Calculating Natural Broadening of Emission Lines I'm trying to demonstrate the small effect of Natural Broadening as compared to other types of broadening (Doppler, Stark, van der Waals, etc.) and my calculations don't match the accepted values.  My understanding is that the calculation for Natural Broadening is:
$$ \Delta  \lambda = \frac{ \lambda^{2}}{2 \pi c}  \big( \sum_{n<u}  A_{un}  + \sum_{n<l}  A_{nl}\big) $$
Where $\Delta  \lambda$ is the Full Width at Half Maximum (FWHM), and $A_{un}$ and $A_{nl}$ being the Einstein coefficients for the upper and lower levels.  The sums are equal to the radiative lifetime of the state $n$ in question.
For instance, for the $H_{ \beta }$ line from $n=4$ to $n=2$, I interpret the equation to mean that the necessary Einstein coefficients are $A_{43}, A_{42}, A_{41},$ and $A_{21}$.  I found the values for these on the NIST Atomic Spectral Database (I can list them here if no one wants to look them up), however, when I perform the calculation, the result is not correct.  The accepted value from what I have found in the literature for
$H_{ \beta }$ is $ 6.2 \times 10^{-5} {\buildrel _{\circ} \over {\mathrm{A}}}$, and my result isn't even close ($ 7.9 \times 10^{-3} {\buildrel _{\circ} \over {\mathrm{A}}}$).
Am I using the correct Einstein coefficients for this?
 A: I am not sure whether you have solved this question already. This problem occurred to me recently as well, and I think leaving what I got might be helpful to people that need help with this in the future.
Your understanding of adding up the Einstein A values of A$_{41}$, A$_{43}$, A$_{42}$, A$_{21}$ is correct. I took values from Wiese W L, Smith M W and Glennon B M 1996 Atomic
Transition Probabilities. Vol. 1. Hydrogen through Neon
(US National Bureau of Standards National Standard
Reference Series, Washington, DC NSRDS-NBS), with
A$_{41}$ = 1.278e7, A$_{42}$ = 8.419e6, A$_{43}$ = 8.986e6, A$_{21}$ = 4.699e8.
Inserting these into the formula you gave I got 6.3$\times 10^{-5}$ angstrom.
A further comment on the possible confusions of the Einstein A values is that, those "A" I adopted above are the "average" transition probabilities that are for the transitions between the lower state of principal quantum number n$_l$ and the upper state, n$_u$. Einstein A of different orbital ($l$) quantum numbers degenerate with the same principal, $n$ (see Section E of the book I referred to above).
