Bremsstrahlung Power2
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In
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, bremsstrahlung (; ; ) is
electromagnetic radiation In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength ...
produced by the
deceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are vector quantities (in that they have magnit ...
of a
charged particle In physics, a charged particle is a particle with an electric charge. For example, some elementary particles, like the electron or quarks are charged. Some composite particles like protons are charged particles. An ion, such as a molecule or atom ...
when deflected by another charged particle, typically an
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
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 at the Department_of_Physics_and_Astronomy,_University_of_Manchester , University of Manchester ...
. The moving particle loses
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
, which is converted into radiation (i.e.,
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 particles that can ...
s), thus satisfying the
law of conservation of energy 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. In the case of a closed system, the principle says that the total amount of energy within the sy ...
. The term is also used to refer to the process of producing the radiation. Bremsstrahlung has a
continuous spectrum In the physical sciences, the term ''spectrum'' was introduced first into optics by Isaac Newton in the 17th century, referring to the range of colors observed when white light was dispersion (optics), dispersed through a prism (optics), prism. ...
, 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 (positive or negative) of a charged particle, which includes
synchrotron radiation Synchrotron radiation (also known as magnetobremsstrahlung) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity (). It is produced artificially in some types ...
(i.e., photon emission by a
relativistic particle In particle physics, a relativistic particle is an elementary particle with kinetic energy greater than or equal to its rest-mass energy given by Einstein's relation, E=m_0c^2, or specifically, of which the velocity is comparable to the speed of l ...
),
cyclotron radiation In particle physics, 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 lin ...
(i.e. photon emission by a non-relativistic particle), and the emission of electrons and
positron The positron or antielectron is the particle with an electric charge of +1''elementary charge, e'', a Spin (physics), spin of 1/2 (the same as the electron), and the same Electron rest mass, mass as an electron. It is the antiparticle (antimatt ...
s during
beta decay In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which an atomic nucleus emits a beta particle (fast energetic electron or positron), transforming into an isobar of that nuclide. For example, beta decay of a neutron ...
. 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 Free may refer to: Concept * Freedom, the ability to act or change without constraint or restriction * Emancipate, attaining civil and political rights or equality * Free (''gratis''), free of charge * Gratis versus libre, the difference betw ...
(i.e., not in an atomic or molecular
bound state A bound state is a composite of two or more fundamental building blocks, such as particles, atoms, or bodies, that behaves as a single object and in which energy is required to split them. In quantum physics, a bound state is a quantum state of a ...
) before, and remain free after, the emission of a photon. In the same parlance, bound–bound radiation refers to discrete
spectral line A spectral line is a weaker or stronger region in an otherwise uniform and continuous spectrum. It may result from emission (electromagnetic radiation), emission or absorption (electromagnetic radiation), absorption of light in a narrow frequency ...
s (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. This article uses SI units, along with the scaled single-particle charge \bar q \equiv q / (4\pi \epsilon_0)^.


Classical description

If
quantum In physics, a quantum (: quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a property can be "quantized" is referred to as "the hypothesis of quantization". This me ...
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 = / is the
Lorentz factor The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...
, \varepsilon_0 is the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
, \dot 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 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 strength of the magnetic field which bends the particle beam i ...
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 propos ...
) 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 accelerator. It was built by the CERN, European Organization for Nuclear Research (CERN) between 1998 and 2008, in collaboration with over 10,000 scientists, ...
) can utilize a circular tunnel. The electrons lose energy due to bremsstrahlung at a rate (m_\text/m_\text)^4 \approx 10^ times higher than protons do.


Angular distribution

The most general formula for radiated power as a function of angle is: \frac = \frac \frac where \hat is a unit vector pointing from the particle towards the observer, and d\Omega is an infinitesimal 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 \boldsymbol and the direction of observation \hat.


