Bremsstrahlung

Bremsstrahlung (), from "to brake" and "radiation"; i.e., "braking radiation" or "deceleration radiation", 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. "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 deceleration (negative acceleration) 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 one—to the radiative combination process, in which a free electron recombines with an ion.
Classical description.
This section is written from a purely classical perspective, with quantum effects neglected. An accelerating charged particle radiates power, as described by the Larmor formula and its relativistic generalizations.
Total radiated power.
The total radiated power is
where formula_2 (the velocity of the particle divided by the speed of light), formula_3 is the Lorentz factor, formula_4 signifies a time derivative of formula_5, and "q" is the charge of the particle.
In the case where velocity is parallel to acceleration (i.e., linear motion), the expression reduces to
where formula_7 is the acceleration. For the case of acceleration perpendicular to the velocity (formula_8), for example in synchrotrons, the total power is
Power radiated in the two limiting cases is proportional to formula_10 formula_11 or formula_12 formula_13. Since formula_14, we see that for particles with the same energy formula_15 the total radiated power goes as formula_16 or formula_17, 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 formula_18 times higher than protons do.
Angular distribution.
The most general formula for radiated power as a function of angle is:
where formula_20 is a unit vector pointing from the particle towards the observer, and formula_21 is an infinitesimal bit of solid angle.
In the case where velocity is parallel to acceleration (for example, linear motion), this simplifies to
where formula_23 is the angle between formula_24 and the direction of observation.
Simplified quantum description.
This section gives a quantum-mechanical analog of the prior section, but with some simplifications. We give a non-relativistic treatment of the special case of an electron of mass formula_25, charge formula_26, and initial speed formula_27 decelerating in the Coulomb field of a gas of heavy ions of charge formula_28 and number density formula_29. The emitted radiation is a photon of frequency formula_30 and energy formula_31. We wish to find the emissivity formula_32 which is the power emitted per (solid angle in photon velocity space * photon frequency), summed over both transverse photon polarizations. We follow the common astrophysical practice of writing this result in terms of an approximate classical result times the free-free emission Gaunt factor "g"ff which incorporates quantum and other corrections:
formula_33
A general, quantum-mechanical formula for formula_34 exists but is very complicated, and usually is found by numerical calculations. We present some approximate results with the following additional assumptions:
With these assumptions, two unitless parameters characterize the process: formula_39, which measures the strength of the electron-ion Coulomb interaction, and formula_40, which measures the photon "softness" and we assume is always small (the choice of the factor 2 is for later convenience). In the limit formula_41, the quantum-mechanical Born approximation gives:
formula_42
In the opposite limit formula_43, the full quantum-mechanical result reduces to the purely classical result
formula_44
where formula_45 is the Euler–Mascheroni constant. Note that formula_46 which is a purely classical expression without Planck's constant formula_47.
A semi-classical, heuristic way to understand the Gaunt factor is to write it as formula_48 where formula_49 and formula_50 are maximum and minimum "impact parameters" for the electron-ion collision, in the presence of the photon electric field. With our assumptions, formula_51: for larger impact parameters, the sinusoidal oscillation of the photon field provides "phase mixing" that strongly reduces the interaction. formula_50 is the larger of the quantum-mechanical deBroglie wavelength formula_53 and the classical distance of closest approach formula_54 where the electron-ion Coulomb potential energy is comparable to the electron's initial kinetic energy.
The above results generally apply as long as the argument of the logarithm is large, and break down when it is less than unity. Namely, the Gaunt factor becomes negative in this case, which is unphysical. A rough approximation to the full calculations, with the appropriate Born and classical limits, is
formula_55
Thermal bremsstrahlung: emission and absorption.
This section discusses bremsstrahlung emission and the inverse absorption process (called inverse bremsstrahlung) in a macroscopic medium. We start with the equation of radiative transfer, which applies to general processes and not just bremsstrahlung:
formula_56
formula_57 is the radiation spectral intensity, or power per (area * solid angle in photon velocity space * photon frequency) summed over both polarizations. formula_58 is the emissivity, analogous to formula_32defined above, and formula_60 is the absorptivity. formula_58 and formula_60 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 formula_63 is uniform in space and time, then the left-hand side of the transfer equation is zero, and we find
formula_64
If the matter and radiation are also in thermal equilibrium at some temperature, then formula_63must be the blackbody spectrum:
formula_66
Since formula_58 and formula_60 are independent of formula_63, this means that formula_70 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 formula_58 and formula_60 once one is known – for matter in equilibrium.
