Angular-frequency Einstein coefficients ($A_{21}/B_{21}=\hslash {\omega }_0^3/({\pi }^2{c}^3)$)

When $B_{12}$ and $B_{21}$ multiply radiation energy density per unit angular frequency, they differ by a factor $2\pi$ from coefficients multiplying the density per ordinary frequency. For nondegenerate levels, Boltzmann distribution populations and thermal equilibrium give $B_{12}=B_{21}$ and the free-space ratio in the title. Specifically $B_{\omega }=2\pi B_{\nu }$ while $A$ is unchanged. Polarization restrictions and level degeneracy must be specified separately.

Weak incoherent frequency components contribute probabilities. The long-time sinc-squared kernel gives the displayed stimulated absorption rate for a fixed polarization and spectral energy density per angular frequency. Isotropic unpolarized radiation instead contributes $|{\mathbf{d}}_{21}{|}^2/3$. Converting to density per ordinary frequency changes the coefficient by $2\pi$.