# Bremsstrahlung

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Bremsstrahlung (German pronunciation:()), from bremsen "to brake" and Strahlung "radiation"; i.e., "braking radiation" or "deceleration radiation", is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic energy, which is converted into radiation (i.e., a photon), thus satisfying the law of conservation of energy. The term is also used to refer to the process of producing the radiation. Bremsstrahlung has a continuous spectrum, which becomes more intense and whose peak intensity shifts toward higher frequencies as the change of the energy of the decelerated particles increases.

In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays.

The electron is a subatomic particle, symbol
e
or
β
, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, ħ. As it is a fermion, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy.

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 in 1932, models for a nucleus composed of protons and neutrons were quickly developed by Dmitri Ivanenko and Werner Heisenberg. An atom is composed of a positively-charged nucleus, with a cloud of negatively-charged electrons surrounding it, bound together by electrostatic force. Almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. Protons and neutrons are bound together to form a nucleus by the nuclear force.

## Contents

Broadly speaking, bremsstrahlung or braking radiation is any radiation produced due to the deceleration (negative acceleration) of a charged particle, which includes synchrotron radiation (i.e. photon emission by a relativistic particle), cyclotron radiation (i.e. photon emission by a non-relativistic particle), and the emission of electrons and positrons during beta decay. However, the term is frequently used in the more narrow sense of radiation from electrons (from whatever source) slowing in matter.

Synchrotron radiation is the electromagnetic radiation emitted when charged particles are accelerated radially, i.e., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, then the emission is called cyclotron emission. If, on the other hand, the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum which is also called continuum radiation.

Cyclotron radiation is electromagnetic radiation emitted by 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' motion through them, creating an acceleration of charged particles that causes them to emit radiation as a result of the acceleration they undergo as they spiral around the lines of the magnetic field.

In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta ray is emitted from an atomic nucleus. For example, beta decay of a neutron transforms it into a proton by the emission of an electron accompanied by an antineutrino, or conversely a proton is converted into a neutron by the emission of a positron with a neutrino, thus changing the nuclide type. Neither the beta particle nor its associated (anti-)neutrino exist within the nucleus prior to beta decay, but are created in the decay process. By this process, unstable atoms obtain a more stable ratio of protons to neutrons. The probability of a nuclide decaying due to beta and other forms of decay is determined by its nuclear binding energy. The binding energies of all existing nuclides form what is called the nuclear band or valley of stability. For either electron or positron emission to be energetically possible, the energy release or Q value must be positive.

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 charged particles that are free; i.e., not part of an ion, atom or molecule, both before and after the deflection (acceleration) that caused the emission.

Plasma is one of the four fundamental states of matter, and was first described by chemist Irving Langmuir in the 1920s. Plasma can be artificially generated by heating or subjecting a neutral gas to a strong electromagnetic field to the point where an ionized gaseous substance becomes increasingly electrically conductive, and long-range electromagnetic fields dominate the behaviour of the matter.

In physics, acceleration is the rate of change of velocity of an object with respect to time. An object's acceleration is the net result of all forces acting on the object, as described by Newton's Second Law. The SI unit for acceleration is metre per second squared (m⋅s−2). Accelerations are vector quantities and add according to the parallelogram law. The vector of the net force acting on a body has the same direction as the vector of the body's acceleration, and its magnitude is proportional to the magnitude of the acceleration, with the object's mass as proportionality constant.

## Particle in vacuum

A charged particle accelerating in a vacuum radiates power, as described by the Larmor formula and its relativistic generalizations. Although the term, bremsstrahlung, is usually reserved for charged particles accelerating in matter, not vacuum, the formulas are similar.[ citation needed ] (In this respect, bremsstrahlung differs from Cherenkov radiation, another kind of braking radiation which occurs only in matter, and not in a vacuum.)

The Larmor formula is used to calculate the total power radiated by a non relativistic point charge as it accelerates or decelerates. This is used in the branch of physics known as electrodynamics and is not to be confused with the Larmor precession from classical nuclear magnetic resonance. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.

Cherenkov radiation is an electromagnetic radiation emitted when a charged particle passes through a dielectric medium at a speed greater than the phase velocity of light in that medium. The characteristic blue glow of an underwater nuclear reactor is due to Cherenkov radiation.

