 where <formula> is the rate constant for emitting a photon spontaneously, and <formula> is the rate constant for emissions in response to ambient photons (induced or stimulated emission). In thermodynamic equilibrium, the number of atoms in state i and those in state j must, on average, be constant; hence, the rates <formula> and <formula> must be equal. Also, by arguments analogous to the derivation of Boltzmann statistics, the ratio of <formula> and <formula> is <formula> where <formula> are the degeneracy of the state i and that of j, respectively, <formula> their energies, k the Boltzmann constant and T the system's temperature. From this, it is readily derived that <formula> and
