Bremsstrahlung

Field lines and modulus of the electric field generated by a (negative) charge first moving at constant speed and then stopping quickly to show the generated Bremsstrahlung radiation.

This section is written from a purely classical perspective, with quantum effects neglected. An accelerating charged particle radiates power, as described by the Larmor formula and its relativistic generalizations.

This section gives a quantum-mechanical analog of the prior section, but with some simplifications. We give a non-relativistic treatment of the special case of an electron of mass ${\displaystyle m_{e}}$, charge ${\displaystyle -e}$, and initial speed ${\displaystyle v}$ decelerating in the Coulomb field of a gas of heavy ions of charge ${\displaystyle Ze}$ and number density ${\displaystyle n_{i}}$. The emitted radiation is a photon of frequency ${\displaystyle \nu =c/\lambda }$ and energy ${\displaystyle h\nu }$. We wish to find the emissivity ${\displaystyle j(v,\nu )}$ 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:

${\displaystyle j(v,\nu )={8\pi \over 3{\sqrt {3}}}\left({e^{2} \over 4\pi \epsilon _{0}}\right)^{3}{Z^{2}n_{i} \over c^{3}m_{e}^{2}v}g_{\rm {ff}}(v,\nu )}$

A general, quantum-mechanical formula for ${\displaystyle g_{\rm {ff}}}$ 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 ${\displaystyle \nu _{\rm {pe}}\propto n_{\rm {e}}^{1/2}}$with ${\displaystyle n_{e}}$ the plasma electron density. Note that light waves are evanescent for ${\displaystyle \nu <\nu _{\rm {pe}}}$ and a significantly different approach would be needed.
• Soft photons: ${\displaystyle h\nu \ll m_{e}v^{2}/2}$, that is, the photon energy is much less than the initial electron kinetic energy.

With these assumptions, two unitless parameters characterize the process: ${\displaystyle \eta _{Z}\equiv Ze^{2}/\hbar v}$, which measures the strength of the electron-ion Coulomb interaction, and ${\displaystyle \eta _{\nu }\equiv h\nu /2m_{e}v^{2}}$, which measures the photon "softness" and we assume is always small (the choice of the factor 2 is for later convenience). In the limit ${\displaystyle \eta _{Z}\ll 1}$, the quantum-mechanical Born approximation gives:

${\displaystyle g_{\rm {ff,Born}}={{\sqrt {3}} \over \pi }\ln {1 \over \eta _{\nu }}}$

In the opposite limit ${\displaystyle \eta _{Z}\gg 1}$, the full quantum-mechanical result reduces to the purely classical result

${\displaystyle g_{\rm {ff,class}}={{\sqrt {3}} \over \pi }\left[\ln \left({1 \over \eta _{Z}\eta _{\nu }}\right)-\gamma \right]}$

where ${\displaystyle \gamma \approx 0.577}$ is the Euler–Mascheroni constant. Note that ${\displaystyle 1/\eta _{Z}\eta _{\nu }=m_{e}v^{3}/\pi Ze^{2}\nu }$ which is a purely classical expression without Planck's constant ${\displaystyle h}$.

A semi-classical, heuristic way to understand the Gaunt factor is to write it as ${\displaystyle g_{\rm {ff}}\approx \ln(b_{\rm {max}}/b_{\rm {min}})}$ where ${\displaystyle b_{max}}$ and ${\displaystyle b_{\rm {min}}}$ are maximum and minimum "impact parameters" for the electron-ion collision, in the presence of the photon electric field. With our assumptions, ${\displaystyle b_{\rm {max}}=v/\nu }$: for larger impact parameters, the sinusoidal oscillation of the photon field provides "phase mixing" that strongly reduces the interaction. ${\displaystyle b_{\rm {min}}}$ is the larger of the quantum-mechanical deBroglie wavelength ${\displaystyle \approx h/m_{e}v}$ and the classical distance of closest approach ${\displaystyle \approx e^{2}/4\pi \epsilon _{0}m_{e}v^{2}}$ where the electron-ion Coulomb potential energy is comparable to the electron's initial kinetic energy.

The above results generally apply as long as the argument of the logarithm is large, and break down when it is less than unity. Namely, the Gaunt factor becomes negative in this case, which is unphysical. A rough approximation to the full calculations, with the appropriate Born and classical limits, is

${\displaystyle g_{\rm {ff}}\approx \max \left[1,{{\sqrt {3}} \over \pi }\ln \left[{1 \over \eta _{\nu }\max(1,e^{\gamma }\eta _{Z})}\right]\right]}$

The bremsstrahlung power spectrum rapidly decreases for large ${\displaystyle \omega }$, and is also suppressed near ${\displaystyle \omega =\omega _{\rm {p}}}$. This plot is for the quantum case ${\displaystyle T_{e}>Z^{2}E_{\rm {h}}}$, and ${\displaystyle \hbar \omega _{\rm {p}}/T_{e}=0.1}$.