The most established relativistic formula for total radiated power is given by [1]

${\displaystyle P={\frac {q^{2}\gamma ^{4}}{6\pi \varepsilon _{0}c}}\left({\dot {\beta }}^{2}+{\frac {\left({\vec {\beta }}\cdot {\dot {\vec {\beta }}}\right)^{2}}{1-\beta ^{2}}}\right),}$

where ${\displaystyle {\vec {\beta }}={\frac {\vec {v}}{c}}}$ (the velocity of the particle divided by the speed of light), ${\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}}$ is the Lorentz factor, ${\displaystyle {\dot {\vec {\beta }}}}$ signifies a time derivative of ${\displaystyle {\vec {\beta }}}$, and q is the charge of the particle. This is commonly written in the mathematically equivalent form [2] using ${\displaystyle \left({\vec {\beta }}\cdot {\dot {\vec {\beta }}}\right)^{2}={\dot {\beta }}^{2}\beta ^{2}-\left({\vec {\beta }}\times {\dot {\vec {\beta }}}\right)^{2}}$:

The Lorentz factor or Lorentz term is the factor by which time, length, and relativistic mass change for an object while that object is moving. The expression appears in several equations in special relativity, and it arises in derivations of the Lorentz transformations. The name originates from its earlier appearance in Lorentzian electrodynamics – named after the Dutch physicist Hendrik Lorentz.

${\displaystyle P={\frac {q^{2}\gamma ^{6}}{6\pi \varepsilon _{0}c}}\left(({\dot {\vec {\beta }}})^{2}-\left({\vec {\beta }}\times {\dot {\vec {\beta }}}\right)^{2}\right).}$

In the case where velocity is parallel to acceleration (for example, linear motion), the formula simplifies to [3]

${\displaystyle P_{a\parallel v}={\frac {q^{2}a^{2}\gamma ^{6}}{6\pi \varepsilon _{0}c^{3}}},}$

where ${\displaystyle a\equiv {\dot {v}}={\dot {\beta }}c}$ is the acceleration. For the case of acceleration perpendicular to the velocity ${\displaystyle \left({\vec {\beta }}\cdot {\dot {\vec {\beta }}}=0\right)}$ (a case that arises in circular particle accelerators known as synchrotrons), the total power radiated reduces to

${\displaystyle P_{a\perp v}={\frac {q^{2}a^{2}\gamma ^{4}}{6\pi \varepsilon _{0}c^{3}}}.}$

radiated in the two limiting cases is proportional to ${\displaystyle \gamma ^{4}}$${\displaystyle \left(a\perp v\right)}$ or ${\displaystyle \gamma ^{6}}$${\displaystyle \left(a\parallel v\right)}$. Since ${\displaystyle E=\gamma mc^{2}}$, we see that the total radiated power goes as ${\displaystyle m^{-4}}$ or ${\displaystyle m^{-6}}$, which accounts for why electrons lose energy to bremsstrahlung radiation much more rapidly than heavier charged particles (e.g., muons, protons, alpha particles). This is the reason a TeV energy electron-positron collider (such as the proposed International Linear Collider) cannot use a circular tunnel (requiring constant acceleration), while a proton-proton collider (such as the Large Hadron Collider) can utilize a circular tunnel. The electrons lose energy due to bremsstrahlung at a rate ${\displaystyle (m_{p}/m_{e})^{4}\approx 10^{13}}$ times higher than protons do.

### Angular distribution

The most general formula for radiated power as a function of angle is: [2]

${\displaystyle {\frac {dP}{d\Omega }}={\frac {q^{2}}{16\pi ^{2}\varepsilon _{0}c}}{\frac {\left|{\hat {n}}\times \left(\left({\hat {n}}-{\vec {\beta }}\right)\times {\dot {\vec {\beta }}}\right)\right|^{2}}{\left(1-{\hat {n}}\cdot {\vec {\beta }}\right)^{5}}}}$

where ${\displaystyle {\hat {n}}}$ is a unit vector pointing from the particle towards the observer, and ${\displaystyle d\Omega }$ is an infinitesimal bit of solid angle.

