by Menzel, D.H. of Harvard College Observatory and Smithsonian Institution Astrophysical Observatory1970, in Spectrum Formation in Stars with Steady-State Extended Atmospheres, eds. Groth,H.G, Wellmann,P., NBS special publication 332, p.134.
mu dI/dr = -(n1-n2 g1/g2)I alpha + (2hv^3/c^2)(n2 g1/g2)alpha (1)where I is the specific intensity and alpha is the atomic absorption coefficient at frequency, v. The quantities n1 and n2 are the atomic populations of the lower and upper levels, whose respective statistical weights are g1 and g2.
In thermodynamic equilibrium, a single parameter T, the absolute temperature, governs the Boltzmann and Planck formulas, so that
n2/n1 = (g2/g1) exp(-hv/kT) (2) I = (2hv^3/c^2) / [exp(hv/kT) - 1]When these equations are substituted into (1) the right-hand side vanishes, as it must for thermodynamic equilibrium, since the intensity is isotropic and independent of position.
In equation (1), the second term in the parenthesis represents the stimulated emission or, as it should more properly be called, the negative absorption. Too many astrophysicist either combine it with the random emission, the second term on the right-hand side, or neglect it altogether. The true source function takes the form of the second term on the right-hand side of (1), which in no sense resembles a Planck function except when hv is much greater than kT.
The quantities n1 and n2 are to be calculated from the equations of statistical equilibrium, which involve collisional excitation and de-excitation as well as radiative processes.
Many years ago, in problems related to gaseous nebulae, I introduced a dimensionless parameter, b, to indicate the degree of departure of a gas from thermodynamic equilibrium at temperature T of the electron gas. This parameter equalled unity for thermodynamic equilibrium. Thus I could write
n2/n1 = [ b2 g2 /(b1 g1) ] exp(-hv/kT) (3)with which expression (1) becomes
dI/dh cos theta = - n1 alpha [(1 - (b2/b1)exp(-hv/kT)) I + 2hv^3/c^2(b2/b1)exp(-hv/kT)] (4)As long as hv/kT was much greater than 1, we can usually neglect the term representing the stimulated emission. And, as long as b2/b1 does not depart too far from unity, the second term on the right hand side is approximately equal to the source function.
Many studies have shown that the b's exhibit the following behavior for nebulae and, presumably, also for stars with highly distended atmospheres. First of all, b's for ground or metastable levels tend to be high. Second, the first excited levels directly above the ground level tend to have b's much less than unity. Third, the b's for still higher levels slowly tend to unity at the series limit.
For all lines except those from the ground or metastable levels, then, b2/b1 will exceed unity. And when the temperature is high enough, the stimulated emission may exceed the ordinary absorption so that
dI/dh cos theta = n1 alpha(b2/b1)exp(-hv/kT)[ 2hv^3/c^2 + I ] (5)Since this term is essentially positive, the emission line increases in intensity with depth. (exponentially)
The process is self-limiting, however, since an increase in the intensity of the incident radiation causes the medium to approach local thermodynamic equilibrium. However, as long as the energy is 'diluted', some sort of laser action will occur.