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''Bremsstrahlung'' (), from "to brake" and "radiation"; i.e., "braking radiation" or "deceleration radiation", is
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
produced by the
deceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
of a charged particle when deflected by another charged particle, typically an
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
by an
atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron ...
. The moving particle loses
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
, which is converted into radiation (i.e.,
photons A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
), thus satisfying the
law of conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means that ...
. The term is also used to refer to the process of producing the radiation. ''Bremsstrahlung'' has a
continuous spectrum In physics, a continuous spectrum usually means a set of attainable values for some physical quantity (such as energy or wavelength) that is best described as an interval of real numbers, as opposed to a discrete spectrum, a set of attainable ...
, 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 Synchrotron radiation (also known as magnetobremsstrahlung radiation) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity (). It is produced artificially in ...
(i.e., photon emission by a relativistic particle),
cyclotron radiation Cyclotron radiation is electromagnetic radiation emitted by non-relativistic accelerating charged particles deflected by a magnetic field. The Lorentz force on the particles acts perpendicular to both the magnetic field lines and the particles' mot ...
(i.e. photon emission by a non-relativistic particle), and the emission of electrons and positrons during
beta decay In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar of that nuclide. For ...
. 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 Bound or bounds may refer to: Mathematics * Bound variable * Upper and lower bounds, observed limits of mathematical functions Physics * Bound state, a particle that has a tendency to remain localized in one or more regions of space Geography * ...
) 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 In physics, a quantum (plural quanta) is the minimum amount of any physical entity ( physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizat ...
effects are negligible, an accelerating charged particle radiates power as described by the
Larmor formula In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light. When any charge ...
and its relativistic generalization.


Total radiated power

The total radiated power is :P = \frac \left( \dot^2 + \frac\right), where \boldsymbol\beta = \frac (the velocity of the particle divided by the speed of light), \gamma = \frac is the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
, \dot signifies a time derivative of \boldsymbol\beta, 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 :P_ = \frac, where a \equiv \dot = \dotc is the acceleration. For the case of acceleration perpendicular to the velocity (\boldsymbol \cdot \dot = 0), for example in
synchrotron A synchrotron is a particular type of cyclic particle accelerator, descended from the cyclotron, in which the accelerating particle beam travels around a fixed closed-loop path. The magnetic field which bends the particle beam into its closed ...
s, the total power is :P_ = \frac. Power radiated in the two limiting cases is proportional to \gamma^4 \left(a \perp v\right) or \gamma^6 \left(a \parallel v\right). Since E = \gamma m c^2, we see that for particles with the same energy E the total radiated power goes as m^ or m^, 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 The International Linear Collider (ILC) is a proposed linear particle accelerator. It is planned to have a collision energy of 500  GeV initially, with the possibility for a later upgrade to 1000 GeV (1 TeV). Although early propose ...
) cannot use a circular tunnel (requiring constant acceleration), while a proton-proton collider (such as the
Large Hadron Collider The Large Hadron Collider (LHC) is the world's largest and highest-energy particle collider. It was built by the European Organization for Nuclear Research (CERN) between 1998 and 2008 in collaboration with over 10,000 scientists and hundr ...
) can utilize a circular tunnel. The electrons lose energy due to bremsstrahlung at a rate (m_p/m_e)^4 \approx 10^ times higher than protons do.