In the case where velocity is parallel to acceleration (for example, linear motion), this simplifies to [2]

${\displaystyle {\frac {dP_{a\parallel v}}{d\Omega }}={\frac {q^{2}a^{2}}{16\pi ^{2}\varepsilon _{0}c^{3}}}{\frac {\sin ^{2}\theta }{(1-\beta \cos \theta )^{5}}}}$

where ${\displaystyle \theta }$ is the angle between ${\displaystyle {\vec {a}}}$ and the direction of observation.

## Thermal bremsstrahlung

NOTE: this article currently gives formulas that apply in the Rayleigh-Jeans limit ${\displaystyle \hbar \omega \ll k_{B}T_{e}}$, and does not use a quantized (Planck) treatment of radiation. Thus a usual factor like ${\displaystyle \exp(-\hbar \omega /k_{B}T_{e})}$ does not appear. The appearance of ${\displaystyle \hbar \omega /k_{B}T_{e}}$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, [4] while a simplified one is given by Ichimaru. [5] In this section we follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber, ${\displaystyle k_{m}}$.

Consider a uniform plasma, with thermal electrons distributed according to the Maxwell–Boltzmann distribution with the temperature ${\displaystyle T_{e}}$. Following Bekefi, the power spectral density (power per angular frequency interval per volume, integrated over the whole ${\displaystyle 4\pi }$ sr of solid angle, and in both polarizations) of the bremsstrahlung radiated, is calculated to be

${\displaystyle {dP_{\mathrm {Br} } \over d\omega }={8{\sqrt {2}} \over 3{\sqrt {\pi }}}\left[{e^{2} \over 4\pi \varepsilon _{0}}\right]^{3}{1 \over (m_{e}c^{2})^{3/2}}\left[1-{\omega _{p}^{2} \over \omega ^{2}}\right]^{1/2}{Z_{i}^{2}n_{i}n_{e} \over (k_{B}T_{e})^{1/2}}E_{1}(y),}$

where ${\displaystyle \omega _{p}\equiv (n_{e}e^{2}/\varepsilon _{0}m_{e})^{1/2}}$ is the electron plasma frequency, ${\displaystyle \omega }$ is the photon frequency, ${\displaystyle n_{e},n_{i}}$ is the number density of electrons and ions, and other symbols are physical constants. The second bracketed factor is the index of refraction of a light wave in a plasma, and shows that emission is greatly suppressed for ${\displaystyle \omega <\omega _{p}}$ (this is the cutoff condition for a light wave in a plasma; in this case the light wave is evanescent). This formula thus only applies for ${\displaystyle \omega >\omega _{p}}$. This formula should be summed over ion species in a multi-species plasma.

The special function ${\displaystyle E_{1}}$ is defined in the exponential integral article, and the unitless quantity ${\displaystyle y}$ is

${\displaystyle y={1 \over 2}{\omega ^{2}m_{e} \over k_{m}^{2}k_{B}T_{e}}}$

${\displaystyle k_{m}}$ is a maximum or cutoff wavenumber, arising due to binary collisions, and can vary with ion species. Roughly, ${\displaystyle k_{m}=1/\lambda _{B}}$ when ${\displaystyle k_{B}T_{e}>Z_{i}^{2}E_{h}}$ (typical in plasmas that are not too cold), where ${\displaystyle E_{h}\approx 27.2}$ eV is the Hartree energy, and ${\displaystyle \lambda _{B}=\hbar /(m_{e}k_{B}T_{e})^{1/2}}$[ clarification needed ] is the electron thermal de Broglie wavelength. Otherwise, ${\displaystyle k_{m}\propto 1/l_{c}}$ where ${\displaystyle l_{c}}$ is the classical Coulomb distance of closest approach.

For the usual case ${\displaystyle k_{m}=1/\lambda _{B}}$, we find

${\displaystyle y={1 \over 2}\left[{\frac {\hbar \omega }{k_{B}T_{e}}}\right]^{2}.}$

The formula for ${\displaystyle dP_{\mathrm {Br} }/d\omega }$ is approximate, in that it neglects enhanced emission occurring for ${\displaystyle \omega }$ slightly above ${\displaystyle \omega _{p}}$.

