Bremsstrahlung

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 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. This is a non-relativistic treatment of the special case of an electron of mass ${\displaystyle m_{\text{e}}}$, 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 ${\displaystyle h\nu }$. The goal is 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. It is expressed as an approximate classical result times the free−free emission Gaunt factor g_ff accounting for quantum and other corrections: $${\displaystyle j(v,\nu )=\frac{8\pi }{3\sqrt{3}}\frac{Z^2{\overline{e}}^6{n}_i}{c^3{m}_{\text{e}}^{2}v}{g}_{\mathrm{f}\mathrm{f}}(v,\nu )}$$ ${\displaystyle j(\nu ,v)=0}$ if ${\displaystyle h\nu >m{v}^2/2}$, that is, the electron does not have enough kinetic energy to emit the photon. A general, quantum-mechanical formula for ${\displaystyle g_{\mathrm{f}\mathrm{f}}}$ exists but is very complicated, and usually is found by numerical calculations. The following additional assumptions may be used to present approximate results:

• Vacuum interaction: the calculation neglects any effects of the background medium, such as plasma screening effects. This is reasonable for photon frequency much greater than the plasma frequency ${\displaystyle {\nu }_{\mathrm{p}\mathrm{e}}\propto n_{\mathrm{e}}^{1/2}}$with ${\displaystyle n_{\text{e}}}$ the plasma electron density. Note that light waves are evanescent for ${\displaystyle \nu <{\nu }_{\mathrm{p}\mathrm{e}}}$ and a significantly different approach would be needed.
• Soft photons: ${\displaystyle h\nu \ll m_{\text{e}}{v}^2/2}$, that is, the photon energy is much less than the initial electron kinetic energy.

With these assumptions, two unitless parameters characterize the process: ${\displaystyle {\eta }_Z\equiv Z{\overline{e}}^2/\hslash v}$, which measures the strength of the electron-ion Coulomb interaction, and ${\displaystyle {\eta }_{\nu }\equiv h\nu /2m_{\text{e}}{v}^2}$, which measures the photon "softness" and which is assumedly 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: $${\displaystyle g_{\text{ff,Born}}=\frac{\sqrt{3}}{\pi }\ln \frac{1}{{\eta }_{\nu }}}$$

In the opposite limit ${\displaystyle {\eta }_Z\gg 1}$, the full quantum-mechanical result reduces to the purely classical result $${\displaystyle g_{\text{ff,class}}=\frac{\sqrt{3}}{\pi }\left[\ln \left(\frac{1}{{\eta }_Z{\eta }_{\nu }}\right)-\gamma \right]}$$ where ${\displaystyle \gamma \approx 0.577}$ is the Euler–Mascheroni constant. Note that ${\displaystyle 1/{\eta }_Z{\eta }_{\nu }=m_{\text{e}}{v}^3/\pi Z{\overline{e}}^2\nu }$ which is a purely classical expression without the Planck constant ${\displaystyle h}$.

A semi-classical, heuristic way to understand the Gaunt factor is to write it as ${\displaystyle g_{\text{ff}}\approx \ln (b_{\text{max}}/b_{\text{min}})}$ where ${\displaystyle b_{max}}$ and ${\displaystyle b_{min}}$ are maximum and minimum "impact parameters" for the electron-ion collision, in the presence of the photon electric field. With our assumptions, ${\displaystyle b_{\mathrm{m}\mathrm{a}\mathrm{x}}=v/\nu }$: for larger impact parameters, the sinusoidal oscillation of the photon field provides "phase mixing" that strongly reduces the interaction. ${\displaystyle b_{\mathrm{m}\mathrm{i}\mathrm{n}}}$ is the larger of the quantum-mechanical de Broglie wavelength ${\displaystyle \approx h/m_{\text{e}}v}$ and the classical distance of closest approach ${\displaystyle \approx {\overline{e}}^2/m_{\text{e}}{v}^2}$ 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 $${\displaystyle g_{\text{ff}}\approx max\left[1,\frac{\sqrt{3}}{\pi }\ln \left[\frac{1}{{\eta }_{\nu }max(1,e^{\gamma }{\eta }_Z)}\right]\right]}$$