Angular distribution

The most general formula for radiated power as a function of angle is:Jackson, ''Classical Electrodynamics'', Sections 14.2–3 :\frac = \frac \frac where \hat is a unit vector pointing from the particle towards the observer, and d\Omega is an infinitesimal bit of solid angle. In the case where velocity is parallel to acceleration (for example, linear motion), this simplifies to :\frac = \frac\frac where \theta is the angle between \mathbf 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 m_e, charge -e, and initial speed v decelerating in the Coulomb field of a gas of heavy ions of charge Ze and number density n_i. The emitted radiation is a photon of frequency \nu=c/\lambda and energy h\nu. We wish to find the emissivity j(v,\nu) 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: j(v,\nu) = \left(\right)^3 g_(v,\nu) j(\nu,v)=0 if h\nu > mv^2/2, that is the electron does not have enough kinetic energy to emit the photon. A general, quantum-mechanical formula for g_ 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 Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability i ...
\nu_ \propto n_^with n_e the plasma electron density. Note that light waves are evanescent for \nu<\nu_ and a significantly different approach would be needed. * Soft photons: h\nu\ll m_ev^2/2, that is, the photon energy is much less than the initial electron kinetic energy. With these assumptions, two unitless parameters characterize the process: \eta_Z \equiv Ze^2/\hbar v, which measures the strength of the electron-ion Coulomb interaction, and \eta_\nu \equiv h\nu/2m_ev^2, which measures the photon "softness" and we assume is always small (the choice of the factor 2 is for later convenience). In the limit \eta_Z\ll 1, the quantum-mechanical Born approximation gives: g_ = \ln In the opposite limit \eta_Z\gg 1, the full quantum-mechanical result reduces to the purely classical result g_ = \left ln\left(\right)- \gamma \right/math> where \gamma\approx 0.577 is the
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...
. Note that 1/\eta_Z\eta_\nu=m_ev^3/\pi Ze^2\nu which is a purely classical expression without Planck's constant h. A semi-classical, heuristic way to understand the Gaunt factor is to write it as g_ \approx \ln(b_/b_) where b_ and b_ are maximum and minimum "impact parameters" for the electron-ion collision, in the presence of the photon electric field. With our assumptions, b_=v/\nu: for larger impact parameters, the sinusoidal oscillation of the photon field provides "phase mixing" that strongly reduces the interaction. b_ is the larger of the quantum-mechanical deBroglie wavelength \approx h/m_ev and the classical distance of closest approach \approx e^2/4\pi\epsilon_0m_ev^2 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 g_ \approx \max\left ,_\ln\left[\right\right.html"_;"title="right.html"_;"title=",_\ln\left[\right">,_\ln\left[\right\right">right.html"_;"title=",_\ln\left[\right">,_\ln\left[\right\right/math>


_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: \frac_\partial_t_I_\nu_+_\hat_\mathbf_n\cdot\nabla_I_\nu_=_j_\nu-k_\nu_I_\nu I_\nu(t,\mathbf_x)_is_the_radiation_spectral_intensity,_or_power_per_(area_*_solid_angle_in_photon_velocity_space_*_photon_frequency)_summed_over_both_polarizations._j_\nu_is_the_emissivity,_analogous_to_j(v,\nu)defined_above,_and_k_\nu_is_the_absorptivity._j_\nu_and_k_\nu_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_I_\nu_is_uniform_in_space_and_time,_then_the_left-hand_side_of_the_transfer_equation_is_zero,_and_we_find I_\nu= If_the_matter_and_radiation_are_also_in_thermal_equilibrium_at_some_temperature,_then_I_\numust_be_the_Black-body_radiation.html" ;"title="right">,_\ln\left[\right\right.html" ;"title="right.html" ;"title=", \ln\left[\right">, \ln\left[\right\right">right.html" ;"title=", \ln\left[\right">, \ln\left[\right\right/math>


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: \frac \partial_t I_\nu + \hat \mathbf n\cdot\nabla I_\nu = j_\nu-k_\nu I_\nu I_\nu(t,\mathbf x) is the radiation spectral intensity, or power per (area * solid angle in photon velocity space * photon frequency) summed over both polarizations. j_\nu is the emissivity, analogous to j(v,\nu)defined above, and k_\nu is the absorptivity. j_\nu and k_\nu 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 I_\nu is uniform in space and time, then the left-hand side of the transfer equation is zero, and we find I_\nu= If the matter and radiation are also in thermal equilibrium at some temperature, then I_\numust be the Black-body radiation">blackbody spectrum: B_\nu(\nu, T_e) = \frac\frac Since j_\nu and k_\nu are independent of I_\nu, this means that j_\nu/k_\nu 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 j_\nu and k_\nu once one is known – for matter in equilibrium.