In the limit ${\displaystyle y\ll 1}$, we can approximate E1 as ${\displaystyle E_{1}(y)\approx -\ln[ye^{\gamma }]+O(y)}$ where ${\displaystyle \gamma \approx 0.577}$ is the Euler–Mascheroni constant. The leading, logarithmic term is frequently used, and resembles the Coulomb logarithm that occurs in other collisional plasma calculations. For ${\displaystyle y>e^{-\gamma }}$ 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

{\displaystyle {\begin{aligned}P_{\mathrm {Br} }&=\int _{\omega _{p}}^{\infty }d\omega {dP_{\mathrm {Br} } \over d\omega }={16 \over 3}\left[{e^{2} \over 4\pi \varepsilon _{0}}\right]^{3}{1 \over m_{e}^{2}c^{3}}Z_{i}^{2}n_{i}n_{e}k_{m}G(y_{p})\\G(y_{p})&={1 \over 2{\sqrt {\pi }}}\int _{y_{p}}^{\infty }dy\,y^{-{\frac {1}{2}}}\left[1-{y_{p} \over y}\right]^{\frac {1}{2}}E_{1}(y)\\y_{p}&=y(\omega =\omega _{p})\end{aligned}}}
${\displaystyle G(y_{p}=0)=1}$ and decreases with ${\displaystyle y_{p}}$; it is always positive. For ${\displaystyle k_{m}=1/\lambda _{B}}$, we find
${\displaystyle P_{\mathrm {Br} }={16 \over 3}{\left({\frac {e^{2}}{4\pi \varepsilon _{0}}}\right)^{3} \over (m_{e}c^{2})^{\frac {3}{2}}\hbar }Z_{i}^{2}n_{i}n_{e}(k_{B}T_{e})^{\frac {1}{2}}G(y_{p})}$

Note the appearance of ${\displaystyle \hbar }$ due to the quantum nature of ${\displaystyle \lambda _{B}}$. In practical units, a commonly used version of this formula for ${\displaystyle G=1}$ is [6]

${\displaystyle P_{\mathrm {Br} }[{\textrm {W}}/{\textrm {m}}^{3}]={Z_{i}^{2}n_{i}n_{e} \over \left[7.69\times 10^{18}{\textrm {m}}^{-3}\right]^{2}}T_{e}[{\textrm {eV}}]^{\frac {1}{2}}.}$

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 ${\displaystyle g_{B}}$, e.g. in [7] one finds

${\displaystyle \varepsilon _{\mathrm {ff} }=1.4\times 10^{-27}T^{\frac {1}{2}}n_{e}n_{i}Z^{2}g_{B},\,}$

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 ${\displaystyle k_{B}T_{e}/m_{e}c^{2}\,.}$ [8]

### Bremsstrahlung cooling

If the plasma is optically thin, the bremsstrahlung radiation leaves the plasma, carrying part of the internal plasma energy. This effect is known as the bremsstrahlung cooling. It is a type of radiative cooling. The energy carried away by bremsstrahlung is called bremsstrahlung losses and represents a type of radiative losses. One generally uses the term bremsstrahlung losses in the context when the plasma cooling is undesired, as e.g. in fusion plasmas.

## Polarizational bremsstrahlung

Polarizational bremsstrahlung (sometimes referred to as "atomic bremsstrahlung") is the radiation emitted by the target's atomic electrons as the target atom is polarized by the Coulomb field of the incident charged particle. [9] [10] Polarizational bremsstrahlung contributions to the total bremsstrahlung spectrum have been observed in experiments involving relatively massive incident particles, [11] resonance processes, [12] and free atoms. [13] 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. [14] [15]

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. [16]

## Sources of bremsstrahlung

### X-ray tube

In an X-ray tube, electrons are accelerated in a vacuum by an electric field towards a piece of metal called the "target". X-rays are emitted as the electrons slow down (decelerate) in the metal. The output spectrum consists of a continuous spectrum of X-rays, with additional sharp peaks at certain energies. The continuous spectrum is due to bremsstrahlung, while the sharp peaks are characteristic X-rays associated with the atoms in the target. For this reason, bremsstrahlung in this context is also called continuous X-rays. [17]

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) ${\displaystyle I}$ against the wavelength ${\displaystyle \lambda }$ of the emitted radiation: [18]

${\displaystyle I(\lambda )\,d\lambda =K\left({\frac {\lambda }{\lambda _{\min }}}-1\right){\frac {1}{\lambda ^{2}}}\,d\lambda }$

The constant K is proportional to the atomic number of the target element, and ${\displaystyle \lambda _{\min }}$ is the minimum wavelength given by the Duane–Hunt law.

