Einstein coefficients ($A_{ji},B_{ij},B_{ji}$)

Einstein coefficients describe spontaneous emission, stimulated emission and absorption between atomic levels. With $B$ multiplying radiation energy density per frequency, thermal detailed balance gives $g_i{B}_{ij}=g_j{B}_{ji}$ and $A_{ji}/B_{ji}=8\pi h{\nu }^3/c^3$. Coefficients defined using mean specific intensity instead have a different normalization; specify the radiation convention.

Absorption of incident radiation that promotes an emitter from a lower to an upper state. In an Einstein-rate description the rate is proportional to the relevant spectral radiation energy density. The degeneracy-weighted absorption coefficient and stimulated-emission coefficient satisfy detailed balance.

For an ideal emitter able to exchange one fermion with a zero-chemical potential bath, Pauli blocking gives downward rate $A(1-f)$ rather than the bosonic $A(1+f)$. If $u_{\nu }=h\nu {\mathcal{D}}_{\nu }f$ and $C=A/(h\nu {\mathcal{D}}_{\nu })>0$, the rate is $A-C{u}_{\nu }$. Detailed balance with a Fermi-Dirac distribution implies $g_1{B}_{12}=g_{2}C$ and $A/C=h\nu {\mathcal{D}}_{\nu }$. For $d$ massless helicity states, ${\mathcal{D}}_{\nu }=d4\pi {\nu }^2/c^3$. Thus an additive induced-emission coefficient would be negative, expressing blocking rather than a negative total rate. The emitter must change its fermion parity; two ordinary electronic energy levels cannot exchange a single neutrino this way.

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 $2\pi$.

Let $A_{21}$ be the spontaneous emission coefficient and let $B\rho$ be the induced transition rate per atom under a resonant radiation spectral energy density $\rho$. For nondegenerate levels the Einstein coefficients obey $B_{12}=B_{21}=B$. In a closed two-level system with no collisional or other population transfer, steady state requiresFor nonzero upper population and drive, division gives the ratio of total spontaneous and stimulated emission rates in the title. In particular $N_1>N_2$ when $A_{21}>0$; continuous driving of this closed model approaches equal populations but cannot create a population inversion. No thermal population law is assumed.