In plasma

NOTE: this section currently gives formulas that apply in the Rayleigh–Jeans limit \hbar \omega \ll k_T_e, and does not use a quantized (Planck) treatment of radiation. Thus a usual factor like \exp(-\hbar\omega/k_T_e) does not appear. The appearance of \hbar \omega / k_T_e in y 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, k_. Consider a uniform plasma, with thermal electrons distributed according to the Maxwell–Boltzmann distribution with the temperature T_e. Following Bekefi, the power spectral density (power per angular frequency interval per volume, integrated over the whole 4\pi steradian, sr of solid angle, and in both polarizations) of the bremsstrahlung radiated, is calculated to be : = \left \right3 \left -\right E_1(y), where \omega_p \equiv (n_ee^2/\varepsilon_0m_e)^ is the electron plasma frequency, \omega is the photon frequency, n_e, n_i is the number density of electrons and ions, and other symbols are
physical constants A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, ...
. The second bracketed factor is the index of refraction of a light wave in a plasma, and shows that emission is greatly suppressed for \omega < \omega_ (this is the cutoff condition for a light wave in a plasma; in this case the light wave is
evanescent Evanescent may refer to: * Evanescent (dermatology) Evanescent skin lesions, like wheals, are those that last for less than 24 hours before resolving.James, William; Berger, Timothy; Elston, Dirk (2005). ''Andrews' Diseases of the Skin: Clinical ...
). This formula thus only applies for \omega>\omega_. This formula should be summed over ion species in a multi-species plasma. The special function E_1 is defined in the
exponential integral In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument. Definitions For real non-zero values of  ...
article, and the unitless quantity y is :y = k_ is a maximum or cutoff wavenumber, arising due to binary collisions, and can vary with ion species. Roughly, k_ = 1/\lambda_ when k_ T_ > Z_i^2 E_ (typical in plasmas that are not too cold), where E_ \approx 27.2 eV is the Hartree energy, and \lambda_ = \hbar/(m_ k_ T_)^ is the electron thermal de Broglie wavelength. Otherwise, k_ \propto 1/l_ where l_ is the classical Coulomb distance of closest approach. For the usual case k_m = 1/\lambda_B, we find : y = \left frac\right2. The formula for dP_\mathrm/d\omega is approximate, in that it neglects enhanced emission occurring for \omega slightly above \omega_. In the limit y\ll 1, we can approximate E_1 as E_1(y) \approx -\ln e^\gamma+ O(y) where \gamma\approx 0.577 is the
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...
. The leading, logarithmic term is frequently used, and resembles the Coulomb logarithm that occurs in other collisional plasma calculations. For y>e^ 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 :\begin P_\mathrm &= \int_^\infty d\omega = \left \right3 Z_i^2 n_i n_e k_ G(y_) \\ G(y_p) &= \int_^\infty dy \, y^ \left - \right\frac E_1(y) \\ y_ &= y(\omega=\omega_) \end :G(y_=0)=1 and decreases with y_; it is always positive. For k_ = 1/\lambda_, we find :P_\mathrm = Z_i^2 n_i n_e (k_ T_e)^\frac G(y_) Note the appearance of \hbar due to the quantum nature of \lambda_. In practical units, a commonly used version of this formula for G=1 is : P_\mathrm textrm/\textrm^3= T_e textrm\frac. 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 g_, e.g. in one finds :\varepsilon_\mathrm = 1.4\times 10^ T^\frac n_e n_i Z^2 g_,\, 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 k_ T_e/m_e c^2\,.


