Bremsstrahlung

If quantum effects are negligible, an accelerating charged particle radiates power as described by the Larmor formula and its relativistic generalization. Total radiated power The total radiated power is P = \frac{2 \bar q^2 \gamma^4}{3 c} \left( \dot{\beta}^2 + \frac{\left(\boldsymbol{\beta} \cdot \dot{\boldsymbol{\beta}}\right)^2}{1 - \beta^2}\right), where \boldsymbol\beta = \frac{\mathbf v}{c} (the velocity of the particle divided by the speed of light), \gamma = {1}/{\sqrt{1-\beta^2}} is the Lorentz factor, \varepsilon_0 is the vacuum permittivity, \dot{\boldsymbol\beta} 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_{a \parallel v} = \frac{2 \bar q^2 a^2 \gamma^6}{3 c^3}, where a \equiv \dot{v} = \dot{\beta}c is the acceleration. For the case of acceleration perpendicular to the velocity (\boldsymbol{\beta} \cdot \dot{\boldsymbol{\beta}} = 0), for example in synchrotrons, the total power is P_{a \perp v} = \frac{2 \bar q^2 a^2 \gamma^4 }{3c^3}. 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^{-4} or 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 (m_\text{p}/m_\text{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: \frac{d P}{d\Omega} = \frac{\bar q^2}{4\pi c} \frac{\left|\hat{\mathbf n} \times \left(\left(\hat{\mathbf n} - \boldsymbol{\beta}\right) \times \dot{\boldsymbol{\beta}}\right)\right|^2}{\left(1 - \hat{\mathbf n}\cdot\boldsymbol{\beta}\right)^5} where \hat{\mathbf n} 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{dP_{a \parallel v}}{d\Omega} = \frac{\bar q^2a^2}{4\pi c^3}\frac{\sin^2 \theta}{(1 - \beta \cos\theta)^5} where \theta is the angle between \boldsymbol{\beta} and the direction of observation \hat{\mathbf n}. == 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. We give a non-relativistic treatment of the special case of an electron of mass m_\text{e}, charge -e, and initial speed v decelerating in the Coulomb field of a gas of heavy ions of charge Ze and number density n_i. The emitted radiation is a photon of frequency \nu=c/\lambda and energy h\nu. We wish to find the emissivity j(v,\nu) which is the power emitted per (solid angle in photon velocity space * photon frequency), summed over both transverse photon polarizations. We express it as an approximate classical result times the free−free emission Gaunt factor gff accounting for quantum and other corrections: j(v,\nu) = {8\pi\over 3\sqrt3} {Z^2\bar e^6 n_i \over c^3m_\text{e}^2v}g_{\rm ff}(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_{\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 \nu_{\rm pe} \propto n_{\rm e}^{1/2}with n_\text{e} the plasma electron density. Note that light waves are evanescent for \nu and a significantly different approach would be needed. • Soft photons: 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: \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_\text{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 \eta_Z\ll 1, the quantum-mechanical Born approximation gives: g_\text{ff,Born} = {\sqrt3 \over \pi}\ln{1\over\eta_\nu} In the opposite limit \eta_Z\gg 1, the full quantum-mechanical result reduces to the purely classical result g_\text{ff,class} = {\sqrt3\over\pi}\left[\ln\left({1\over \eta_Z\eta_\nu}\right)- \gamma \right] where \gamma\approx 0.577 is the Euler–Mascheroni constant. Note that 1/\eta_Z\eta_\nu=m_\text{e}v^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{ff} \approx \ln(b_\text{max}/b_\text{min}) where b_{\max} and b_{\min} are maximum and minimum "impact parameters" for the electron-ion collision, in the presence of the photon electric field. With our assumptions, b_{\rm max}=v/\nu: for larger impact parameters, the sinusoidal oscillation of the photon field provides "phase mixing" that strongly reduces the interaction. b_{\rm min} is the larger of the quantum-mechanical de Broglie wavelength \approx h/m_\text{e} v and the classical distance of closest approach \approx \bar 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 g_\text{ff} \approx \max\left[1, {\sqrt3\over\pi} \ln\left[{1\over \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. We start with the equation of radiative transfer, which applies to general processes and not just bremsstrahlung: \frac{1}{c} \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 × × 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={j_\nu \over k_\nu} If the matter and radiation are also in thermal equilibrium at some temperature, then I_\nu must be the blackbody spectrum: 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 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{B} T_\text{e}, and does not use a quantized (Planck) treatment of radiation. Thus a usual factor like \exp(-\hbar\omega/k_{\rm B}T_\text{e}) does not appear. The appearance of \hbar \omega / k_\text{B} T_\text{e} in y below is due to the quantum-mechanical treatment of collisions. In a plasma, the free electrons continually collide with the ions, producing bremsstrahlung. A complete analysis requires accounting for both binary Coulomb collisions as well as collective (dielectric) behavior. A detailed treatment is given by Bekefi, while a simplified one is given by Ichimaru. In this section we follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber, {{nowrap|k_\text{max}.}} Consider a uniform plasma, with thermal electrons distributed according to the Maxwell–Boltzmann distribution with the temperature T_\text{e}. Following Bekefi, the power spectral density (power per angular frequency interval per volume, integrated over the whole 4\pi sr of solid angle and summed over the polarizations) of the bremsstrahlung radiated, is calculated to be {dP_\mathrm{Br} \over d\omega} = \frac{8\sqrt 2}{3\sqrt\pi} {\bar e^6 \over (m_\text{e} c^2)^{3/2}} \left[1-{\omega_{\rm p}^2 \over \omega^2}\right]^{1/2} {Z_i^2 n_i n_\text{e} \over (k_{\rm B} T_\text{e})^{1/2}} E_1(y), where \omega_p \equiv (n_\text{e} e^2/\varepsilon_0m_\text{e})^{1/2} is the electron plasma frequency, \omega is the photon frequency, 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 \omega (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 \omega>\omega_{\rm p}. This formula should be summed over ion species in a multi-species plasma. The function E_1 is the exponential integral, and the unitless quantity y is y = \frac{1}{2} {\omega^2 m_\text{e} \over k_\text{max}^2 k_\text{B} T_\text{e}} k_\text{max} is a maximum or cutoff wavenumber, arising due to binary collisions, and can vary with ion species. Roughly, k_\text{max} = 1 / \lambda_\text{B} when k_\text{B} T_\text{e} > Z_i^2 E_\text{h} (typical in plasmas that are not too cold), where E_\text{h} \approx 27.2 eV is the Hartree energy, and \lambda_\text{B} = \hbar / (m_\text{e} k_\text{B} T_\text{e})^{1/2} is the electron thermal de Broglie wavelength. Otherwise, k_\text{max} \propto 1/l_\text{C} where l_\text{C} is the classical Coulomb distance of closest approach. For the usual case k_m = 1/\lambda_B, we find y = \frac{1}{2} \left[\frac{\hbar\omega}{k_\text{B} T_\text{e}}\right]^2. The formula for dP_\mathrm{Br} / d\omega is approximate, in that it neglects enhanced emission occurring for \omega slightly above {{nowrap|\omega_\text{p}.}} In the limit y\ll 1, we can approximate E_1 as E_1(y) \approx -\ln [y e^\gamma] + O(y) where \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 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 \begin{align} P_\mathrm{Br} &= \int_{\omega_\text{p}}^\infty d\omega \frac{dP_\mathrm{Br}}{d\omega} = \frac{16}{3} \frac{\bar e^6}{m_\text{e}^2c^3} Z_i^2 n_i n_\text{e} k_\text{max} G(y_\text{p}) \\[1ex] G(y_p) &= \frac{1}{2\sqrt{\pi}} \int_{y_\text{p}}^\infty dy \, y^{-{1}/{2}} \left[1 - {y_\text{p} \over y}\right]^{1/2} E_1(y) \\[1ex] y_\text{p} &= y({\omega\!=\!