Bremsstrahlung
X-ray tube

In an X-ray tube, electrons are accelerated in a vacuum by an electric field towards a piece of metal called the "target". X-rays are emitted as the electrons slow down (decelerate) in the metal. The output spectrum consists of a continuous spectrum of X-rays, with additional sharp peaks at certain energies. The continuous spectrum is due to bremsstrahlung, while the sharp peaks are characteristic X-rays associated with the atoms in the target. For this reason, bremsstrahlung in this context is also called continuous X-rays.

The shape of this continuum spectrum is approximately described by Kramers' law.

The formula for Kramers' law is usually given as the distribution of intensity (photon count) I {\displaystyle I} against the wavelength λ {\displaystyle \lambda } of the emitted radiation: I (λ) d λ = K (λ λ min − 1) d λ λ 2 {\displaystyle I(\lambda)\,d\lambda =K\left({\frac {\lambda }{\lambda _{\min }}}-1\right){\frac {d\lambda }{\lambda ^{2}}}}

The constant K is proportional to the atomic number of the target element, and λ min {\displaystyle \lambda _{\min }} is the minimum wavelength given by the Duane–Hunt law.

The spectrum has a sharp cutoff at λ min {\displaystyle \lambda _{\min }}, which is due to the limited energy of the incoming electrons. For example, if an electron in the tube is accelerated through 60 kV, then it will acquire a kinetic energy of 60 keV, and when it strikes the target it can create X-rays with energy of at most 60 keV, by conservation of energy. (This upper limit corresponds to the electron coming to a stop by emitting just one X-ray photon. Usually the electron emits many photons, and each has an energy less than 60 keV.) A photon with energy of at most 60 keV has wavelength of at least 21 pm, so the continuous X-ray spectrum has exactly that cutoff, as seen in the graph. More generally the formula for the low-wavelength cutoff, the Duane–Hunt law, is: λ min = h c e V ≈ 1239.8 V p m / k V {\displaystyle \lambda _{\min }={\frac {hc}{eV}}\approx {\frac {1239.8}{V}}\,\mathrm {pm/kV} } where h is the Planck constant, c is the speed of light, V is the voltage that the electrons are accelerated through, e is the elementary charge, and pm is picometres.
startend:end_checker