In plasma.
NOTE: this section currently gives formulas that apply in the Rayleigh-Jeans limit formula_73, and does not use a quantized (Planck) treatment of radiation. Thus a usual factor like formula_74 does not appear. The appearance of formula_75 in formula_76 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. 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, formula_77.
Consider a uniform plasma, with thermal electrons distributed according to the Maxwell–Boltzmann distribution with the temperature formula_78. Following Bekefi, the power spectral density (power per angular frequency interval per volume, integrated over the whole formula_79 sr of solid angle, and in both polarizations) of the bremsstrahlung radiated, is calculated to be
where formula_81 is the electron plasma frequency, formula_82 is the photon frequency, formula_83 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 formula_84 (this is the cutoff condition for a light wave in a plasma; in this case the light wave is evanescent). This formula thus only applies for formula_85. This formula should be summed over ion species in a multi-species plasma.
The special function formula_86 is defined in the exponential integral article, and the unitless quantity formula_76 is
formula_77 is a maximum or cutoff wavenumber, arising due to binary collisions, and can vary with ion species. Roughly, formula_90 when formula_91 (typical in plasmas that are not too cold), where formula_92 eV is the Hartree energy, and formula_93 is the electron thermal de Broglie wavelength. Otherwise, formula_94 where formula_95 is the classical Coulomb distance of closest approach.
For the usual case formula_96, we find
The formula for formula_98 is approximate, in that it neglects enhanced emission occurring for formula_82 slightly above formula_100.
In the limit formula_101, we can approximate formula_102 as formula_103
where formula_45 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 formula_105 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 formula_111 due to the quantum nature of formula_112. In practical units, a commonly used version of this formula for formula_113 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 formula_115, e.g. in one finds
where everything is expressed in the CGS units.
Relativistic corrections.
For very high temperatures there are relativistic corrections to this formula, that is, additional terms of the order of formula_117
Bremsstrahlung cooling.
If the plasma is optically thin, the bremsstrahlung radiation leaves the plasma, carrying part of the internal plasma energy. This effect is known as the "bremsstrahlung cooling". It is a type of radiative cooling. The energy carried away by bremsstrahlung is called "bremsstrahlung losses" and represents a type of radiative losses. One generally uses the term "bremsstrahlung losses" in the context when the plasma cooling is undesired, as e.g. in fusion plasmas.
Polarizational bremsstrahlung.
Polarizational bremsstrahlung (sometimes referred to as "atomic bremsstrahlung") is the radiation emitted by the target's atomic electrons as the target atom is polarized by the Coulomb field of the incident charged particle. Polarizational bremsstrahlung contributions to the total bremsstrahlung spectrum have been observed in experiments involving relatively massive incident particles, resonance processes, and free atoms. 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.
It is worth noting that the term "polarizational" is not meant to imply that the emitted bremsstrahlung is polarized. Also, the angular distribution of polarizational bremsstrahlung is theoretically quite different than ordinary bremsstrahlung.
Sources.
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) formula_118 against the wavelength formula_119 of the emitted radiation:
The constant "K" is proportional to the atomic number of the target element, and formula_121 is the minimum wavelength given by the Duane–Hunt law.
The spectrum has a sharp cutoff at formula_121, 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:
where "h" is Planck's 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.
Beta decay.
Beta particle-emitting substances sometimes exhibit a weak radiation with continuous spectrum that is due to bremsstrahlung (see the "outer bremsstrahlung" below). In this context, bremsstrahlung is a type of "secondary radiation", in that it is produced as a result of stopping (or slowing) the primary radiation (beta particles). It is very similar to X-rays produced by bombarding metal targets with electrons in X-ray generators (as above) except that it is produced by high-speed electrons from beta radiation.
Inner and outer bremsstrahlung.
