 Solution

Specify the radiation normalization first. Let $u_{\nu }$ be radiation energy density per ordinary frequency, with SI units $\mathrm{J}{\mathrm{m}}^{-3}\mathrm{H}{\mathrm{z}}^{-1}$. Then $A_{21}$ is the spontaneous emission rate per upper-state chemical atom, while $B_{12}{u}_{\nu }$ and $B_{21}{u}_{\nu }$ are the induced absorption and stimulated emission rates, evaluated at a sharp transition frequency. For a finite spectral line profile, replace $u_{\nu }$ by its profile-weighted average.

The Einstein coefficients obeyHere $g_i$ are the statistical weights of the two energy levels, and the field is unpolarized and isotropic. The units and physical dimensions areThe apparently time-independent dimension of $B$ follows because a per-hertz density includes an extra unit of time. If instead $B$ multiplies the mean intensity $J_{\nu }$, use $B^{(J)}=(4\pi /c)B$ and $A/B^{(J)}=2h{\nu }_0^3/c^2$. These distinct normalizations must not be mixed in the radiative transfer equation.