Bremsstrahlung
In plasma: approximate Classical Results

NOTE: this section currently gives formulas that apply in the Rayleigh–Jeans limit ℏ ω ≪ k B T e {\displaystyle \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 (− ℏ ω / k B T e) {\displaystyle \exp(-\hbar \omega /k_{\rm {B}}T_{\text{e}})} does not appear. The appearance of ℏ ω / k B T e {\displaystyle \hbar \omega /k_{\text{B}}T_{\text{e}}} in y {\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. In this section we follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber, k max {\displaystyle k_{\text{max}}}.

Consider a uniform plasma, with thermal electrons distributed according to the Maxwell–Boltzmann distribution with the temperature T e {\displaystyle T_{\text{e}}}. Following Bekefi, the power spectral density (power per angular frequency interval per volume, integrated over the whole 4 π {\displaystyle 4\pi } sr of solid angle, and in both polarizations) of the bremsstrahlung radiated, is calculated to be d P B r d ω = 8 2 3 π e ¯ 6 (m e c 2) 3 / 2 1 / 2 Z i 2 n i n e (k B T e) 1 / 2 E 1 (y), {\displaystyle {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^{1/2}{Z_{i}^{2}n_{i}n_{\text{e}} \over (k_{\rm {B}}T_{\text{e}})^{1/2}}E_{1}(y),} where ω p ≡ (n e e 2 / ε 0 m e) 1 / 2 {\displaystyle \omega _{p}\equiv (n_{\text{e}}e^{2}/\varepsilon _{0}m_{\text{e}})^{1/2}} is the electron plasma frequency, ω {\displaystyle \omega } is the photon frequency, n e, n i {\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 ω < ω p {\displaystyle \omega <\omega _{\rm {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 ω > ω p {\displaystyle \omega >\omega _{\rm {p}}}. This formula should be summed over ion species in a multi-species plasma.

The special function E 1 {\displaystyle E_{1}} is defined in the exponential integral article, and the unitless quantity y {\displaystyle y} is y = 1 2 ω 2 m e k max 2 k B T e {\displaystyle y={\frac {1}{2}}{\omega ^{2}m_{\text{e}} \over k_{\text{max}}^{2}k_{\text{B}}T_{\text{e}}}}

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

For the usual case k m = 1 / λ B {\displaystyle k_{m}=1/\lambda _{B}}, we find y = 1 2 2. {\displaystyle y={\frac {1}{2}}\left^{2}.}

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

In the limit y ≪ 1 {\displaystyle y\ll 1}, we can approximate E 1 {\displaystyle E_{1}} as E 1 (y) ≈ − ln + O (y) {\displaystyle E_{1}(y)\approx -\ln+O(y)} where γ ≈ 0.577 {\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 y > e − γ {\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 P B r = ∫ ω p ∞ d ω d P B r d ω = 16 3 e ¯ 6 m e 2 c 3 Z i 2 n i n e k max G (y p) G (y p) = 1 2 π ∫ y p ∞ d y y − 1 / 2 1 / 2 E 1 (y) y p = y (ω = ω p) {\displaystyle {\begin{aligned}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}}^{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}}}^{\infty }dy\,y^{-{1}/{2}}\left^{1/2}E_{1}(y)\\y_{\text{p}}&=y({\omega \!=\!\omega _{\text{p}}})\end{aligned}}}

P B r = 16 3 e ¯ 6 (m e c 2) 3 2 ℏ Z i 2 n i n e (k B T e) 1 2 G (y p) {\displaystyle 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 ℏ {\displaystyle \hbar } due to the quantum nature of λ B {\displaystyle \lambda _{\rm {B}}}. In practical units, a commonly used version of this formula for G = 1 {\displaystyle G=1} is P B r = Z i 2 n i n e 2 T e 1 2. {\displaystyle P_{\mathrm {Br} }={Z_{i}^{2}n_{i}n_{\text{e}} \over \left^{2}}T_{\text{e}}^{\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 B {\displaystyle g_{\rm {B}}}, e.g. in one finds ε ff = 1.4 × 10 − 27 T 1 2 n e n i Z 2 g B, {\displaystyle \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.
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