Bremsstrahlung

However, the term is frequently used in the more narrow sense of radiation from electrons (from whatever source) slowing in matter.

If quantum effects are negligible, an accelerating charged particle radiates power as described by the Larmor formula and its relativistic generalization.

The "vacuum case" of the interaction of one electron, one ion, and one photon, using the pure Coulomb potential, has an exact solution that was probably first published by Arnold Sommerfeld in 1931.

[5] This analytical solution involves complicated mathematics, and several numerical calculations have been published, such as by Karzas and Latter.

We express it as an approximate classical result times the free−free emission Gaunt factor gff accounting for quantum and other corrections:

are maximum and minimum "impact parameters" for the electron-ion collision, in the presence of the photon electric field.

: for larger impact parameters, the sinusoidal oscillation of the photon field provides "phase mixing" that strongly reduces the interaction.

The above approximations generally apply as long as the argument of the logarithm is large, and break down when it is less than unity.

are properties of the matter, not the radiation, and account for all the particles in the medium – not just a pair of one electron and one ion as in the prior section.

below is due to the quantum-mechanical treatment of collisions.In a plasma, the free electrons continually collide with the ions, producing bremsstrahlung.

A complete analysis requires accounting for both binary Coulomb collisions as well as collective (dielectric) behavior.

[10] In this section we follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber,

The second bracketed factor is the index of refraction of a light wave in a plasma, and shows that emission is greatly suppressed for

is a maximum or cutoff wavenumber, arising due to binary collisions, and can vary with ion species.

Bekefi gives corrected expressions for the logarithmic term that match detailed binary-collision calculations.

[18] However, there is still some debate as to whether or not there are significant polarizational bremsstrahlung contributions in experiments involving fast electrons incident on solid targets.

[19][20][21] It is worth noting that the term "polarizational" is not meant to imply that the emitted bremsstrahlung is polarized.

[22] In an X-ray tube, electrons are accelerated in a vacuum by an electric field towards a piece of material called the "target".

Already in the early 20th century physicists found out that X-rays consist of two components, one independent of the target material and another with characteristics of fluorescence.

[24] The German term itself was introduced in 1909 by Arnold Sommerfeld in order to explain the nature of the first variety of X-rays.

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.

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.

[27] In some cases, such as the decay of 32P, the bremsstrahlung produced by shielding the beta radiation with the normally used dense materials (e.g. lead) is itself dangerous; in such cases, shielding must be accomplished with low density materials, such as Plexiglas (Lucite), plastic, wood, or water;[28] as the atomic number is lower for these materials, the intensity of bremsstrahlung is significantly reduced, but a larger thickness of shielding is required to stop the electrons (beta radiation).

These photons become manifest in terrestrial gamma-ray flashes and are the source for beams of electrons, positrons, neutrons and protons.

[29] The appearance of Bremsstrahlung photons also influences the propagation and morphology of discharges in nitrogen–oxygen mixtures with low percentages of oxygen.

[31] They assumed plane waves for electrons which scatter at the nucleus of an atom, and derived a cross section which relates the complete geometry of that process to the frequency of the emitted photon.

The quadruply differential cross section, which shows a quantum mechanical symmetry to pair production, is where

The absolute value of the virtual photon between the nucleus and electron is The range of validity is given by the Born approximation

For practical applications (e.g. in Monte Carlo codes) it can be interesting to focus on the relation between the frequency

Köhn and Ebert integrated the quadruply differential cross section by Bethe and Heitler over

Bremsstrahlung produced by a high-energy electron deflected in the electric field of an atomic nucleus.
Field lines and modulus of the electric field generated by a (negative) charge first moving at a constant speed and then stopping quickly to show the generated Bremsstrahlung radiation.
Bekefi's classical result for the bremsstrahlung emission power spectrum from a Maxwellian electron distribution. It rapidly decreases for large , and is also suppressed near . This plot is for the quantum case , and . The blue curve is the full formula with , the red curve is the approximate logarithmic form for .
Relativistic corrections to the emission of a 30 keV photon by an electron impacting on a proton.
Spectrum of the X-rays emitted by an X-ray tube with a rhodium target, operated at 60 kV . The continuous curve is due to bremsstrahlung, and the spikes are characteristic K lines for rhodium. The curve goes to zero at 21 pm in agreement with the Duane–Hunt law , as described in the text.