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 Arnold Johannes Wilhelm Sommerfeld (; 5 December 1868 – 26 April 1951) was a German Theoretical physics, theoretical physicist who pioneered developments in Atomic physics, atomic and Quantum mechanics, quantum physics, and also educated and ...
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_\text, 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 express it as an approximate classical result times the free−free emission
Gaunt factor The Gaunt factor (or Kramers–Gaunt factor) is a correction factor that accounts for the effect of quantum mechanics on an object's continuous x-ray absorption or emission spectrum. In cases where classical physics provides a close approximation to ...
''g''ff accounting for quantum and other corrections: j(v,\nu) = 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 ...
\nu_ \propto n_^with n_\text 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_\textv^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 Z \bar e^2/\hbar v, which measures the strength of the electron-ion Coulomb interaction, and \eta_\nu \equiv h\nu/2m_\textv^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_\text = \ln In the opposite limit \eta_Z\gg 1, the full quantum-mechanical result reduces to the purely classical result g_\text = \left ln\left(\right)- \gamma \right/math> where \gamma\approx 0.577 is the
Euler–Mascheroni constant Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...
. Note that 1/\eta_Z\eta_\nu=m_\textv^3/\pi Z\bar e^2\nu which is a purely classical expression without the Planck constant h. A semi-classical, heuristic way to understand the Gaunt factor is to write it as g_\text \approx \ln(b_\text/b_\text) 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 de Broglie wavelength \approx h/m_\text v and the classical distance of closest approach \approx \bar e^2 / m_\text v^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_\text \approx \max\left , \ln\left[\right\right">right.html" ;"title=", \ln\left[\right">, \ln\left[\right\right/math>


Thermal bremsstrahlung in a medium: 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 Ã— × 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_\nu must be the Black-body radiation">blackbody spectrum A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The radiation emitted by a black body in thermal equilibrium with its environment is ...
: B_\nu(\nu, T_\text) = \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: approximate classical results

NOTE: this section currently gives formulas that apply in the Rayleigh–Jeans limit \hbar \omega \ll k_\text T_\text, and does not use a quantized (Planck) treatment of radiation. Thus a usual factor like \exp(-\hbar\omega/k_T_\text) does not appear. The appearance of \hbar \omega / k_\text T_\text 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, Consider a uniform plasma, with thermal electrons distributed according to the Maxwell–Boltzmann distribution with the temperature T_\text. 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 = \frac \left -\right E_1(y), where \omega_p \equiv (n_\text e^2/\varepsilon_0m_\text)^ is the electron plasma frequency, \omega is the photon frequency, n_\text, 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 cannot be explained by a theory and therefore must be measured experimentally. It is distinct from a mathematical constant, which has a ...
. 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), a class of skin lesions * "Evanescent" (song), a song by Vamps * Evanescent wave In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic f ...
). 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&nb ...
article, and the unitless quantity y is y = \frac k_\text is a maximum or cutoff wavenumber, arising due to binary collisions, and can vary with ion species. Roughly, k_\text = 1 / \lambda_\text when k_\text T_\text > Z_i^2 E_\text (typical in plasmas that are not too cold), where E_\text \approx 27.2 eV is the
Hartree energy The hartree (symbol: ''E''h), also known as the Hartree energy, is the unit of energy in the atomic units system, named after the British physicist Douglas Hartree. Its CODATA recommended value is = The hartree is approximately the negative ...
, and \lambda_\text = \hbar / (m_\text k_\text T_\text)^ is the electron thermal de Broglie wavelength. Otherwise, k_\text \propto 1/l_\text where l_\text is the classical Coulomb distance of closest approach. For the usual case k_m = 1/\lambda_B, we find y = \frac \left frac\right2. The formula for dP_\mathrm / d\omega is approximate, in that it neglects enhanced emission occurring for \omega slightly above 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 called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...
. 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 \frac = \frac \frac Z_i^2 n_i n_\text k_\text G(y_\text) \\ ex G(y_p) &= \frac \int_^\infty dy \, y^ \left - \right E_1(y) \\ ex y_\text &= y() \end : G(y_\text=0) = 1 and decreases with y_\text; it is always positive. For k_\text = 1/\lambda_\text, we find P_\mathrm = Z_i^2 n_i n_\text (k_ T_\text)^\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 mathrm= T_\text mathrm\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 The Gaunt factor (or Kramers–Gaunt factor) is a correction factor that accounts for the effect of quantum mechanics on an object's continuous x-ray absorption or emission spectrum. In cases where classical physics provides a close approximation to ...
g_, e.g. in one finds \varepsilon_\text = 1.4\times 10^ T^\frac n_\text n_i Z^2 g_\text,\, 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


