of the tube. The thin-target spectrum (electron passes through without losing much energy) has a different shape, worked out quantum-mechanically by Hans Bethe and Walter Heitler in 1934. The total bremsstrahlung power radiated per electron scales as Z²/m², which is why heavy-nucleus targets (tungsten, Z = 74) produce efficient bremsstrahlung for electrons but lightweight particles (positrons, electrons) are the preferred projectile. Astrophysical bremsstrahlung is responsible for the thermal X-ray emission from hot ionised gas in galaxy clusters (T ∼ 10⁷ K, kT ∼ 1 keV), where free-free transitions of thermal electrons in the proton Coulomb field produce a characteristic exponential bremsstrahlung spectrum whose slope directly measures the cluster temperature.