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Free-free absorption is the inverse of bremsstrahlung: a free electron absorbs a photon during an encounter with an ion, which permits conservation of momentum. In local thermodynamic equilibrium its absorption coefficient and the thermal free-free emissivity obey Kirchhoff's law of thermal radiation, $j_{\nu }={\alpha }_{\nu }{B}_{\nu }(T)$. For a nonrelativistic plasma at $h\nu \ll k_{B}T$, ${\alpha }_{\nu }\propto n_e{\sum }_i{Z}_i^2{n}_i{T}^{-3/2}{\nu }^{-2}{g}_{\mathrm{ff}}$, including the stimulated-emission correction in the net coefficient.

For $h\nu \ll k_{B}T$, expand the exponential in the thermal free-free emissivity:The energy emitted per unit frequency is therefore approximately flat, with the weak Gaunt factor dependence superposed. At these low photon energies, most of the thermal electrons can participate in emission; there is no exponentially small population threshold.

For $h\nu >k_{B}T$, the spectrum turns down approximately as $e^{-h\nu /(k_{B}T)}$, apart from the Gaunt factor. Conservation of energy requires an emitting electron to supply at least $h\nu$. The high-energy tail of its Maxwell-Boltzmann distribution is exponentially rare, which produces this cutoff.

“Flat” refers to $j_{\nu }$, an energy spectrum per unit frequency. The emitted photon number spectrum instead contains $1/(h\nu )$, while energy per logarithmic frequency interval is $\nu j_{\nu }$. At sufficiently low frequency the plasma can also become optically thick to free-free absorption; the given optically thin emissivity is then not the escaping spectrum.

Neglect electron mass and nuclear binding corrections when converting mass density $\rho$ into hydrogen and helium number densities. With $x=x_H$,The thermal free-free emissivity adds over ion species with weights $Z_i^2$. ConsequentlyWith each Gaunt factor set to one, the mixture hasAt fixed $n_e$, pure hydrogen gives the composition factor $1$, and pure helium gives $2$. The latter check follows independently from $n_{\mathrm{He}}=n_e/2$ and $Z^2=4$; simply replacing $Z$ by an average charge would lose this weighting.