Bremsstrahlung

In particle physics, bremsstrahlung is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic energy, which is converted into radiation, thus satisfying the law of conservation of energy. The term is also used to refer to the process of producing the radiation. Bremsstrahlung has a continuous spectrum, which becomes more intense and whose peak intensity shifts toward higher frequencies as the change of the energy of the decelerated particles increases.
Broadly speaking, bremsstrahlung or braking radiation is any radiation produced due to the acceleration of a charged particle. This includes synchrotron radiation, cyclotron radiation, and the emission of electrons and positrons during beta decay. However, the term is frequently used in the more narrow sense of radiation produced when electrons decelerate in matter.
Bremsstrahlung emitted from plasma is sometimes referred to as free–free radiation that is, created by electrons that are free before, and remain free after, the emission of a photon. In the same parlance, bound–bound radiation refers to discrete spectral lines, while free–bound radiation refers to the radiative combination process, in which a free electron recombines with an ion.
This article uses SI units, along with the scaled single-particle charge.

Classical description

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

Total radiated power

The total radiated power is
where , is the Lorentz factor, is the vacuum permittivity, signifies a time derivative of and is the charge of the particle.
In the case where velocity is parallel to acceleration, the expression reduces to
where is the acceleration. For the case of acceleration perpendicular to the velocity, for example in synchrotrons, the total power is
Power radiated in the two limiting cases is proportional to or . Since, we see that for particles with the same energy the total radiated power goes as or, which accounts for why electrons lose energy to bremsstrahlung radiation much more rapidly than heavier charged particles. This is the reason a TeV energy electron-positron collider cannot use a circular tunnel, while a proton-proton collider can utilize a circular tunnel. The electrons lose energy due to bremsstrahlung at a rate times higher than protons do.

Angular distribution

The most general formula for radiated power as a function of angle is:
where is a unit vector pointing from the particle towards the observer, and is an infinitesimal solid angle.
In the case where velocity is parallel to acceleration, this simplifies to
where is the angle between and the direction of observation.

Simplified quantum-mechanical description

The full quantum-mechanical treatment of bremsstrahlung is very involved. 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. This analytical solution involves complicated mathematics, and several numerical calculations have been published, such as by Karzas and Latter. Other approximate formulas have been presented, such as in recent work by Weinberg and Pradler and Semmelrock.
This section gives a quantum-mechanical analog of the prior section, but with some simplifications to illustrate the important physics. We give a non-relativistic treatment of the special case of an electron of mass, charge, and initial speed decelerating in the Coulomb field of a gas of heavy ions of charge and number density. The emitted radiation is a photon of frequency and energy. We wish to find the emissivity which is the power emitted per, summed over both transverse photon polarizations. We express it as an approximate classical result times the free−free emission Gaunt factor g_ff accounting for quantum and other corrections:
if, that is, the electron does not have enough kinetic energy to emit the photon. A general, quantum-mechanical formula for exists but is very complicated, and usually is found by numerical calculations. We present some approximate results with the following additional assumptions:

Vacuum interaction: we neglect any effects of the background medium, such as plasma screening effects. This is reasonable for photon frequency much greater than the plasma frequency with the plasma electron density. Note that light waves are evanescent for and a significantly different approach would be needed.
Soft photons:, that is, the photon energy is much less than the initial electron kinetic energy.

With these assumptions, two unitless parameters characterize the process:, which measures the strength of the electron-ion Coulomb interaction, and, which measures the photon "softness" and we assume is always small. In the limit, the quantum-mechanical Born approximation gives:
In the opposite limit, the full quantum-mechanical result reduces to the purely classical result
where is the Euler–Mascheroni constant. Note that which is a purely classical expression without the Planck constant.
A semi-classical, heuristic way to understand the Gaunt factor is to write it as where and are maximum and minimum "impact parameters" for the electron-ion collision, in the presence of the photon electric field. With our assumptions, : for larger impact parameters, the sinusoidal oscillation of the photon field provides "phase mixing" that strongly reduces the interaction. is the larger of the quantum-mechanical de Broglie wavelength and the classical distance of closest approach where the electron-ion Coulomb potential energy is comparable to the electron's initial kinetic energy.
The above approximations generally apply as long as the argument of the logarithm is large, and break down when it is less than unity. Namely, these forms for the Gaunt factor become negative, which is unphysical. A rough approximation to the full calculations, with the appropriate Born and classical limits, is

Thermal bremsstrahlung in a medium: emission and absorption

This section discusses bremsstrahlung emission and the inverse absorption process in a macroscopic medium. We start with the equation of radiative transfer, which applies to general processes and not just bremsstrahlung:
is the radiation spectral intensity, or power per summed over both polarizations. is the emissivity, analogous to defined above, and is the absorptivity. and 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. If is uniform in space and time, then the left-hand side of the transfer equation is zero, and we find
If the matter and radiation are also in thermal equilibrium at some temperature, then must be the blackbody spectrum:
Since and are independent of, this means that must be the blackbody spectrum whenever the matter is in equilibrium at some temperature – regardless of the state of the radiation. This allows us to immediately know both and once one is known – for matter in equilibrium.

In plasma: approximate classical results

NOTE: this section currently gives formulas that apply in the Rayleigh–Jeans limit, and does not use a quantized treatment of radiation. Thus a usual factor like does not appear. The appearance of in 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 behavior. A detailed treatment is given by Bekefi, while a simplified one is given by Ichimaru. In this section we follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber,
Consider a uniform plasma, with thermal electrons distributed according to the Maxwell–Boltzmann distribution with the temperature. Following Bekefi, the power spectral density of the bremsstrahlung radiated, is calculated to be
where is the electron plasma frequency, is the photon frequency, is the number density of electrons and ions, and other symbols are physical constants. The second bracketed factor is the index of refraction of a light wave in a plasma, and shows that emission is greatly suppressed for . This formula thus only applies for. This formula should be summed over ion species in a multi-species plasma.
The function is the exponential integral, and the unitless quantity is
is a maximum or cutoff wavenumber, arising due to binary collisions, and can vary with ion species. Roughly, when , where eV is the Hartree energy, and is the electron thermal de Broglie wavelength. Otherwise, where is the classical Coulomb distance of closest approach.
For the usual case, we find
The formula for is approximate, in that it neglects enhanced emission occurring for slightly above
In the limit, we can approximate as where is the Euler–Mascheroni constant. The leading, logarithmic term is frequently used, and resembles the Coulomb logarithm that occurs in other collisional plasma calculations. For the log term is negative, and the approximation is clearly inadequate. Bekefi gives corrected expressions for the logarithmic term that match detailed binary-collision calculations.
The total emission power density, integrated over all frequencies, is
Note the appearance of due to the quantum nature of. In practical units, a commonly used version of this formula for is
This formula is 1.59 times the one given above, with the difference due to details of binary collisions. Such ambiguity is often expressed by introducing Gaunt factor, e.g. in one finds
where everything is expressed in the CGS units.