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 In the study of heat transfer, radiative cooling is the process by which a body loses heat by thermal radiation. As Planck's law describes, every physical body spontaneously and continuously emits electromagnetic radiation. Radiative coolin ...
. 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 An X-ray tube is a vacuum tube that converts electrical input power into X-rays. The availability of this controllable source of X-rays created the field of radiography, the imaging of partly opaque objects with penetrating radiation. In contrast ...
, electrons are accelerated in a vacuum by an
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical 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 against the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
\lambda of the emitted radiation: : I(\lambda) \, d\lambda = K \left( \frac - 1 \right)\frac \, d\lambda The constant ''K'' is proportional to the
atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of ever ...
of the target element, and \lambda_ is the minimum wavelength given by the Duane–Hunt law. The spectrum has a sharp cutoff at \lambda_, 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 Kev can refer to: Given name * Kev Adams, French comedian, actor, screenwriter and film producer born Kevin Smadja in 1991 * Kevin Kev Carmody (born 1946), Indigenous Australian singer-songwriter * Kev Coghlan (born 1988), Scottish Grand Prix moto ...
, and when it strikes the target it can create X-rays with energy of at most 60 keV, by
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...
. (This upper limit corresponds to the electron coming to a stop by emitting just one X-ray
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
. 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: :\lambda_\min = \frac \approx \frac\text where ''h'' is Planck's constant, ''c'' is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
, ''V'' is the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
that the electrons are accelerated through, ''e'' is the
elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundam ...
, and ''pm'' is
picometre The picometre (international spelling as used by the International Bureau of Weights and Measures; SI symbol: pm) or picometer (American spelling) is a unit of length in the International System of Units (SI), equal to , or one trillionth of ...
s.


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 particle A beta particle, also called beta ray or beta radiation (symbol β), is a high-energy, high-speed electron or positron emitted by the radioactive decay of an atomic nucleus during the process of beta decay. There are two forms of beta decay, � ...
s). It is very similar to X-rays produced by bombarding metal targets with electrons in
X-ray generator An X-ray generator is a device that produces X-rays. Together with an X-ray detector, it is commonly used in a variety of applications including medicine, X-ray fluorescence, electronic assembly inspection, and measurement of material thicknes ...
s (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 An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical 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 The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collide ...
emission by beta decay the photon's energy comes from the electron-
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number). Until the 1960s, nucleons were ...
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 A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
, 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 A gamma ray, also known as gamma radiation (symbol γ or \gamma), is a penetrating form of electromagnetic radiation arising from the radioactive decay of atomic nuclei. It consists of the shortest wavelength electromagnetic waves, typically s ...
, 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 Lead is a chemical element with the symbol Pb (from the Latin ) and atomic number 82. It is a heavy metal that is denser than most common materials. Lead is soft and malleable, and also has a relatively low melting point. When freshly cut, ...
) is itself dangerous; in such cases, shielding must be accomplished with low density materials, ''e.g.''
Plexiglas Poly(methyl methacrylate) (PMMA) belongs to a group of materials called engineering plastics. It is a transparent thermoplastic. PMMA is also known as acrylic, acrylic glass, as well as by the trade names and brands Crylux, Plexiglas, Acrylite ...
(
Lucite Poly(methyl methacrylate) (PMMA) belongs to a group of materials called engineering plastics. It is a transparent thermoplastic. PMMA is also known as acrylic, acrylic glass, as well as by the trade names and brands Crylux, Plexiglas, Acrylite ...
),
plastic Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adapta ...
,
wood Wood is a porous and fibrous structural tissue found in the stems and roots of trees and other woody plants. It is an organic materiala natural composite of cellulose fibers that are strong in tension and embedded in a matrix of lignin ...
, or
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
; 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 In astronomy, the intracluster medium (ICM) is the superheated plasma that permeates a galaxy cluster. The gas consists mainly of ionized hydrogen and helium and accounts for most of the baryonic material in galaxy clusters. The ICM is heated to t ...
. 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 The Chandra X-ray Observatory (CXO), previously known as the Advanced X-ray Astrophysics Facility (AXAF), is a Flagship-class space telescope launched aboard the during STS-93 by NASA on July 23, 1999. Chandra is sensitive to X-ray sources ...
,
XMM-Newton ''XMM-Newton'', also known as the High Throughput X-ray Spectroscopy Mission and the X-ray Multi-Mirror Mission, is an X-ray space observatory launched by the European Space Agency in December 1999 on an Ariane 5 rocket. It is the second cornerst ...
,
ROSAT ROSAT (short for Röntgensatellit; in German X-rays are called Röntgenstrahlen, in honour of Wilhelm Röntgen) was a German Aerospace Center-led satellite X-ray telescope, with instruments built by West Germany, the United Kingdom and the Uni ...
, ASCA, EXOSAT, Suzaku,
RHESSI Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI, originally High Energy Solar Spectroscopic Imager or HESSI or Explorer 81) was a NASA solar flare observatory. It was the sixth mission in the Small Explorer program (SMEX), selected ...
and future missions like IXObr>
and Astro-

