PX282 - C9d - free-free absorption

it follows kramer's law:

κ \propto ρ T^{- 7 / 2}

this shows that the constant depends on the composition of the material
consider a free electron near an ion that absorbs an electron
the ion is needed to provide energy and momentum, without which conservation of energy-momentum becomes impossible
conserving momentum:

p_{γ} = p_{e}

conserving energy:

\begin{matrix} m_{e} c^{2} + p_{γ} c = \sqrt{m_{e}^{2} c^{4} + p_{e}^{2} c^{2}} \\ m_{e}^{2} c^{4} + p_{γ}^{2} c^{2} + 2 m_{e} p_{γ} c^{3} = m_{e}^{2} c^{4} + p_{e}^{2} c^{2} \\ ⟹ 2 m_{e} p_{γ} c^{3} = 0 \end{matrix}

it is only solvable if $m_{e} = 0$ , or $p_{γ} = 0$ , neither is physically possible
thus, additonal energy/momentum is needed to allow this interaction
the time an electron is in proximity to an ion $\propto v^{- 3}$
from the boltzmann distribution:

v = \sqrt{\frac{2 k_{B} T}{m}} ⟹ v \propto T^{1 / 2}

therefore, the number of systems able to absorb $\propto ρ T^{- 1 / 2}$
the absorption coefficient for a single system $\propto v_{γ}^{- 3}$
compiling above relations:

κ \propto v^{- 3} ρ T^{- 1 / 2}

most absorption happens at the peak of the planck function, $B (T) :$

∴ κ \propto ρ T^{- 7 / 2}