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. The equation of radiative transfer, which applies to general processes and not just bremsstrahlung is: $${\displaystyle \frac{1}{c}{\mathrm{\partial }}_t{I}_{\nu }+\hat{\mathbf{n}}\cdot \mathrm{\nabla }{I}_{\nu }=j_{\nu }-k_{\nu }{I}_{\nu }}$$

${\displaystyle I_{\nu }(t,\mathbf{x})}$ is the radiation spectral intensity, or power per (area × solid angle in photon velocity space × photon frequency) summed over both polarizations. ${\displaystyle j_{\nu }}$ 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 thus: $${\displaystyle I_{\nu }=\frac{j_{\nu }}{k_{\nu }}}$$

If the matter and radiation are also in thermal equilibrium at some temperature, then ${\displaystyle I_{\nu }}$ must be the blackbody spectrum: $${\displaystyle B_{\nu }(\nu ,T_{\text{e}})=\frac{2h{\nu }^3}{c^2}\frac{1}{e^{h\nu /k_{\text{B}}{T}_{\text{e}}}-1}}$$ 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.

In plasma: approximate classical results

NOTE: this section currently gives formulas that apply in the Rayleigh–Jeans limit ${\displaystyle \hslash \omega \ll k_{\text{B}}{T}_{\text{e}}}$, and does not use a quantized (Planck) treatment of radiation. Thus a usual factor like ${\displaystyle \exp (-\hslash \omega /k_{\mathrm{B}}{T}_{\text{e}})}$ does not appear. The appearance of ${\displaystyle \hslash \omega /k_{\text{B}}{T}_{\text{e}}}$ 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, while a simplified one is given by Ichimaru. These calculations follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber, ${\displaystyle k_{\text{max}}}$.

Consider a uniform plasma, with thermal electrons distributed according to the Maxwell–Boltzmann distribution with the temperature ${\displaystyle T_{\text{e}}}$. 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 summed over the polarizations) of the bremsstrahlung radiated, is calculated to be $${\displaystyle \frac{d{P}_{\mathrm{B}\mathrm{r}}}{d\omega }=\frac{8\sqrt{2}}{3\sqrt{\pi }}\frac{{\overline{e}}^6}{(m_{\text{e}}{c}^2{)}^{3/2}}{\left[1-\frac{{\omega }_{\mathrm{p}}^2}{{\omega }^2}\right]}^{1/2}\frac{Z_i^2{n}_i{n}_{\text{e}}}{(k_{\mathrm{B}}{T}_{\text{e}}{)}^{1/2}}{E}_1(y),}$$ where ${\displaystyle {\omega }_p\equiv (n_{\text{e}}{e}^2/{\epsilon }_0{m}_{\text{e}}{)}^{1/2}}$ is the electron plasma frequency, ${\displaystyle \omega }$ is the photon frequency, ${\displaystyle n_{\text{e}},n_i}$ 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 ${\displaystyle \omega <{\omega }_{\mathrm{p}}}$ (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 ${\displaystyle \omega >{\omega }_{\mathrm{p}}}$. This formula should be summed over ion species in a multi-species plasma.

The function ${\displaystyle E_1}$ is the exponential integral, and the unitless quantity ${\displaystyle y}$ is $${\displaystyle y=\frac{1}{2}\frac{{\omega }^2{m}_{\text{e}}}{k_{\text{max}}^2{k}_{\text{B}}{T}_{\text{e}}}}$$

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

For the usual case ${\displaystyle k_m=1/{\lambda }_B}$, $${\displaystyle y=\frac{1}{2}{\left[\frac{\hslash \omega }{k_{\text{B}}{T}_{\text{e}}}\right]}^2.}$$