The spectrum has a sharp cutoff at ${\displaystyle \lambda _{\min }}$, which is due to the limited energy of the incoming electrons. For example, if an electron in the tube is accelerated through 60 kV, then it will acquire a kinetic energy of 60 keV, and when it strikes the target it can create X-rays with energy of at most 60 keV, by conservation of energy. (This upper limit corresponds to the electron coming to a stop by emitting just one X-ray photon. Usually the electron emits many photons, and each has an energy less than 60 keV.) A photon with energy of at most 60 keV has wavelength of at least 21 pm, so the continuous X-ray spectrum has exactly that cutoff, as seen in the graph. More generally the formula for the low-wavelength cutoff, the Duane-Hunt law, is: [19]

${\displaystyle \lambda _{\min }={\frac {hc}{eV}}\approx {\frac {1239.8}{V}}{\text{ pm/kV}}}$

where h is Planck's constant, c is the speed of light, V is the voltage that the electrons are accelerated through, e is the elementary charge, and pm is picometres.

### Beta decay

Beta particle-emitting substances sometimes exhibit a weak radiation with continuous spectrum that is due to bremsstrahlung (see the "outer bremsstrahlung" below). In this context, bremsstrahlung is a type of "secondary radiation", in that it is produced as a result of stopping (or slowing) the primary radiation (beta particles). It is very similar to X-rays produced by bombarding metal targets with electrons in X-ray generators (as above) except that it is produced by high-speed electrons from beta radiation.

#### Inner and outer bremsstrahlung

The "inner" bremsstrahlung (also known as "internal bremsstrahlung") arises from the creation of the electron and its loss of energy (due to the strong electric field in the region of the nucleus undergoing decay) as it leaves the nucleus. Such radiation is a feature of beta decay in nuclei, but it is occasionally (less commonly) seen in the beta decay of free neutrons to protons, where it is created as the beta electron leaves the proton.

In electron and positron emission by beta decay the photon's energy comes from the electron-nucleon pair, with the spectrum of the bremsstrahlung decreasing continuously with increasing energy of the beta particle. In electron capture, the energy comes at the expense of the neutrino, and the spectrum is greatest at about one third of the normal neutrino energy, decreasing to zero electromagnetic energy at normal neutrino energy. Note that in the case of electron capture, bremsstrahlung is emitted even though no charged particle is emitted. Instead, the bremsstrahlung radiation may be thought of as being created as the captured electron is accelerated toward being absorbed. Such radiation may be at frequencies that are the same as soft gamma radiation, but it exhibits none of the sharp spectral lines of gamma decay, and thus is not technically gamma radiation.

The internal process is to be contrasted with the "outer" bremsstrahlung due to the impingement on the nucleus of electrons coming from the outside (i.e., emitted by another nucleus), as discussed above. [20]

In some cases, e.g., the bremsstrahlung produced by shielding the beta radiation with the normally used dense materials (e.g. lead) is itself dangerous; in such cases, shielding must be accomplished with low density materials, e.g. Plexiglas (Lucite), plastic, wood, or water; [21] as the atomic number is lower for these materials, the intensity of bremsstrahlung is significantly reduced, but a larger thickness of shielding is required to stop the electrons (beta radiation).

### In astrophysics

The dominant luminous component in a cluster of galaxies is the 107 to 108 kelvin intracluster medium. The emission from the intracluster medium is characterized by thermal bremsstrahlung. This radiation is in the energy range of X-rays and can be easily observed with space-based telescopes such as Chandra X-ray Observatory, XMM-Newton, ROSAT, ASCA, EXOSAT, Suzaku, RHESSI and future missions like IXO and Astro-H .

Bremsstrahlung is also the dominant emission mechanism for H II regions at radio wavelengths.

