Problem 6.131

A beam of light of frequency \(\omega\), equal to the resonant frequency of transition of atoms of gas, passes through that gas heated to temperature \(T .\) In this case \(\hbar \omega \gg k T .\) Taking into account induced radiation, demonstrate that the absorption coefficient of the gas \(x\) varies as \(x=x_{0}\left(1-\mathrm{e}^{-\hbar \omega / k T}\right),\) where \(x_{0}\) is the absorption coefficient for \(T \rightarrow 0\)