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

In the limit ${\displaystyle y\ll 1}$, ${\displaystyle E_1}$ can be approximated as ${\displaystyle E_1(y)\approx -\ln [y{e}^{\gamma }]+O(y)}$ 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 $${\displaystyle \begin{array}{cc}{P}_{\mathrm{B}\mathrm{r}}& ={\int }_{{\omega }_{\text{p}}}^{\mathrm{\infty }}\frac{d{P}_{\mathrm{B}\mathrm{r}}}{d\omega }d\omega =\frac{16}{3}\frac{{\overline{e}}^6}{m_{\text{e}}^2{c}^3}{Z}_i^2{n}_i{n}_{\text{e}}{k}_{\text{max}}G(y_{\text{p}})\\ G(y_p)& =\frac{1}{2\sqrt{\pi }}{\int }_{y_{\text{p}}}^{\mathrm{\infty }}{\left[1-\frac{y_{\text{p}}}{y}\right]}^{1/2}{E}_1(y)\frac{dy}{y^{1/2}}\\ y_{\text{p}}& =y(\omega ={\omega }_{\text{p}})\end{array}}$$ ${\displaystyle G(y_{\text{p}}=0)=1}$ and decreases with ${\displaystyle y_{\text{p}}}$; it is always positive. For ${\displaystyle k_{\text{max}}=1/{\lambda }_{\text{B}}}$, $${\displaystyle P_{\mathrm{B}\mathrm{r}}=\frac{16}{3}\frac{{\overline{e}}^6}{(m_{\text{e}}{c}^2{)}^{\frac{3}{2}}\hslash }{Z}_i^2{n}_i{n}_{\text{e}}(k_{\mathrm{B}}{T}_{\text{e}}{)}^{\frac{1}{2}}G(y_{\mathrm{p}})}$$

Note the appearance of ${\displaystyle \hslash }$ due to the quantum nature of ${\displaystyle {\lambda }_{\mathrm{B}}}$. In practical units, a commonly used version of this formula for ${\displaystyle G=1}$ is $${\displaystyle P_{\mathrm{B}\mathrm{r}}[\mathrm{W}/{\mathrm{m}}^3]=\frac{Z_i^2{n}_i{n}_{\text{e}}}{{\left[7.69\times {10}^{18}{\mathrm{m}}^{-3}\right]}^2}{T}_{\text{e}}[\mathrm{e}\mathrm{V}{]}^{\frac{1}{2}}.}$$

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 ${\displaystyle g_{\mathrm{B}}}$, e.g. in one finds $${\displaystyle {\epsilon }_{\text{ff}}=1.4\times {10}^{-27}{T}^{\frac{1}{2}}{n}_{\text{e}}{n}_i{Z}^2{g}_{\text{B}},}$$ 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 ${\displaystyle k_{\text{B}}{T}_{\text{e}}/m_{\text{e}}{c}^2}$.

Bremsstrahlung cooling

If the plasma is optically thin, the bremsstrahlung radiation leaves the plasma, carrying part of the internal plasma energy. This effect is known as the bremsstrahlung cooling. It is a type of radiative cooling. 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.