The "inner" bremsstrahlung (also known as "internal bremsstrahlung") arises from the creation of the electron and its loss of energy (due to the strong electric field in the region of the nucleus undergoing decay) as it leaves the nucleus. Such radiation is a feature of beta decay in nuclei, but it is occasionally (less commonly) seen in the beta decay of free neutrons to protons, where it is created as the beta electron leaves the proton.
In electron and positron emission by beta decay the photon's energy comes from the electron-nucleon pair, with the spectrum of the bremsstrahlung decreasing continuously with increasing energy of the beta particle. In electron capture, the energy comes at the expense of the neutrino, and the spectrum is greatest at about one third of the normal neutrino energy, decreasing to zero electromagnetic energy at normal neutrino energy. Note that in the case of electron capture, bremsstrahlung is emitted even though no charged particle is emitted. Instead, the bremsstrahlung radiation may be thought of as being created as the captured electron is accelerated toward being absorbed. Such radiation may be at frequencies that are the same as soft gamma radiation, but it exhibits none of the sharp spectral lines of gamma decay, and thus is not technically gamma radiation.
The internal process is to be contrasted with the "outer" bremsstrahlung due to the impingement on the nucleus of electrons coming from the outside (i.e., emitted by another nucleus), as discussed above.
Radiation safety.
In some cases, "e.g." , 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, "e.g." Plexiglas (Lucite), plastic, wood, or water; 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).
In astrophysics.
The dominant luminous component in a cluster of galaxies is the 107 to 108 kelvin intracluster medium. The emission from the intracluster medium is characterized by thermal bremsstrahlung. This radiation is in the energy range of X-rays and can be easily observed with space-based telescopes such as Chandra X-ray Observatory, XMM-Newton, ROSAT, ASCA, EXOSAT, Suzaku, RHESSI and future missions like IXO and Astro-H .
Bremsstrahlung is also the dominant emission mechanism for H II regions at radio wavelengths.
In electric discharges.
In electric discharges, for example as laboratory discharges between two electrodes or as lightning discharges between cloud and ground or within clouds, electrons produce Bremsstrahlung photons while scattering off air molecules. These photons become manifest in terrestrial gamma-ray flashes and are the source for beams of electrons, positrons, neutrons and protons. The appearance of Bremsstrahlung photons also influences the propagation and morphology of discharges in nitrogen-oxygen mixtures with low percentages of oxygen.
Quantum mechanical description.
The complete quantum mechanical description was first performed by Bethe and Heitler. 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:
There formula_125 is the atomic number, formula_126 the fine structure constant, formula_111 the reduced Planck's constant and formula_128 the speed of light. The kinetic energy formula_129 of the electron in the initial and final state is connected to its total energy formula_130 or its momenta formula_131 via
where formula_25 is the mass of an electron. Conservation of energy gives
where formula_135 is the photon energy. The directions of the emitted photon and the scattered electron are given by
where formula_137 is the momentum of the photon.
The differentials are given as
The absolute value of the virtual photon between the nucleus and electron is
The range of validity is given by the Born approximation
where this relation has to be fulfilled for the velocity formula_141 of the electron in the initial and final state.
For practical applications (e.g. in Monte Carlo codes) it can be interesting to focus on the relation between the frequency formula_82 of the emitted photon and the angle between this photon and the incident electron. Köhn and Ebert integrated the quadruply differential cross section by Bethe and Heitler over formula_143 and formula_144 and obtained:
with
and
However, a much simpler expression for the same integral can be found in (Eq. 2BN) and in (Eq. 4.1).
An analysis of the doubly differential cross section above shows that electrons whose kinetic energy is larger than the rest energy (511 keV) emit photons in forward direction while electrons with a small energy emit photons isotropically.
Electron–electron bremsstrahlung.
One mechanism, considered important for small atomic numbers formula_125, is the scattering of a free electron at the shell electrons of an atom or molecule. Since electron–electron bremsstrahlung is a function of formula_125 and the usual electron-nucleus bremsstrahlung is a function of formula_150, electron–electron bremsstrahlung is negligible for metals. For air, however, it plays an important role in the production of terrestrial gamma-ray flashes.