Derivation optical depth cloud of atoms According to Wikipedia, "the optical depth $\tau$ of a cloud of atoms is given by
$$
\tau = \frac{d^2 \nu N} {2 c \hbar \epsilon_0 A \gamma},
$$
where $d$ denotes the transition dipole moment, $\gamma$ the natural linewidth of the transition, $\nu$ the frequency, $N$ the number of atoms, and $A$ the cross-section of the beam".
I have tried finding a derivation of this formula, but I cannot. Can anyone provide me with a good reference for this, or maybe give me the highlights of its derivation?
 A: The definition of optical depth is $\tau = n \sigma x$, where $n$ is a number density, $\sigma$ is the absorption cross-section and $x$ a distance travelled. If the beam has area $A$, then $n x = N/A$.
So the question boils down to can one show that 
$$\sigma =  \frac{d^2 \nu}{2 c \hbar \epsilon_0 \gamma}$$
If we take the classical cross-section
$$\sigma_c = \left[\frac{\gamma/2\pi}{(\omega - \omega_{ij})^2 + \gamma^{2}/4}\right] \left(\frac{\pi e^2}{2\epsilon_0 m_e c}\right) $$
The actual cross-section is given by this multiplied by the oscillator strength $f_{ij}$.
If $d$ is the modulus of the electric dipole matrix element for the transition(?), then I believe that
$$f_{ij} = \frac{8 \pi^2 m_e \nu_{ij}}{3e^2 h} d^2$$ 
This gives (assuming we take the case of $\omega = \omega_{ij}$)
$$ \sigma = f_{ij} \sigma_c = \left(\frac{4\pi}{3}\right)\left[\frac{d^2 \nu}{c \hbar \epsilon_0 \gamma}\right]     $$
So I am "out" by a factor $8\pi/3$.
Pointing out flaws in my calculation would be greatly appreciated, as I've been staring at this for an hour and I'm beginning to suspect there could be an error on the wikipedia page.
A: According to http://scienceworld.wolfram.com/physics/OpticalDepth.html (the reference given at the bottom of the Wiki article) the optical depth is given by
$$\tau =  N \sigma$$
where $N$ is the column density and $\sigma$ is the cross section. There is a helpful discussion of the mathematics of the cross section in this document. Equation (34) in that document looks like it might reduce to the one you have above in the limit where $\omega = \omega_0$ - although I admit I didn't work through the details. Some useful references in that document as well - in particular their ref. 2: A.Corney, Atomic and Laser Spectroscopy (Clarendon, Oxford, 1977).