Quantum mechanical description

The complete quantum mechanical description was first performed by Bethe and Heitler. They assumed plane waves for electrons which scatter at the nucleus of an atom, and derived a cross section which relates the complete geometry of that process to the frequency of the emitted photon. The quadruply differential cross section, which shows a quantum mechanical symmetry to pair production, is: $${\displaystyle \begin{array}{cc}{d}^4\sigma =& \frac{Z^2{\alpha }_{\text{fine}}^3{\hslash }^2}{(2\pi {)}^2}\frac{\left|{\mathbf{p}}_f\right|}{\left|{\mathbf{p}}_i\right|}\frac{d\omega }{\omega }\frac{d{\mathrm{\Omega }}_{i}d{\mathrm{\Omega }}_{f}d\mathrm{\Phi }}{{\left|\mathbf{q}\right|}^4}\\ & \times [\frac{{\mathbf{p}}_f^2{\sin }^2{\mathrm{\Theta }}_f}{{\left(E_f-c\left|{\mathbf{p}}_f\right|\cos {\mathrm{\Theta }}_f\right)}^2}\left(4E_i^2-c^2{\mathbf{q}}^2\right)+\frac{{\mathbf{p}}_i^2{\sin }^2{\mathrm{\Theta }}_i}{{\left(E_i-c\left|{\mathbf{p}}_i\right|\cos {\mathrm{\Theta }}_i\right)}^2}\left(4E_f^2-c^2{\mathbf{q}}^2\right)\\ & +2{\hslash }^2{\omega }^2\frac{{\mathbf{p}}_i^2{\sin }^2{\mathrm{\Theta }}_i+{\mathbf{p}}_f^2{\sin }^2{\mathrm{\Theta }}_f}{(E_f-c\left|{\mathbf{p}}_f\right|\cos {\mathrm{\Theta }}_f)\left(E_i-c\left|{\mathbf{p}}_i\right|\cos {\mathrm{\Theta }}_i\right)}\\ & -2\frac{\left|{\mathbf{p}}_i\right|\left|{\mathbf{p}}_f\right|\sin {\mathrm{\Theta }}_i\sin {\mathrm{\Theta }}_f\cos \mathrm{\Phi }}{\left(E_f-c\left|{\mathbf{p}}_f\right|\cos {\mathrm{\Theta }}_f\right)\left(E_i-c\left|{\mathbf{p}}_i\right|\cos {\mathrm{\Theta }}_i\right)}\left(2E_i^2+2E_f^2-c^2{\mathbf{q}}^2\right)],\end{array}}$$ where ${\displaystyle Z}$ is the atomic number, ${\displaystyle {\alpha }_{\text{fine}}\approx 1/137}$ the fine-structure constant, ${\displaystyle \hslash }$ the reduced Planck 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 $${\displaystyle E_{i,f}=E_{\text{kin},i/f}+m_{\text{e}}{c}^2=\sqrt{m_{\text{e}}^2{c}^4+{\mathbf{p}}_{i,f}^2{c}^2},}$$ where ${\displaystyle m_{\text{e}}}$ is the mass of an electron. Conservation of energy gives $${\displaystyle E_f=E_i-\hslash \omega ,}$$ where ${\displaystyle \hslash \omega }$ is the photon energy. The directions of the emitted photon and the scattered electron are given by $${\displaystyle \begin{array}{cc}{\mathrm{\Theta }}_i& =\sphericalangle ({\mathbf{p}}_i,\mathbf{k}),\\ {\mathrm{\Theta }}_f& =\sphericalangle ({\mathbf{p}}_f,\mathbf{k}),\\ \mathrm{\Phi }& =\text{Angle between the planes }({\mathbf{p}}_i,\mathbf{k})\text{ and }({\mathbf{p}}_f,\mathbf{k}),\end{array}}$$ where ${\displaystyle \mathbf{k}}$ is the momentum of the photon.

The differentials are given as $${\displaystyle \begin{array}{cc}d{\mathrm{\Omega }}_i& =\sin {\mathrm{\Theta }}_{i}d{\mathrm{\Theta }}_i,\\ d{\mathrm{\Omega }}_f& =\sin {\mathrm{\Theta }}_{f}d{\mathrm{\Theta }}_f.\end{array}}$$

The absolute value of the virtual photon between the nucleus and electron is $${\displaystyle \begin{array}{cc}-{\mathbf{q}}^2=& -{\left|{\mathbf{p}}_i\right|}^2-{\left|{\mathbf{p}}_f\right|}^2-{\left(\frac{\hslash }{c}\omega \right)}^2+2\left|{\mathbf{p}}_i\right|\frac{\hslash }{c}\omega \cos {\mathrm{\Theta }}_i-2\left|{\mathbf{p}}_f\right|\frac{\hslash }{c}\omega \cos {\mathrm{\Theta }}_f\\ & +2\left|{\mathbf{p}}_i\right|\left|{\mathbf{p}}_f\right|\left(\cos {\mathrm{\Theta }}_f\cos {\mathrm{\Theta }}_i+\sin {\mathrm{\Theta }}_f\sin {\mathrm{\Theta }}_i\cos \mathrm{\Phi }\right).\end{array}}$$

