Bremsstrahlung - Thermal Bremsstrahlung

Thermal Bremsstrahlung

In a plasma the free electrons constantly produce bremsstrahlung in collisions with the ions. A complete analysis requires accounting for both binary Coulomb collisions as well as collective (dielectric) behavior. A detailed treatment is given in, while a simplified one is given in. In this section we follow Bekefi's dielectric treatment, with collisions included approximately via the cutoff wavenumber .

Consider a uniform plasma, with thermal electrons (distributed according to the Maxwell–Boltzmann distribution with the temperature ). Following Bekefi, the power spectral density (power per angular frequency interval per volume, integrated over the whole sr of solid angle, and in both polarizations) of the bremsstrahlung radiated, is calculated to be

${dP_\mathrm{Br} \over d\omega} = {8\sqrt 2 \over 3\sqrt\pi} \left^{1/2} \left \left E_1(y),$

where is the electron plasma frequency, is the number density of electrons and ions, is the classical radius of electron, is its mass, is the Boltzmann constant, and is the speed of light. The first bracketed factor is the index of refraction of a light wave in a plasma, and shows that emission is greatly suppressed for (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 . Note that the second bracketed factor has units of 1/volume and the third factor has units of energy, giving the correct total units of energy/volume. This formula should be summed over ion species in a multi-species plasma.

The special function is defined in the exponential integral article, and the unitless quantity is

is a maximum or cutoff wavenumber, arising due to binary collisions, and can vary with ion species. Roughly, when (typical in plasmas that are not too cold), where eV is the Hartree energy, and is the electron thermal de Broglie wavelength. Otherwise, where is the classical Coulomb distance of closest approach.

For the usual case, we find

The formula for is approximate, in that it neglects enhanced emission occurring for slightly above .

In the limit, we can approximate E1 as where 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 the log term is negative, and the approximation is clearly inadequate. Bekefi gives corrected expressions for the logarthmic term that match detailed binary-collision calculations.

The total emission power density, integrated over all frequencies, is

$\begin{align} P_\mathrm{Br} &= \int_{\omega_p}^\infty d\omega {dP_\mathrm{Br}\over d\omega} = {16 \over 3} \left \left k_m G(y_p) \\ G(y_p) &= {1 \over 2\sqrt{\pi}} \int_{y_p}^\infty dy y^{-1/2} \left^{1/2} E_1(y) \\ y_p &= y(\omega=\omega_p) \end{align}$

and decreases with ; it is always positive. For, we find

$P_\mathrm{Br} = {16 \over 3} \left \left \alpha G(y_p)$

The first bracketed factor has units of 1/volume, while the second has units of power. Note the appearance of the fine-structure constant due to the quantum nature of . In practical units, a commonly used version of this formula for is

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, e.g. in one finds

$\varepsilon_\mathrm{ff} = 1.4\times 10^{-27} T^{1/2} n_{e} n_{i} Z^{2} g_B,\,$

where everything is expressed in the CGS units.

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