Opacity

Now, assume Thermodynamical Equilibrium!N u /N l =g u /g l exp(-hν/kT)Boltzmann distribution, g u and g l statistical weightsAlso, number of downward transitions = number of upward transitions per time unit.Collision transitions up and down cancel in TE => N uA ul+ N uB ulI ν= N lB luI νSolve for I νand use the Boltzmann distribtution:Ι ν=A ul(g l /g u ) B luexp (hν/kT) - B ulcf Planck’s law:Ι ν=2hν 3 1c 2 exp (hν/kT) - 1This obviously suggests{ AulB ul= (g l /g u ) B lu= 2hν 3 /c 2 B ulNote: Planck’s law can actually be proven this way, if oneuses the low energy limit (Rayleigh-Jean´s law),see Gray, page 115