\omega_\text{p}}) \end{align} : G(y_\text{p}=0) = 1 and decreases with y_\text{p}; it is always positive. For k_\text{max} = 1/\lambda_\text{B}, we find P_\mathrm{Br} = {16 \over 3} {\bar e^6 \over (m_\text{e} c^2)^\frac{3}{2}\hbar} Z_i^2 n_i n_\text{e} (k_{\rm B} T_\text{e})^\frac{1}{2} G(y_{\rm p}) Note the appearance of \hbar due to the quantum nature of \lambda_{\rm B}. In practical units, a commonly used version of this formula for G=1 is P_\mathrm{Br} [\mathrm{W/m^3}] = {Z_i^2 n_i n_\text{e} \over \left[7.69 \times 10^{18} \mathrm{m^{-3}}\right]^2} T_\text{e}[\mathrm{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 g_{\rm B}, e.g. in one finds \varepsilon_\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 {{nowrap|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. == Sources ==

The complete quantum mechanical description was first performed by Bethe and Heitler. They assumed plane waves for electrons which scatter at the nucleus of an atom, and derived a cross section which relates the complete geometry of that process to the frequency of the emitted photon. The quadruply differential cross section, which shows a quantum mechanical symmetry to pair production, is : \begin{align} 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. \\ & {} \qquad+ 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)} \\ & {} \qquad- 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|\cos\Theta_i\right)} \left(2E_i^2 + 2E_f^2 - c^2\mathbf{q}^2\right) \right], \end{align} where Z is the atomic number, \alpha_\text{fine}\approx 1/137 the fine-structure constant, \hbar the reduced Planck constant and c the speed of light. The kinetic energy E_{\text{kin},i/f} of the electron in the initial and final state is connected to its total energy E_{i,f} or its momenta \mathbf{p}_{i,f} via 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 m_\text{e} is the mass of an electron. Conservation of energy 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{align} \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{align} where \mathbf{k} is the momentum of the photon. The differentials are given as \begin{align} d\Omega_i &= \sin\Theta_i\ d\Theta_i,\\ d\Omega_f &= \sin\Theta_f\ d\Theta_f. \end{align} The absolute value of the virtual photon between the nucleus and electron is : \begin{align} -\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{align} The range of validity is given by the Born approximation v \gg \frac{Zc}{137} where this relation has to be fulfilled for the velocity v of the electron in the initial and final state. For practical applications (e.g. in Monte Carlo codes) it can be interesting to focus on the relation between the frequency \omega of the emitted photon and the angle between this photon and the incident electron. Köhn and Ebert integrated the quadruply differential cross section by Bethe and Heitler over \Phi and \Theta_f and obtained: \frac{d^2\sigma (E_i, \omega, \Theta_i)}{d\omega \, d\Omega_i} = \sum\limits_{j=1}^6 I_j with : \begin{align} 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^2p_f^2\sin^2\Theta_i - \sqrt{\Delta_2^2 + 4p_i^2p_f^2\sin^2\Theta_i}\left(\Delta_1 + \Delta_2\right) + \Delta_1\Delta_2} {-\Delta_2^2 - 4p_i^2p_f^2\sin^2\Theta_i - \sqrt{\Delta_2^2 + 4p_i^2p_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_fc}{E_f - p_fc}\right), \\ I_3 = {} & \frac{2\pi A}{\sqrt{\left(\Delta_2E_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_fc\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_2E_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 A p_f c\left(\Delta_2 E_f + \Delta_1 p_f c\right)} {\left(\Delta_2E_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_2E_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 - 4 p_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_2p_fc + \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{align} and : \begin{align} 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{align} 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. == See also ==