Editing Bremsstrahlung (section)

== 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.<ref>{{Cite journal |last=Sommerfeld |first=A. |date=1931 |title=Über die Beugung und Bremsung der Elektronen |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19314030302 |journal=Annalen der Physik |language=de |volume=403 |issue=3 |pages=257–330 |doi=10.1002/andp.19314030302 | bibcode=1931AnP...403..257S }}</ref> This analytical solution involves complicated mathematics, and several numerical calculations have been published, such as by Karzas and Latter.<ref>{{Cite journal |last1=Karzas |first1=W. J. |last2=Latter |first2=R. |date=May 1961 |title=Electron Radiative Transitions in a Coulomb Field. |url=http://adsabs.harvard.edu/doi/10.1086/190063 |journal=The Astrophysical Journal Supplement Series | language=en |volume=6 |pages=167 |doi=10.1086/190063 |bibcode=1961ApJS....6..167K |issn=0067-0049}}</ref>  Other approximate formulas have been presented, such as in recent work by Weinberg <ref>{{Cite journal |last=Weinberg |first=Steven |date=2019-04-30 |title=Soft bremsstrahlung |url=https://link.aps.org/doi/10.1103/PhysRevD.99.076018 |journal=Physical Review D |language=en |volume=99 |issue=7 |pages=076018 |doi=10.1103/PhysRevD.99.076018 |arxiv=1903.11168 |bibcode=2019PhRvD..99g6018W |s2cid=85529161 |issn=2470-0010}}</ref> and Pradler and Semmelrock.<ref>{{Cite journal |last1=Pradler |first1=Josef |last2=Semmelrock |first2=Lukas |date=2021-11-01 |title=Nonrelativistic Electron–Ion Bremsstrahlung: An Approximate Formula for All Parameters |journal=The Astrophysical Journal |volume=922 |issue=1 |pages=57 |doi=10.3847/1538-4357/ac24a8 |arxiv=2105.13362 |bibcode=2021ApJ...922...57P | s2cid=235248150 |issn=0004-637X |doi-access=free }}</ref>

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 <math>m_\text{e}</math>, charge <math>-e</math>, and initial speed <math>v</math> decelerating in the Coulomb field of a gas of heavy ions of charge <math>Ze</math> and number density <math>n_i</math>. The emitted radiation is a photon of frequency <math>\nu=c/\lambda</math> and energy <math>h\nu</math>. We wish to find the emissivity <math>j(v,\nu)</math> 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]] ''g''<sub>ff</sub> accounting for quantum and other corrections:
<math display="block">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)</math>
<math>j(\nu,v) = 0</math> if <math>h\nu > mv^2/2</math>, that is, the electron does not have enough kinetic energy to emit the photon.  A general, quantum-mechanical formula for <math>g_{\rm ff}</math> 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 oscillation|plasma frequency]] <math>\nu_{\rm pe} \propto n_{\rm e}^{1/2}</math>with <math>n_\text{e}</math> the plasma electron density. Note that light waves are evanescent for <math>\nu<\nu_{\rm pe}</math> and a significantly different approach would be needed.
* Soft photons: <math>h\nu\ll m_\text{e}v^2/2</math>, that is, the photon energy is much less than the initial electron kinetic energy.

With these assumptions, two unitless parameters characterize the process: <math>\eta_Z \equiv Z \bar e^2/\hbar v</math>, which measures the strength of the electron-ion Coulomb interaction, and <math>\eta_\nu \equiv h\nu/2m_\text{e}v^2</math>, which measures the photon "softness" and we assume is always small (the choice of the factor 2 is for later convenience). In the limit <math>\eta_Z\ll 1</math>, the quantum-mechanical Born approximation gives:
<math display="block">g_\text{ff,Born} = {\sqrt3 \over \pi}\ln{1\over\eta_\nu}</math>

In the opposite limit <math>\eta_Z\gg 1</math>, the full quantum-mechanical result reduces to the purely classical result
<math display="block">g_\text{ff,class} = {\sqrt3\over\pi}\left[\ln\left({1\over \eta_Z\eta_\nu}\right)- \gamma \right]</math>
where <math>\gamma\approx 0.577</math> is the [[Euler–Mascheroni constant]]. Note that <math>1/\eta_Z\eta_\nu=m_\text{e}v^3/\pi Z\bar e^2\nu</math> which is a purely classical expression without the Planck constant <math>h</math>.

A semi-classical, heuristic way to understand the Gaunt factor is to write it as <math>g_\text{ff} \approx \ln(b_\text{max}/b_\text{min})</math> where <math>b_{\max}</math> and <math>b_{\min}</math> are maximum and minimum "impact parameters" for the electron-ion collision, in the presence of the photon electric field. With our assumptions, <math>b_{\rm max}=v/\nu</math>: for larger impact parameters, the sinusoidal oscillation of the photon field provides "phase mixing" that strongly reduces the interaction. <math>b_{\rm min}</math> is the larger of the quantum-mechanical de&nbsp;Broglie wavelength <math>\approx h/m_\text{e} v</math> and the classical distance of closest approach <math>\approx \bar e^2 / m_\text{e} v^2</math> 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
<math display="block">g_\text{ff} \approx \max\left[1, {\sqrt3\over\pi} \ln\left[{1\over \eta_\nu\max(1,e^\gamma\eta_Z)}\right] \right]</math>