Bremsstrahlung is also the dominant emission mechanism for
H II region An H II region or HII region is a region of interstellar atomic hydrogen that is ionized. It is typically in a molecular cloud of partially ionized gas in which star formation has recently taken place, with a size ranging from one to hundreds ...
s 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: :\begin d^4\sigma = &\frac\frac \frac\frac \\ &\times \left[ \frac\left(4E_i^2 - c^2\mathbf^2\right) + \frac\left(4E_f^2 - c^2\mathbf^2\right) \right. \\ & \qquad+ 2\hbar^2\omega^2 \frac \\ & \qquad- 2\left. \frac \left(2E_i^2 + 2E_f^2 - c^2\mathbf^2\right) \right]. \end There Z is the
atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of ever ...
, \alpha_\text\approx 1/137 the
fine-structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...
, \hbar the reduced Planck's constant and c the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
. The kinetic energy E_ of the electron in the initial and final state is connected to its total energy E_ or its
momenta Momenta is an autonomous driving company headquartered in Beijing, China that aims to build the 'Brains' for autonomous vehicles. In December 2021, Momenta and BYD established a 100 million yuan ($15.7 million) joint venture to deploy autonomous ...
\mathbf_ via : E_ = E_ + m_e c^2 = \sqrt, where m_e is the
mass of an electron The electron mass (symbol: ''m''e) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about or about , which has an energy-equivalent of ab ...
.
Conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...
gives : E_f = E_i - \hbar\omega, where \hbar\omega is the photon energy. The directions of the emitted photon and the scattered electron are given by : \begin \Theta_i &= \sphericalangle(\mathbf_i, \mathbf),\\ \Theta_f &= \sphericalangle(\mathbf_f, \mathbf),\\ \Phi &= \text (\mathbf_i, \mathbf) \text (\mathbf_f, \mathbf), \end where \mathbf is the momentum of the photon. The differentials are given as : \begin d\Omega_i &= \sin\Theta_i\ d\Theta_i,\\ d\Omega_f &= \sin\Theta_f\ d\Theta_f. \end The
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of the
virtual photon A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle. The concept of virtual particles arises in the perturb ...
between the nucleus and electron is : \begin -\mathbf^2 = & -\left, \mathbf_i\^2 - \left, \mathbf_f\^2 - \left(\frac\omega\right)^2 + 2\left, \mathbf_i\\frac \omega\cos\Theta_i - 2\left, \mathbf_f\\frac \omega\cos\Theta_f \\ & + 2\left, \mathbf_i\ \left, \mathbf_f\ \left(\cos\Theta_f\cos\Theta_i + \sin\Theta_f\sin\Theta_i\cos\Phi\right). \end The range of validity is given by the Born approximation : v \gg \frac where this relation has to be fulfilled for the velocity v 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 \omega 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 \Phi and \Theta_f and obtained: : \frac = \sum\limits_^6 I_j with :\begin I_1 = &\frac \ln\left(\frac \right) \\ & \times\left 1 + \frac - \frac - \frac \right \\ I_2 = &-\frac\ln\left(\frac\right), \\ I_3 = & \frac \times \ln\left left(\left[E_f + p_fc\rightright.\right. \\ & \left.\left[4p_i^2 p_f^2 \sin^2\Theta_i\left(E_f - p_f c\right) + \left(\Delta_1 + \Delta_2\right)\left(\left Delta_2 E_f + \Delta_1 p_f c\right- \sqrt\right)\right]\right) \\ &\left[\left(E_f - p_f c\right)\left(4p_i^2 p_f^2 \sin^2\Theta_i\left[-E_f - p_f c\right]\right.\right. \\ & + \left.\left.\left(\Delta_1 - \Delta_2\right)\left(\left Delta_2 E_f + \Delta_1 p_f c\right- \sqrt\right]\right)\right]^ \\ & \times \left \frac \right.\\ & -\frac - \frac \\ & + \left.\frac \right \\ I_4 = & -\frac - \frac , \\ I_5 = & \frac \\ & \times\left[\frac\right.\\ & \times\frac \\ & + \frac \\ & + \frac \\ & + \left.\frac\right], \\ I_6 = & \frac, \end and : \begin A &= \frac \frac \frac \\ \Delta_1 &= -\mathbf_i^2 - \mathbf_f^2 - \left(\frac\omega\right)^2 + 2\frac \omega\left, \mathbf_i\\cos\Theta_i, \\ \Delta_2 &= -2\frac\omega\left, \mathbf_f\ + 2\left, \mathbf_i\\left, \mathbf_f\\cos\Theta_i. \end 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 Z, is the scattering of a free electron at the shell electrons of an atom or molecule. Since electron–electron bremsstrahlung is a function of Z and the usual electron-nucleus bremsstrahlung is a function of Z^2, electron–electron bremsstrahlung is negligible for metals. For air, however, it plays an important role in the production of terrestrial gamma-ray flashes.


