= Solution
{c}
For a closed absorbing <two-level system>, write $N=N_1+N_2$, let $A=1/\tau$, and use $\sigma_a=\sigma_{12}$ for the <absorption cross section>. This differs from the stimulated-<emission cross section> convention in part (a). With $r=g_2/g_1$, the <rate equations> are
$$
\boxed{\dot N_2=\frac{\sigma_a I}{\hbar\omega}\left(N_1-\frac{N_2}{r}\right)-AN_2,\qquad
\dot N_1=-\dot N_2.}
$$
At steady state, $N_2=W_aN/[A+(1+1/r)W_a]$ for $W_a=\sigma_aI/(\hbar\omega)$. Hence the positive <absorption coefficient> is
$$
\kappa=\sigma_a\left(N_1-\frac{N_2}{r}\right)
=\frac{N\sigma_a}{1+I/I_{s,a}},\qquad
I_{s,a}=\frac{\hbar\omega}{(1+1/r)\sigma_a\tau}.
$$
For the given <sodium> fine-structure transition, the <statistical weights> are $g_1=2$, $g_2=4$, so $r=2$ and $1+1/r=3/2$. The small-signal transmission gives the <optical depth>
$$
D_0=N\sigma_aL=-\ln0.90=0.1053605.
$$
To account for the finite decrease of <intensity> through the sample, integrate $dI/dz=-\kappa I$ rather than treating the stronger beam as exactly uniform. If $T=I_{\rm out}/I_{\rm in}$, this gives
$$
\ln T+\frac{I_{\rm in}(T-1)}{I_{s,a}}=-D_0,\qquad
I_{s,a}=\frac{I_{\rm in}(1-T)}{D_0+\ln T}.
$$
Taking $270\,\mathrm{W\,m^{-2}}$ as the incident <intensity> and $T=0.967$,
$$
I_{s,a}=\frac{270(0.033)}{0.1053605+\ln0.967}
\simeq124.09\,\mathrm{W\,m^{-2}}.
$$
Using the <photon energy> $hc/\lambda$ then gives
$$
\boxed{\sigma_a=\frac{hc}{(3/2)\lambda\tau I_{s,a}}
\simeq1.13\times10^{-13}\,\mathrm{m^2}.}
$$
Ignoring the small variation of the stronger <intensity> would instead give about $1.11\times10^{-13}\,\mathrm{m^2}$; the integrated value uses the stated transmissions more consistently. The length is not needed to eliminate $N$, but gives the check $N\simeq9.3\times10^{14}\,\mathrm{m^{-3}}$.
The resonant <absorption cross section> is of optical-<wavelength>-squared size, much larger than a typical atomic geometric collision area. <Pressure broadening> <cross sections> involve phase-changing interatomic encounters and need not equal hard-sphere collision areas; their numerical value depends on collision partners and speed. No precise ratio follows from the supplied data. A useful distinction is that resonant <absorption> involves coherent response on the optical scale, so its <cross section> need not resemble an <chemical atom>'s geometric area.
For <homogeneous spectral broadening>, all participating <chemical atoms> must share the same <resonance> and single-atom response width. In a dilute sample dominated by <natural linewidth>, the weak-beam requirement is
$$
\Delta\nu_D=\frac1\lambda\sqrt{\frac{8\ln2\,k_BT}{m_{\rm Na}}}
\ll\frac1{2\pi\tau}\simeq9.95\,\mathrm{MHz}.
$$
Using the <sodium> <atomic mass>, equality occurs near $T=0.017\,\mathrm K$, so $T\ll17\,\mathrm{mK}$ is a sufficient thermal condition. Spatial <Zeeman energy> shifts, light shifts and transit effects should also be small, the <laser> should address the same selected component, and the interaction should last long enough for steady populations. Uniform impact <pressure broadening> can also be homogeneous, but would change the width and effective <cross section>; it is not included in the lifetime-only numerical model.
Real <sodium> also has <hyperfine structure> and <optical pumping> among <magnetic sublevels>. The inferred $\sigma_a$ is the effective cross section of the stated $g_J=2J+1$ <two-level system> model, assuming statistical sublevel populations and no loss into unaddressed states. A measurement on a resolved, polarized closed <hyperfine structure> component requires its actual <transition dipole moment> and <selection rules>; a universal polarization-independent <sodium> cross section cannot be inferred from this simplified model. The dependence of resonant cross sections and <saturation intensities> on sublevel preparation is tabulated explicitly for the real <sodium> D lines.<Sodium D Line Data by Daniel A. Steck>[Section on saturation intensity and resonant scattering cross section, and Table 7]{ref}
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