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 cooling has b ...
. 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 contras ...
, electrons are accelerated in a vacuum by an
electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
towards a piece of material called the "target". X-rays are emitted as the electrons hit 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 Fluorescence is one of two kinds of photoluminescence, the emission of light by a substance that has absorbed light or other electromagnetic radiation. When exposed to ultraviolet radiation, many substances will glow (fluoresce) with colore ...
. Now we say that the output spectrum consists of a continuous spectrum of X-rays with additional sharp peaks at certain energies. The former is due to bremsstrahlung, while the latter 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 German term itself was introduced in 1909 by
Arnold Sommerfeld Arnold Johannes Wilhelm Sommerfeld (; 5 December 1868 – 26 April 1951) was a German Theoretical physics, theoretical physicist who pioneered developments in Atomic physics, atomic and Quantum mechanics, quantum physics, and also educated and ...
in order to explain the nature of the first variety of 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 and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...
\lambda of the emitted radiation: I(\lambda) \, d\lambda = K \left( \frac - 1 \right)\frac The constant is proportional to the
atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of its atomic nucleus. For ordinary nuclei composed of protons and neutrons, this is equal to the proton number (''n''p) or the number of pro ...
of the target element, and \lambda_ is the minimum wavelength given by the
Duane–Hunt law The Duane–Hunt law, named after the American physicists William Duane and Franklin L. Hunt, gives the maximum frequency of X-rays that can be emitted by Bremsstrahlung in an X-ray tube by accelerating electrons through an excitation voltage ''V ...
. The spectrum has a sharp cutoff at 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 In physics, an electronvolt (symbol eV), also written electron-volt and electron volt, is the measure of an amount of kinetic energy gained by a single electron accelerating through an electric potential difference of one volt in vacuum. When us ...
, and when it strikes the target it can create X-rays with energy of at most 60 keV, by
conservation of energy The law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be Conservation law, ''conserved'' over time. In the case of a Closed system#In thermodynamics, closed system, the principle s ...
. (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 particles that can ...
. 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 , 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\,\mathrm where is the
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
, is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
, is the
voltage Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...
that the electrons are accelerated through, is the
elementary charge The elementary charge, usually denoted by , is a fundamental physical constant, defined as the electric charge carried by a single proton (+1 ''e'') or, equivalently, the magnitude of the negative electric charge carried by a single electron, ...
, 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 ...
, and in the rightmost expression the voltage is in units of kilovolts (kV).


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, known as beta decay. There are two forms of beta decay, β− decay and Π...
s). It is very similar to X-rays produced by bombarding metal targets with electrons in
X-ray generator An X-ray machine is a device that uses X-rays for a variety of applications including medicine, X-ray fluorescence, electronic assembly inspection, and measurement of material thickness in manufacturing operations. In medical applications, X-ra ...
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 called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
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 particle with an electric charge of +1''elementary charge, e'', a Spin (physics), spin of 1/2 (the same as the electron), and the same Electron rest mass, mass as an electron. It is the antiparticle (antimatt ...
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. Until the 1960s, nucleons were thought to be ele ...
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 an elementary particle that interacts via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is so small ('' -ino'') that i ...
, 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 ), is a penetrating form of electromagnetic radiation arising from high energy interactions like the radioactive decay of atomic nuclei or astronomical events like solar flares. It consists o ...
, but it exhibits none of the sharp spectral lines of
gamma decay Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
, 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, such as the decay of , the bremsstrahlung produced by shielding the beta radiation with the normally used dense materials (e.g.
lead Lead () is a chemical element; it has Chemical symbol, symbol Pb (from Latin ) and atomic number 82. It is a Heavy metal (elements), heavy metal that is density, denser than most common materials. Lead is Mohs scale, soft and Ductility, malleabl ...
) is itself dangerous; in such cases, shielding must be accomplished with low density materials, such as
Plexiglas Poly(methyl methacrylate) (PMMA) is a synthetic polymer derived from methyl methacrylate. It is a transparent thermoplastic, used as an engineering plastic. PMMA is also known as acrylic, acrylic glass, as well as by the trade names and bra ...
(
Lucite Poly(methyl methacrylate) (PMMA) is a synthetic polymer derived from methyl methacrylate. It is a transparent thermoplastic, used as an engineering plastic. PMMA is also known as acrylic, acrylic glass, as well as by the trade names and brands ...
),
plastic Plastics are a wide range of synthetic polymers, synthetic or Semisynthesis, semisynthetic materials composed primarily of Polymer, polymers. Their defining characteristic, Plasticity (physics), plasticity, allows them to be Injection moulding ...
,
wood Wood is a structural tissue/material found as xylem in the stems and roots of trees and other woody plants. It is an organic materiala natural composite of cellulosic fibers that are strong in tension and embedded in a matrix of lignin t ...
, or
water Water is an inorganic compound with the chemical formula . It is a transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance. It is the main constituent of Earth's hydrosphere and the fluids of all known liv ...
; 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 (physics), 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 ...
. 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 corners ...
,
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 Un ...
,
ASCA ''Asca'' is a genus of mites. Asca or ASCA may also refer to: Organisations * ASCA (news agency), Italy (founded 1969) * Accumulating savings & credit association, a form of microfinance * American Swimming Coaches Association (founded 1958) * A ...
, EXOSAT, Suzaku, RHESSI and future missions like IXObr>
and Astro-

