In particle physics, (; ) 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 (i.e., photons), thus satisfying the law of conservation of energy. The term is also used to refer to the process of producing the radiation. 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, or braking radiation is any radiation produced due to the acceleration (positive or negative) of a charged particle, which includes synchrotron radiation (i.e., photon emission by a relativistic particle), cyclotron radiation (i.e. photon emission by a non-relativistic particle), and the emission of electrons and positrons during beta decay. However, the term is frequently used in the more narrow sense of radiation from electrons (from whatever source) slowing in matter.
Bremsstrahlung emitted from plasma is sometimes referred to as free–free radiation. This refers to the fact that the radiation in this case is created by electrons that are free (i.e., not in an atomic or molecular bound state) before, and remain free after, the emission of a photon. In the same parlance, bound–bound radiation refers to discrete spectral lines (an electron "jumps" between two bound states), while free–bound radiation refers to the radiative combination process, in which a free electron recombines with an ion.
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 (the velocity of the particle divided by the speed of light), 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 (i.e., linear motion), 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 (e.g., muons, protons, alpha particles). This is the reason a TeV energy electron-positron collider (such as the proposed International Linear Collider) cannot use a circular tunnel (requiring constant acceleration), while a proton-proton collider (such as the Large Hadron 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 bit of solid angle.
In the case where velocity is parallel to acceleration (for example, linear motion), 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 A. 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 (solid angle in photon velocity space * photon frequency), summed over both transverse photon polarizations. We express it as an approximate classical result times the free−free emission Gaunt factor gff accounting for quantum and other corrections: