Editing Bremsstrahlung (section)

==In plasma: approximate classical results==
'''NOTE''': this section currently gives formulas that apply in the Rayleigh–Jeans limit <math>\hbar \omega \ll k_\text{B} T_\text{e}</math>, and does not use a quantized (Planck) treatment of radiation. Thus a usual factor like <math>\exp(-\hbar\omega/k_{\rm B}T_\text{e})</math> does not appear. The appearance of <math>\hbar \omega / k_\text{B} T_\text{e}</math> in <math>y</math> below is due to the quantum-mechanical treatment of collisions.[[File:Bremsstrahlung power2.svg|thumb|Bekefi's classical result for the bremsstrahlung emission power spectrum from a Maxwellian electron distribution.  It rapidly decreases for large <math>\omega</math>, and is also suppressed near <math>\omega = \omega_{\rm p}</math>. This plot is for the quantum case <math>T_\text{e} > Z^2 E_\text{h}</math>, and <math>\hbar \omega_\text{p} / T_\text{e} = 0.1</math>.  The blue curve is the full formula with <math>E_1(y)</math>, the red curve is the approximate logarithmic form for <math>y \ll 1</math>.]]

In a [[plasma (physics)|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,<ref>''Radiation Processes in Plasmas,'' G. Bekefi, Wiley, 1st edition (1966)</ref> while a simplified one is given by Ichimaru.<ref>''Basic Principles of Plasmas Physics: A Statistical Approach,'' S. Ichimaru, p. 228.</ref> In this section we follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber, {{nowrap|<math>k_\text{max}</math>.}}

Consider a uniform plasma, with thermal electrons distributed according to the [[Maxwell–Boltzmann distribution]] with the temperature <math>T_\text{e}</math>. Following Bekefi, the power spectral density (power per angular frequency interval per volume, integrated over the whole <math>4\pi</math> [[steradian|sr]] of solid angle, and in both polarizations) of the bremsstrahlung radiated, is calculated to be
<math display="block"> {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),
</math>
where <math>\omega_p \equiv (n_\text{e} e^2/\varepsilon_0m_\text{e})^{1/2}</math> is the electron plasma frequency, <math>\omega</math> is the photon frequency, <math>n_\text{e}, n_i</math> 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 <math>\omega < \omega_{\rm p}</math> (this is the cutoff condition for a light wave in a plasma; in this case the light wave is [[evanescent wave|evanescent]]). This formula thus only applies for <math>\omega>\omega_{\rm p}</math>. This formula should be summed over ion species in a multi-species plasma.

The special function <math>E_1</math> is defined in the [[exponential integral]] article, and the unitless quantity <math>y</math> is
<math display="block">y = \frac{1}{2} {\omega^2 m_\text{e} \over k_\text{max}^2 k_\text{B} T_\text{e}} </math>

<math>k_\text{max}</math> is a maximum or cutoff wavenumber, arising due to binary collisions, and can vary with ion species. Roughly, <math>k_\text{max} = 1 / \lambda_\text{B}</math> when <math>k_\text{B} T_\text{e} > Z_i^2 E_\text{h}</math> (typical in plasmas that are not too cold), where <math>E_\text{h} \approx 27.2</math> eV is the [[atomic units|Hartree energy]], and <math>\lambda_\text{B} = \hbar / (m_\text{e} k_\text{B} T_\text{e})^{1/2}</math>{{clarify |date=May 2016 |reason= That's NOT thermal de Broglie wavelength, a factor of square-root of (2π) is missing on the RHS.}} is the electron [[thermal de Broglie wavelength]]. Otherwise, <math>k_\text{max} \propto 1/l_\text{C}</math> where <math>l_\text{C}</math> is the classical Coulomb distance of closest approach.

For the usual case <math>k_m = 1/\lambda_B</math>, we find
<math display="block">y = \frac{1}{2} \left[\frac{\hbar\omega}{k_\text{B} T_\text{e}}\right]^2. </math>

The formula for <math>dP_\mathrm{Br} / d\omega</math> is approximate, in that it neglects enhanced emission occurring for <math>\omega</math> slightly above {{nowrap|<math>\omega_\text{p}</math>.}}

In the limit <math>y\ll 1</math>, we can approximate <math>E_1 </math> as <math>E_1(y) \approx -\ln [y e^\gamma] + O(y) </math> where <math>\gamma \approx 0.577</math> 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 <math>y > e^{-\gamma}</math> 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
<math display="block">\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}</math>
: <math>G(y_\text{p}=0) = 1</math> and decreases with <math>y_\text{p}</math>; it is always positive. For <math>k_\text{max} = 1/\lambda_\text{B}</math>, we find
<math display="block">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})</math>

Note the appearance of <math>\hbar</math> due to the quantum nature of <math>\lambda_{\rm B}</math>. In practical units, a commonly used version of this formula for <math>G=1</math> is <ref>NRL Plasma Formulary, 2006 Revision, p. 58.</ref>
<math display="block">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}. </math>

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]] <math>g_{\rm B}</math>, e.g. in <ref>''Radiative Processes in Astrophysics'', G.B. Rybicki & A.P. Lightman, p. 162.</ref> one finds
<math display="block">\varepsilon_\text{ff} = 1.4\times 10^{-27} T^\frac{1}{2} n_\text{e} n_i Z^2 g_\text{B},\,</math>
where everything is expressed in the [[CGS]] units.

=== Relativistic corrections ===
[[File:Brem cross section-en.svg|thumb|Relativistic corrections to the emission of a 30&nbsp;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 {{nowrap|<math>k_\text{B} T_\text{e}/m_\text{e} c^2</math>.}}<ref>{{cite thesis |type=PhD thesis |publisher=MIT |title=Fundamental limitations on plasma fusion systems not in thermodynamic equilibrium |first=T. H. |last=Rider |year=1995 |page=25 |hdl=1721.1/11412 }}</ref>

=== Bremsstrahlung cooling ===
If the plasma is [[optical depth|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 loss]]es. One generally uses the term ''bremsstrahlung losses'' in the context when the plasma cooling is undesired, as e.g. in [[nuclear fusion|fusion plasmas]].