 Solution

Use $u$ as a spectroscopic wavenumber, in the same units throughout. Write the mass absorption coefficient of the absorbing species asHere $S$ is the spectral line strength, the integrated mass absorption coefficient; it is not the line-centre coefficient. If ${\kappa }_{\nu }$ has units ${\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}$ and $\nu$ is in $\mathrm{c}{\mathrm{m}}^{-1}$, then $S$ has units ${\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}\mathrm{c}{\mathrm{m}}^{-1}$. For absorber mass density ${\rho }_a$, the volume coefficient in Beer-Lambert law is ${\rho }_a{\kappa }_{\nu }$.

Lifetime broadening. Finite lifetimes of the upper and lower states destroy optical coherence, even for an isolated molecule. Exponential coherence decay gives a Lorentzian line shape. For population decay rates ${\Gamma }_u,{\Gamma }_l$ and no extra dephasing, its wavenumber half-width is $({\Gamma }_u+{\Gamma }_l)/(4\pi c)$, with $c$ expressed in the length unit reciprocal to $\nu$. This sets a natural linewidth and can matter at extremely low collision rates; it is generally much smaller than atmospheric collisional or thermal widths.

Thermal Doppler broadening. A Maxwell-Boltzmann distribution of molecular line-of-sight velocities givesThis Gaussian distribution has a $1/e$ half-width ${\Delta }_D$, determined by absorber molecular mass $m_a$ and temperature. It is especially relevant in the rarefied upper atmosphere, where collisions are infrequent.

Pressure broadening. Collisions perturb transition frequencies and phases. In the impact regime, exponential phase decorrelation givesHere ${\Delta }_L$ is the half-width at half maximum, proportional to the coherence-destroying collision rate. It depends on perturber density, molecular collision properties, and temperature; it is often parameterized by ${\Delta }_L\propto p(T_{\ast }/T{)}^{n_T}$ for a specified broadener. It dominates much of the dense lower atmosphere. The Lorentzian line shape has algebraic $\delta {\nu }^{-2}$ wings, whereas the Doppler profile has rapidly decreasing Gaussian wings. Where both mechanisms matter, their convolution is a Voigt profile. A pure pressure-broadened profile should not be extrapolated to arbitrarily low pressure while discarding the finite Doppler core.