Editing Bremsstrahlung (section)

== Quantum mechanical description ==
The complete quantum mechanical description was first performed by Bethe and Heitler.<ref>{{cite journal |last1=Bethe |first1=H. A. |last2=Heitler |first2=W. |year=1934 |title=On the stopping of fast particles and on the creation of positive electrons |journal= Proceedings of the Royal Society A |volume=146 |issue= 856|pages=83–112 |doi=10.1098/rspa.1934.0140 |bibcode=1934RSPSA.146...83B |doi-access=free }}</ref> 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
: <math>\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}</math>
where <math>Z</math> is the [[atomic number]], <math>\alpha_\text{fine}\approx 1/137</math> the [[fine-structure constant]], <math>\hbar</math> the [[reduced Planck constant]] and <math>c</math> the [[speed of light]]. The kinetic energy <math>E_{\text{kin},i/f}</math> of the electron in the initial and final state is connected to its total energy <math>E_{i,f}</math> or its [[momenta]] <math>\mathbf{p}_{i,f}</math> via
<math display="block">
  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},
</math>
where <math>m_\text{e}</math> is the [[mass of an electron]]. [[Conservation of energy]] gives
<math display="block">  E_f = E_i - \hbar\omega, </math>
where <math> \hbar\omega </math> is the photon energy. The directions of the emitted photon and the scattered electron are given by
<math display="block">\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}</math>
where <math>\mathbf{k}</math> is the momentum of the photon.

The differentials are given as
<math display="block">\begin{align}
  d\Omega_i &= \sin\Theta_i\ d\Theta_i,\\
  d\Omega_f &= \sin\Theta_f\ d\Theta_f.
\end{align}</math>

The [[absolute value]] of the [[virtual photon]] between the nucleus and electron is
: <math>\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}</math>

The range of validity is given by the Born approximation
<math display="block"> v \gg \frac{Zc}{137} </math>
where this relation has to be fulfilled for the velocity <math> v </math> of the electron in the initial and final state.

For practical applications (e.g. in [[Monte Carlo N-Particle Transport Code|Monte Carlo codes]]) it can be interesting to focus on the relation between the frequency <math>\omega</math> of the emitted photon and the angle between this photon and the incident electron. Köhn and [[Ute Ebert|Ebert]] integrated the quadruply differential cross section by Bethe and Heitler over <math>\Phi</math> and <math>\Theta_f</math> and obtained:<ref>{{cite journal |last1=Köhn |first1=C. |last2=Ebert |first2=U.|author2-link= Ute Ebert |title=Angular distribution of bremsstrahlung photons and of positrons for calculations of terrestrial gamma-ray flashes and positron beams |journal= Atmospheric Research|year=2014 |volume=135–136 |pages=432–465 |doi=10.1016/j.atmosres.2013.03.012 |bibcode=2014AtmRe.135..432K |arxiv=1202.4879 |s2cid=10679475 }}</ref>
<math display="block">
  \frac{d^2\sigma (E_i, \omega, \Theta_i)}{d\omega \, d\Omega_i} = \sum\limits_{j=1}^6 I_j
</math>
with
: <math>\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}</math>
and
: <math>\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}</math>

However, a much simpler expression for the same integral can be found in <ref>{{cite journal |first1=H. W. |last1=Koch |first2=J. W. |last2=Motz |title=Bremsstrahlung Cross-Section Formulas and Related Data |journal= Reviews of Modern Physics|volume=31 |issue= 4|year=1959 |pages=920–955 |doi=10.1103/RevModPhys.31.920 |bibcode=1959RvMP...31..920K }}</ref> (Eq. 2BN) and in <ref>{{cite journal |first1=R. L. |last1=Gluckstern | first2=M. H. Jr. |last2=Hull |title=Polarization Dependence of the Integrated Bremsstrahlung Cross Section |journal= Physical Review|volume=90 |issue= 6|pages=1030–1035 |year=1953 |doi=10.1103/PhysRev.90.1030 |bibcode=1953PhRv...90.1030G }}</ref> (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&nbsp;keV) emit photons in forward direction while electrons with a small energy emit photons isotropically.