Bremsstrahlung is also the dominant emission mechanism for
H II region An H II 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 of light year ...
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 A terrestrial gamma-ray flash (TGF), also known as dark lightning, is a burst of gamma rays produced in Earth's atmosphere. TGFs have been recorded to last 0.2 to 3.5 milliseconds, and have energies of up to 20 million electronvolts. It is spec ...
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 Pair production is the creation of a subatomic particle and its antiparticle from a neutral boson. Examples include creating an electron and a positron, a muon and an antimuon, or a proton and an antiproton. Pair production often refers ...
, 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 where Z is the
atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of its atomic nucleus. For ordinary nuclei composed of protons and neutrons, this is equal to the proton number (''n''p) or the number of pro ...
, \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 Alpha, Greek letter ''alpha''), is a Dimensionless physical constant, fundamental physical constant that quantifies the strength of the el ...
, \hbar the
reduced Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
and c the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
. The kinetic energy E_ of the electron in the initial and final state is connected to its total energy E_ or its
momenta Momenta Global Limited (shortened to Momenta) is a developer of intelligent driving technologies, based in Beijing and Suzhou, China. History Momenta was founded in September 2016 led by Cao Xudong, a former scientist at Microsoft Research a ...
\mathbf_ via E_ = E_ + m_\text c^2 = \sqrt, where m_\text is the
mass of an electron In particle physics, the electron mass (symbol: ) 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- ...
.
Conservation of energy The law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be Conservation law, ''conserved'' over time. In the case of a Closed system#In thermodynamics, closed system, the principle s ...
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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of the
virtual photon Virtual photons are a fundamental concept in particle physics and quantum field theory that play a crucial role in describing the interactions between electrically charged particles. Virtual photons are referred to as " virtual" because they do no ...
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 Ebert is a surname of German origin. Notable people with the surname include: * Alex Ebert (born 1978), lead singer for the band Ima Robot * Anton Ebert (1845–1896), Austrian painter * Beanie Ebert (1902–1980), American football player * Blan ...
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 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 electron–electron bremsstrahlung is negligible for metals. For air, however, it plays an important role in the production of
terrestrial gamma-ray flashes A terrestrial gamma-ray flash (TGF), also known as dark lightning, is a burst of gamma rays produced in Earth's atmosphere. TGFs have been recorded to last 0.2 to 3.5 milliseconds, and have energies of up to 20 million electronvolts. It is spec ...
.


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 In particle physics, 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 lin ...
*
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 much as a laser but employs relativistic electrons as a active laser medium, gain medium instead of using ...
* 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 ene ...
* Nuclear fusion: bremsstrahlung losses *
Radiation length In particle physics, the radiation length is a characteristic of a material, related to the energy loss of high energy elementary particle, particles electromagnetically interacting with it. It is defined as the mean length (in cm) into the mate ...
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 othe ...


References


Further reading

*


External links


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