### In electric discharges

In electric discharges, for example as laboratory discharges between two electrodes or as lightning discharges between cloud and ground or within clouds, electrons produce Bremsstrahlung photons while scattering off air molecules. These photons become manifest in terrestrial gamma-ray flashes and are the source for beams of electrons, positrons, neutrons and protons. [22] The appearance of Bremsstrahlung photons also influences the propagation and morphology of discharges in nitrogen-oxygen mixtures with low percentages of oxygen. [23]

## Quantum mechanical description

The complete quantum mechanical description was first performed by Bethe and Heitler. [24] 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:

{\displaystyle {\begin{aligned}d^{4}\sigma ={}&{\frac {Z^{2}\alpha _{\text{fine}}^{3}\hbar ^{2}}{(2\pi )^{2}}}{\frac {\left|\mathbf {p} _{f}\right|}{\left|\mathbf {p} _{i}\right|}}{\frac {d\omega }{\omega }}{\frac {d\Omega _{i}\,d\Omega _{f}\,d\Phi }{\left|\mathbf {q} \right|^{4}}}\\&{}\times \left[{\frac {\mathbf {p} _{f}^{2}\sin ^{2}\Theta _{f}}{\left(E_{f}-c\left|\mathbf {p} _{f}\right|\cos \Theta _{f}\right)^{2}}}\left(4E_{i}^{2}-c^{2}\mathbf {q} ^{2}\right)+{\frac {\mathbf {p} _{i}^{2}\sin ^{2}\Theta _{i}}{\left(E_{i}-c\left|\mathbf {p} _{i}\right|\cos \Theta _{i}\right)^{2}}}\left(4E_{f}^{2}-c^{2}\mathbf {q} ^{2}\right)\right.\\&{}+2\hbar ^{2}\omega ^{2}{\frac {\mathbf {p} _{i}^{2}\sin ^{2}\Theta _{i}+\mathbf {p} _{f}^{2}\sin ^{2}\Theta _{f}}{(E_{f}-c\left|\mathbf {p} _{f}\right|\cos \Theta _{f})\left(E_{i}-c\left|\mathbf {p} _{i}\right|\cos \Theta _{i}\right)}}\\&{}-2\left.{\frac {\left|\mathbf {p} _{i}\right|\left|\mathbf {p} _{f}\right|\sin \Theta _{i}\sin \Theta _{f}\cos \Phi }{\left(E_{f}-c\left|\mathbf {p} _{f}\right|\cos \Theta _{f}\right)\left(E_{i}-c\left|\mathbf {p} _{i}\right|c1\cos \Theta _{i}\right)}}\left(2E_{i}^{2}+2E_{f}^{2}-c^{2}\mathbf {q} ^{2}\right)\right].\end{aligned}}}

There ${\displaystyle Z}$ is the atomic number, ${\displaystyle \alpha _{\text{fine}}\approx 1/137}$ the fine structure constant, ${\displaystyle \hbar }$ the reduced Planck's constant and ${\displaystyle c}$ the speed of light. The kinetic energy ${\displaystyle E_{{\text{kin}},i/f}}$ of the electron in the initial and final state is connected to its total energy ${\displaystyle E_{i,f}}$ or its momenta ${\displaystyle \mathbf {p} _{i,f}}$ via

${\displaystyle E_{i,f}=E_{{\text{kin}},i/f}+m_{e}c^{2}={\sqrt {m_{e}^{2}c^{4}+\mathbf {p} _{i,f}^{2}c^{2}}},}$

where ${\displaystyle m_{e}}$ is the mass of an electron. Conservation of energy gives

${\displaystyle E_{f}=E_{i}-\hbar \omega ,}$

where ${\displaystyle \hbar \omega }$ is the photon energy. The directions of the emitted photon and the scattered electron are given by

{\displaystyle {\begin{aligned}\Theta _{i}&=\sphericalangle (\mathbf {p} _{i},\mathbf {k} ),\\\Theta _{f}&=\sphericalangle (\mathbf {p} _{f},\mathbf {k} ),\\\Phi &={\text{Angle between the planes }}(\mathbf {p} _{i},\mathbf {k} ){\text{ and }}(\mathbf {p} _{f},\mathbf {k} ),\end{aligned}}}

where ${\displaystyle \mathbf {k} }$ is the momentum of the photon.

