stellar - about H$_alpha$ and H$_beta$ lines in astronomical objects' spectra

In a normal stellar atmosphere, where the temperature decreases with height, I think this is not possible.

H-alpha absorption arises from the $n=2$ to $n=3$ transition; H-beta from the $n=2$ to $n=4$ transition. Thus both transitions are governed by the number of atoms in the $n=2$ lower level.

The absorption coefficient for a transition can be written in proportionality terms by (in local thermodynamic equilibrium)
$$ alpha_nu propto nu B_{lu} n_l ( 1 - exp[-hnu/kT]),$$
where $B_{ul}$ is the Einstein absorption coefficient, $n_l$ is the population of the lower energy level and $nu$ is the photon frequency corresponding to the transition.

Using the well-known relationship between the Einstein emission and absorption coefficients and values for the emission coefficients found here, we can estimate a ratio of absorption coefficients for a given temperature.

$$ frac{alpha_{Hbeta}}{alpha_{Halpha}} = frac{g_{n=4}}{g_{n=3}} frac{A_{Hbeta}}{A_{Halpha}} left(frac{nu_{alpha}}{nu_{beta}}right)^2 left(frac{1 - exp(-hnu_{beta}/kT)}{1-exp(-hnu_{alpha}/kT}right),$$
where $A_X$ are the Einstein emission coefficients and the statistical weights $g_n$ are given by $2n^2$.

Thus if we take $nu_alpha = 4.57times10^{14}$ Hs, $nu_{beta}=6.17times10^{14}$ Hz, $A_{Halpha} simeq 10^{8}$ s$^{-1}$, $A_{Hbeta} simeq 3times 10^{7}$ s$^{-1}$, then
$$frac{alpha_{Hbeta}}{alpha_{Halpha}} = 0.29 left( frac{1 - exp(-29642/T)}{1 - exp(-21956/T)}right)$$

So when $T$ (in Kelvin) is small, the ratio is about 0.3. When $T$ becomes large then the ratio increases, but above $T sim 12,000$ K, all the Hydrogen is ionised.

Thus in thermodynamic equilibrium is it very difficult to manufacture a circumstance where the optical depth in the H$beta$ line is larger than that in the H$alpha$ line.

[I'd be grateful if someone can tell me whether the ratio I calculated above is correct, since I could not find it in any reference and need to work it out from scratch].