Bremsstrahlung
Thermal bremsstrahlung in a medium: emission and absorption

This section discusses bremsstrahlung emission and the inverse absorption process (called inverse bremsstrahlung) in a macroscopic medium. We start with the equation of radiative transfer, which applies to general processes and not just bremsstrahlung: 1 c ∂ t I ν + n ^ ⋅ ∇ I ν = j ν − k ν I ν {\displaystyle {\frac {1}{c}}\partial _{t}I_{\nu }+{\hat {\mathbf {n} }}\cdot \nabla I_{\nu }=j_{\nu }-k_{\nu }I_{\nu }}

I ν (t, x) {\displaystyle I_{\nu }(t,\mathbf {x})} is the radiation spectral intensity, or power per (area × solid angle in photon velocity space × photon frequency) summed over both polarizations. j ν {\displaystyle j_{\nu }} is the emissivity, analogous to j (v, ν) {\displaystyle j(v,\nu)} defined above, and k ν {\displaystyle k_{\nu }} is the absorptivity. j ν {\displaystyle j_{\nu }} and k ν {\displaystyle k_{\nu }} are properties of the matter, not the radiation, and account for all the particles in the medium – not just a pair of one electron and one ion as in the prior section. If I ν {\displaystyle I_{\nu }} is uniform in space and time, then the left-hand side of the transfer equation is zero, and we find I ν = j ν k ν {\displaystyle I_{\nu }={j_{\nu } \over k_{\nu }}}

If the matter and radiation are also in thermal equilibrium at some temperature, then I ν {\displaystyle I_{\nu }} must be the blackbody spectrum: B ν (ν, T e) = 2 h ν 3 c 2 1 e h ν / k B T e − 1 {\displaystyle B_{\nu }(\nu,T_{\text{e}})={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /k_{\text{B}}T_{\text{e}}}-1}}} Since j ν {\displaystyle j_{\nu }} and k ν {\displaystyle k_{\nu }} are independent of I ν {\displaystyle I_{\nu }}, this means that j ν / k ν {\displaystyle j_{\nu }/k_{\nu }} must be the blackbody spectrum whenever the matter is in equilibrium at some temperature – regardless of the state of the radiation. This allows us to immediately know both j ν {\displaystyle j_{\nu }} and k ν {\displaystyle k_{\nu }} once one is known – for matter in equilibrium.
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