The differentials are given as

{\displaystyle {\begin{aligned}d\Omega _{i}&=\sin \Theta _{i}\ d\Theta _{i},\\d\Omega _{f}&=\sin \Theta _{f}\ d\Theta _{f}.\end{aligned}}}

The absolute value of the virtual photon between the nucleus and electron is

{\displaystyle {\begin{aligned}-\mathbf {q} ^{2}={}&-\left|\mathbf {p} _{i}\right|^{2}-\left|\mathbf {p} _{f}\right|^{2}-\left({\frac {\hbar }{c}}\omega \right)^{2}+2\left|\mathbf {p} _{i}\right|{\frac {\hbar }{c}}\omega \cos \Theta _{i}-2\left|\mathbf {p} _{f}\right|{\frac {\hbar }{c}}\omega \cos \Theta _{f}\\&{}+2\left|\mathbf {p} _{i}\right|\left|\mathbf {p} _{f}\right|\left(\cos \Theta _{f}\cos \Theta _{i}+\sin \Theta _{f}\sin \Theta _{i}\cos \Phi \right).\end{aligned}}}

The range of validity is given by the Born approximation

${\displaystyle v\gg {\frac {Zc}{137}}}$

where this relation has to be fulfilled for the velocity ${\displaystyle v}$ 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 ${\displaystyle \omega }$ of the emitted photon and the angle between this photon and the incident electron. Köhn and Ebert [25] integrated the quadruply differential cross section by Bethe and Heitler over ${\displaystyle \Phi }$ and ${\displaystyle \Theta _{f}}$ and obtained:

${\displaystyle {\frac {d^{2}\sigma (E_{i},\omega ,\Theta _{i})}{d\omega \,d\Omega _{i}}}=\sum \limits _{j=1}^{6}I_{j}}$

