# Integrated absorption coefficient

The absorption coefficient is usually defined as

$$\alpha(\nu)=\frac{1}{d}\ln\frac{I_0(\nu)}{I(\nu)},$$ where $d$ is the thickness of the sample, $\nu$ is the light frequency, and $I_0$ and $I$ are the initial and transmitted light intensities, respectively. Assume there is a sharp line in the spectrum at a frequency $\nu_{if}$, caused by a transition of an active center from a state $i$ to a state $f$.

I have found in some sources (unfortunately without further references) the following expression (sometimes the factor $1/n$ is missing): $$\int\alpha(\nu)d\nu\sim \frac{N\nu_{if}}{n}|M_{if}|^2$$ where $N$ is the concentration of the active centers in the sample, $n$ is the refractive index of the sample (probably assumed to be frequency independent) and $M_{if}$ is the matrix element for the light induced transition $i\rightarrow f$. The proportionality coefficient presumably depends only on fundamental physical constants. The integration shall be performed over the "line contribution".

How can the expression be derived? Does the result depend on the shape of the line? The proportionality to $N$ and $|M_{if}|^2$ is obvious, but is there a clear physical interpretation for the factors $\nu_{if}$ and $n$?