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 $T \rightarrow 0$