text: NOTE: this section currently gives formulas that apply in the Rayleigh–Jeans limit , and does not use a quantized (Planck) treatment of radiation. Thus a usual factor like  does not appear. The appearance of  in  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, 
  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
  where  is the electron plasma frequency,  is the photon frequency,  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  (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 . 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  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 logarithmic term that match detailed binary-collision calculations.
  The total emission power density, integrated over all frequencies, is
  : and decreases with ; it is always positive. For , we find
  Note the appearance of  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
  where everything is expressed in the CGS units.