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:

${\displaystyle {1 \over c}\partial _{t}I_{\nu }+{\hat {n}}\cdot \nabla I_{\nu }=j_{\nu }-k_{\nu }I_{\nu }}$

${\displaystyle I_{\nu }(t,{\vec {x}})}$ is the radiation spectral intensity, or power per (area * solid angle in photon velocity space * photon frequency) summed over both polarizations. ${\displaystyle j_{\nu }}$ is the emissivity, analogous to ${\displaystyle j(v,\nu )}$defined above, and ${\displaystyle k_{\nu }}$ is the absorptivity. ${\displaystyle j_{\nu }}$ and ${\displaystyle k_{\nu }}$ 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 ${\displaystyle I_{\nu }}$ is uniform in space and time, then the left-hand side of the transfer equation is zero, and we find

${\displaystyle I_{\nu }={j_{\nu } \over k_{\nu }}}$

If the matter and radiation are also in thermal equilibrium at some temperature, then ${\displaystyle I_{\nu }}$must be the blackbody spectrum:

${\displaystyle B_{\nu }(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /kT}-1}}}$

Since ${\displaystyle j_{\nu }}$ and ${\displaystyle k_{\nu }}$ are independent of ${\displaystyle I_{\nu }}$, this means that ${\displaystyle j_{\nu }/k_{\nu }}$ 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 ${\displaystyle j_{\nu }}$ and ${\displaystyle k_{\nu }}$ once one is known – for matter in equilibrium.

NOTE: this section currently gives formulas that apply in the Rayleigh-Jeans limit ${\displaystyle \hbar \omega \ll k_{\rm {B}}T_{e}}$, and does not use a quantized (Planck) treatment of radiation. Thus a usual factor like ${\displaystyle \exp(-\hbar \omega /k_{\rm {B}}T_{e})}$ does not appear. The appearance of ${\displaystyle \hbar \omega /k_{\rm {B}}T_{e}}$ in ${\displaystyle y}$ 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,[5] while a simplified one is given by Ichimaru.[6] In this section we follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber, ${\displaystyle k_{\rm {max}}}$.

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

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

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 ${\displaystyle \omega <\omega _{\rm {p}}}$ (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 ${\displaystyle \omega >\omega _{\rm {p}}}$. 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

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

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

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

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

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

In the limit ${\displaystyle y\ll 1}$, we can approximate ${\displaystyle E_{1}}$ 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

${\displaystyle {\begin{aligned}P_{\mathrm {Br} }&=\int _{\omega _{\rm {p}}}^{\infty }d\omega {dP_{\mathrm {Br} } \over d\omega }={16 \over 3}\left[{e^{2} \over 4\pi \varepsilon _{0}}\right]^{3}{1 \over m_{e}^{2}c^{3}}Z_{i}^{2}n_{i}n_{e}k_{\rm {max}}G(y_{\rm {p}})\\G(y_{p})&={1 \over 2{\sqrt {\pi }}}\int _{y_{\rm {p}}}^{\infty }dy\,y^{-{\frac {1}{2}}}\left[1-{y_{\rm {p}} \over y}\right]^{\frac {1}{2}}E_{1}(y)\\y_{\rm {p}}&=y(\omega =\omega _{\rm {p}})\end{aligned}}}$

${\displaystyle G(y_{\rm {p}}=0)=1}$ and decreases with ${\displaystyle y_{\rm {p}}}$; it is always positive. For ${\displaystyle k_{\rm {max}}=1/\lambda _{\rm {B}}}$, we find

${\displaystyle P_{\mathrm {Br} }={16 \over 3}{\left({\frac {e^{2}}{4\pi \varepsilon _{0}}}\right)^{3} \over (m_{e}c^{2})^{\frac {3}{2}}\hbar }Z_{i}^{2}n_{i}n_{e}(k_{\rm {B}}T_{e})^{\frac {1}{2}}G(y_{\rm {p}})}$

Note the appearance of ${\displaystyle \hbar }$ due to the quantum nature of ${\displaystyle \lambda _{\rm {B}}}$. In practical units, a commonly used version of this formula for ${\displaystyle G=1}$ is [7]

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

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 ${\displaystyle g_{\rm {B}}}$, e.g. in [8] one finds

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

where everything is expressed in the CGS units.

Relativistic corrections

Relativistic corrections to the emission of a 30-keV photon by an electron impacting on a proton.

For very high temperatures there are relativistic corrections to this formula, that is, additional terms of the order of ${\displaystyle k_{\rm {B}}T_{e}/m_{e}c^{2}\,.}$[9]

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.[10][11] Polarizational bremsstrahlung contributions to the total bremsstrahlung spectrum have been observed in experiments involving relatively massive incident particles,[12] resonance processes,[13] and free atoms.[14] 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.[15][16]

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.[17]

The complete quantum mechanical description was first performed by Bethe and Heitler.[25] 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 integrated the quadruply differential cross section by Bethe and Heitler over ${\displaystyle \Phi }$ and ${\displaystyle \Theta _{f}}$ and obtained:[26]

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

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)^{4}+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 [27] (Eq. 2BN) and in [28] (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.

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.[29] 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.[30]

• Cyclotron radiation
• Free-electron laser
• History of X-rays
• List of plasma physics articles
• Nuclear fusion: bremsstrahlung losses
• Radiation length characterising energy loss by bremsstrahlung by high energy electrons in matter
• Synchrotron light source

• Eberhard Haug; Werner Nakel (2004). The elementary process of bremsstrahlung. Scientific Lecture Notes in Physics. 73. River Edge, NJ: World Scientific. ISBN 978-981-238-578-9.

Look up bremsstrahlung in Wiktionary, the free dictionary.

• Index of Early Bremsstrahlung Articles