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 ${\displaystyle \mathrm{\Phi }}$ and ${\displaystyle {\mathrm{\Theta }}_f}$ and obtained: $${\displaystyle \frac{d^2\sigma (E_i,\omega ,{\mathrm{\Theta }}_i)}{d\omega d{\mathrm{\Omega }}_i}=\sum _{j=1}^6{I}_j}$$ with $${\displaystyle \begin{array}{cc}{I}_1=& \frac{2\pi A}{\sqrt{{\mathrm{\Delta }}_2^2+4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i}}\ln \left(\frac{{\mathrm{\Delta }}_2^2+4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i-\sqrt{{\mathrm{\Delta }}_2^2+4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i}\left({\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2\right)+{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2}{-{\mathrm{\Delta }}_2^2-4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i-\sqrt{{\mathrm{\Delta }}_2^2+4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i}\left({\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2\right)+{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2}\right)\\ & \times \left[1+\frac{c{\mathrm{\Delta }}_2}{p_f\left(E_i-c{p}_i\cos {\mathrm{\Theta }}_i\right)}-\frac{p_i^2{c}^2{\sin }^2{\mathrm{\Theta }}_i}{{\left(E_i-c{p}_i\cos {\mathrm{\Theta }}_i\right)}^2}-\frac{2{\hslash }^2{\omega }^2{p}_f{\mathrm{\Delta }}_2}{c\left(E_i-c{p}_i\cos {\mathrm{\Theta }}_i\right)\left({\mathrm{\Delta }}_2^2+4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\right)}\right],\\ I_2=& -\frac{2\pi Ac}{p_f\left(E_i-c{p}_i\cos {\mathrm{\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({\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right)}^4+4m^2{c}^4{p}_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i}}\times \ln [(\left[E_f+p_{f}c\right]\\ & \left[4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\left(E_f-p_{f}c\right)+\left({\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2\right)\left(\left[{\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right]-\sqrt{{\left[{\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right]}^2+4m^2{c}^4{p}_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i}\right)\right])\\ & [\left(E_f-p_{f}c\right)(4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\left[-E_f-p_{f}c\right]\\ & +{\left({\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2\right)\left(\left[{\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right]-\sqrt{{\left({\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right)}^2+4m^2{c}^4{p}_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i}\right])]}^{-1}\\ & \times [-\frac{\left({\mathrm{\Delta }}_2^2+4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\right)\left(E_f^3+E_f{p}_f^2{c}^2\right)+p_{f}c\left(2\left[{\mathrm{\Delta }}_1^2-4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\right]E_f{p}_{f}c+{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2\left[3E_f^2+p_f^2{c}^2\right]\right)}{{\left({\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right)}^2+4m^2{c}^4{p}_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i}\\ & -\frac{c\left({\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right)}{p_f\left(E_i-c{p}_i\cos {\mathrm{\Theta }}_i\right)}-\frac{4E_i^2{p}_f^2\left(2{\left[{\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right]}^2-4m^2{c}^4{p}_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\right)\left({\mathrm{\Delta }}_1{E}_f+{\mathrm{\Delta }}_2{p}_{f}c\right)}{{\left({\left[{\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right]}^2+4m^2{c}^4{p}_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\right)}^2}\\ & +\frac{8p_i^2{p}_f^2{m}^2{c}^4{\sin }^2{\mathrm{\Theta }}_i\left(E_i^2+E_f^2\right)-2{\hslash }^2{\omega }^2{p}_i^2{\sin }^2{\mathrm{\Theta }}_i{p}_{f}c\left({\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right)+2{\hslash }^2{\omega }^2{p}_f{m}^2{c}^3\left({\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right)}{\left(E_i-c{p}_i\cos {\mathrm{\Theta }}_i\right)\left({\left[{\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right]}^2+4m^2{c}^4{p}_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\right)}],\\ I_4=& -\frac{4\pi A{p}_{f}c\left({\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right)}{{\left({\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right)}^2+4m^2{c}^4{p}_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i}-\frac{16\pi E_i^2{p}_f^{2}A{\left({\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right)}^2}{{\left({\left[{\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right]}^2+4m^2{c}^4{p}_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\right)}^2},\\ I_5=& \frac{4\pi A}{\left(-{\mathrm{\Delta }}_2^2+{\mathrm{\Delta }}_1^2-4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\right)\left({\left[{\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right]}^2+4m^2{c}^4{p}_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\right)}\\ & \times [\frac{{\hslash }^2{\omega }^2{p}_f^2}{E_i-c{p}_i\cos {\mathrm{\Theta }}_i}\\ & \times \frac{E_f\left(2{\mathrm{\Delta }}_2^2\left[{\mathrm{\Delta }}_2^2-{\mathrm{\Delta }}_1^2\right]+8p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\left[{\mathrm{\Delta }}_2^2+{\mathrm{\Delta }}_1^2\right]\right)+p_{f}c\left(2{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2\left[{\mathrm{\Delta }}_2^2-{\mathrm{\Delta }}_1^2\right]+16{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2{p}_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\right)}{{\mathrm{\Delta }}_2^2+4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i}\\ & +\frac{2{\hslash }^2{\omega }^2{p}_i^2{\sin }^2{\mathrm{\Theta }}_i\left(2{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2{p}_{f}c+2{\mathrm{\Delta }}_2^2{E}_f+8p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i{E}_f\right)}{E_i-c{p}_i\cos {\mathrm{\Theta }}_i}\\ & +\frac{2E_i^2{p}_f^2\left(2\left[{\mathrm{\Delta }}_2^2-{\mathrm{\Delta }}_1^2\right]{\left[{\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right]}^2+8p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\left[\left({\mathrm{\Delta }}_1^2+{\mathrm{\Delta }}_2^2\right)\left(E_f^2+p_f^2{c}^2\right)+4{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2{E}_f{p}_{f}c\right]\right)}{{\left({\mathrm{\Delta }}_2{E}_f+{\mathrm{\Delta }}_1{p}_{f}c\right)}^2+4m^2{c}^4{p}_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i}\\ & +\frac{8p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\left(E_i^2+E_f^2\right)\left({\mathrm{\Delta }}_2{p}_{f}c+{\mathrm{\Delta }}_1{E}_f\right)}{E_i-c{p}_i\cos {\mathrm{\Theta }}_i}],\\ I_6=& \frac{16\pi E_f^2{p}_i^2{\sin }^2{\mathrm{\Theta }}_{i}A}{{\left(E_i-c{p}_i\cos {\mathrm{\Theta }}_i\right)}^2\left(-{\mathrm{\Delta }}_2^2+{\mathrm{\Delta }}_1^2-4p_i^2{p}_f^2{\sin }^2{\mathrm{\Theta }}_i\right)},\end{array}}$$ and $${\displaystyle \begin{array}{cc}A& =\frac{Z^2{\alpha }_{\text{fine}}^3}{(2\pi {)}^2}\frac{\left|{\mathbf{p}}_f\right|}{\left|{\mathbf{p}}_i\right|}\frac{{\hslash }^2}{\omega }\\ {\mathrm{\Delta }}_1& =-{\mathbf{p}}_i^2-{\mathbf{p}}_f^2-{\left(\frac{\hslash }{c}\omega \right)}^2+2\frac{\hslash }{c}\omega \left|{\mathbf{p}}_i\right|\cos {\mathrm{\Theta }}_i,\\ {\mathrm{\Delta }}_2& =-2\frac{\hslash }{c}\omega \left|{\mathbf{p}}_f\right|+2\left|{\mathbf{p}}_i\right|\left|{\mathbf{p}}_f\right|\cos {\mathrm{\Theta }}_i.\end{array}}$$

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 ${\displaystyle Z}$, is the scattering of a free electron at the shell electrons of an atom or molecule. Since electron–electron bremsstrahlung is a function of ${\displaystyle Z}$ 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.