with

{\displaystyle {\begin{aligned}I_{1}={}&{\frac {2\pi A}{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}}\ln \left({\frac {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}-{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\left(\Delta _{1}+\Delta _{2}\right)+\Delta _{1}\Delta _{2}}{-\Delta _{2}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}-{\sqrt {\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\left(\Delta _{1}-\Delta _{2}\right)+\Delta _{1}\Delta _{2}}}\right)\\&{}\times \left[1+{\frac {c\Delta _{2}}{p_{f}\left(E_{i}-cp_{i}\cos \Theta _{i}\right)}}-{\frac {p_{i}^{2}c^{2}\sin ^{2}\Theta _{i}}{\left(E_{i}-cp_{i}\cos \Theta _{i}\right)^{2}}}-{\frac {2\hbar ^{2}\omega ^{2}p_{f}\Delta _{2}}{c\left(E_{i}-cp_{i}\cos \Theta _{i}\right)\left(\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}}\right],\\I_{2}={}&-{\frac {2\pi Ac}{p_{f}\left(E_{i}-cp_{i}\cos \Theta _{i}\right)}}\ln \left({\frac {E_{f}+p_{f}c}{E_{f}-p_{f}c}}\right),\\I_{3}={}&{\frac {2\pi A}{\sqrt {\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}}\times \ln \left[\left(\left[E_{f}+p_{f}c\right]\right.\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 {\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\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 {\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\right]\right)\right]^{-1}\\&{}\times \left[-{\frac {\left(\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)\left(E_{f}^{3}+E_{f}p_{f}^{2}c^{2}\right)+p_{f}c\left(2\left[\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right]E_{f}p_{f}c+\Delta _{1}\Delta _{2}\left[3E_{f}^{2}+p_{f}^{2}c^{2}\right]\right)}{\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\right.\\&{}-{\frac {c\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)}{p_{f}\left(E_{i}-cp_{i}\cos \Theta _{i}\right)}}-{\frac {4E_{i}^{2}p_{f}^{2}\left(2\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}-4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)\left(\Delta _{1}E_{f}+\Delta _{2}p_{f}c\right)}{\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)^{2}}}\\&{}+\left.{\frac {8p_{i}^{2}p_{f}^{2}m^{2}c^{4}\sin ^{2}\Theta _{i}\left(E_{i}^{2}+E_{f}^{2}\right)-2\hbar ^{2}\omega ^{2}p_{i}^{2}\sin ^{2}\Theta _{i}p_{f}c\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)+2\hbar ^{2}\omega ^{2}p_{f}m^{2}c^{3}\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)}{\left(E_{i}-cp_{i}\cos \Theta _{i}\right)\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}}\right],\\I_{4}={}&{}-{\frac {4\pi Ap_{f}c\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)}{\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}-{\frac {16\pi E_{i}^{2}p_{f}^{2}A\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}}{\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)^{2}}},\\I_{5}={}&{\frac {4\pi A}{\left(-\Delta _{2}^{2}+\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)\left(\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}}\\&{}\times \left[{\frac {\hbar ^{2}\omega ^{2}p_{f}^{2}}{E_{i}-cp_{i}\cos \Theta _{i}}}\right.\\&{}\times {\frac {E_{f}\left(2\Delta _{2}^{2}\left[\Delta _{2}^{2}-\Delta _{1}^{2}\right]+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left[\Delta _{2}^{2}+\Delta _{1}^{2}\right]\right)+p_{f}c\left(2\Delta _{1}\Delta _{2}\left[\Delta _{2}^{2}-\Delta _{1}^{2}\right]+16\Delta _{1}\Delta _{2}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}{\Delta _{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\\&{}+{\frac {2\hbar ^{2}\omega ^{2}p_{i}^{2}\sin ^{2}\Theta _{i}\left(2\Delta _{1}\Delta _{2}p_{f}c+2\Delta _{2}^{2}E_{f}+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}E_{f}\right)}{E_{i}-cp_{i}\cos \Theta _{i}}}\\&{}+{\frac {2E_{i}^{2}p_{f}^{2}\left(2\left[\Delta _{2}^{2}-\Delta _{1}^{2}\right]\left[\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right]^{2}+8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left[\left(\Delta _{1}^{2}+\Delta _{2}^{2}\right)\left(E_{f}^{2}+p_{f}^{2}c^{2}\right)+4\Delta _{1}\Delta _{2}E_{f}p_{f}c\right]\right)}{\left(\Delta _{2}E_{f}+\Delta _{1}p_{f}c\right)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}}}\\&{}+\left.{\frac {8p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\left(E_{i}^{2}+E_{f}^{2}\right)\left(\Delta _{2}p_{f}c+\Delta _{1}E_{f}\right)}{E_{i}-cp_{i}\cos \Theta _{i}}}\right],\\I_{6}={}&{\frac {16\pi E_{f}^{2}p_{i}^{2}\sin ^{2}\Theta _{i}A}{\left(E_{i}-cp_{i}\cos \Theta _{i}\right)^{2}\left(-\Delta _{2}^{2}+\Delta _{1}^{2}-4p_{i}^{2}p_{f}^{2}\sin ^{2}\Theta _{i}\right)}},\end{aligned}}}

and

{\displaystyle {\begin{aligned}A&={\frac {Z^{2}\alpha _{\text{fine}}^{3}}{(2\pi )^{2}}}{\frac {\left|\mathbf {p} _{f}\right|}{\left|\mathbf {p} _{i}\right|}}{\frac {\hbar ^{2}}{\omega }}\\\Delta _{1}&=-\mathbf {p} _{i}^{2}-\mathbf {p} _{f}^{2}-\left({\frac {\hbar }{c}}\omega \right)^{2}+2{\frac {\hbar }{c}}\omega \left|\mathbf {p} _{i}\right|\cos \Theta _{i},\\\Delta _{2}&=-2{\frac {\hbar }{c}}\omega \left|\mathbf {p} _{f}\right|+2\left|\mathbf {p} _{i}\right|\left|\mathbf {p} _{f}\right|\cos \Theta _{i}.\end{aligned}}}

However, a much simpler expression for the same integral can be found in [26] (Eq. 2BN) and in [27] (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 ${\displaystyle Z}$, is the scattering of a free electron at the shell electrons of an atom or molecule. [28] Since electron–electron bremsstrahlung is a function of ${\displaystyle Z}$ and the usual electron-nucleus bremsstrahlung is a function of ${\displaystyle Z^{2}}$, electron–electron bremsstrahlung is negligible for metals. For air, however, it plays an important role in the production of terrestrial gamma-ray flashes. [29]

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