See also

*
Beamstrahlung Beamstrahlung (from beam + bremsstrahlung ) is the radiation from one beam of charged particles in storage rings, linear or circular colliders, namely the synchrotron radiation emitted due to the electromagnetic field of the opposing beam.
*
Cyclotron radiation Cyclotron radiation is electromagnetic radiation emitted by non-relativistic accelerating charged particles deflected by a magnetic field. The Lorentz force on the particles acts perpendicular to both the magnetic field lines and the particles' mot ...
*
Wiggler (synchrotron) A wiggler is an insertion device in a synchrotron. It is a series of magnets designed to periodically laterally deflect ('wiggle') a beam of charged particles (invariably electrons or positrons) inside a storage ring of a synchrotron. These ...
*
Free-electron laser A free-electron laser (FEL) is a (fourth generation) light source producing extremely brilliant and short pulses of radiation. An FEL functions and behaves in many ways like a laser, but instead of using stimulated emission from atomic or molecula ...
* History of X-rays *
Landau–Pomeranchuk–Migdal effect In high-energy physics, the Landau–Pomeranchuk–Migdal effect, also known as the Landau–Pomeranchuk effect and the Pomeranchuk effect, or simply LPM effect, is a reduction of the bremsstrahlung and pair production cross sections at high energi ...
* List of plasma physics articles * Nuclear fusion: bremsstrahlung losses *
Radiation length In physics, the radiation length is a characteristic of a material, related to the energy loss of high energy particles electromagnetically interacting with it. Definition In materials of high atomic number (e.g. W, U, Pu) the electrons of energie ...
characterising energy loss by bremsstrahlung by high energy electrons in matter *
Synchrotron light source A synchrotron light source is a source of electromagnetic radiation (EM) usually produced by a storage ring, for scientific and technical purposes. First observed in synchrotrons, synchrotron light is now produced by storage rings and other ...


References


Further reading

*


External links


Index of Early Bremsstrahlung Articles
{{Authority control Atomic physics Plasma physics Scattering Quantum electrodynamics