4
$\begingroup$

I am interested in the linear absorption of $762\,\rm nm$ light near a transition of molecular oxygen. I need to find some experimental numbers that will tell us how far the $762\,\rm nm$ light will propagate before getting absorbed. Specifically, I want to know the e-folding length, $\gamma^{-1}$ (the length over which the intensity will drop by $e^{-1}$). I believe this is also called the optical depth when using Beer-Lambert law.

My main problem is that I do not know the definitions of experimentally measured quantities and how they relate to the e-folding length. I was reading "Atmospheric Propagation of Radiation" by Frederick Smith and page 61 says that for the inverse wavelength $\lambda^{-1}=13\,120.909\,\rm cm^{-1}$ the Band Intensity is $1.95\times10^{-22}\,\rm cm$. In "Laser Remote Chemical Analysis" they call it the integrated band intensity for this line but with units of cm-molecule (basically the same thing).

Does anyone know how the band intensity relates to the e-folding or absorption length?

Our best guess based on physical and dimensional arguments is that the e-folding length will go like $\gamma^{-1} \propto 1 / (B N \Delta\lambda)$ where $B$ is the band intensity with units of $\rm cm$, $N$ is the number density with units of $\rm cm^{-3}$, and $\Delta\lambda$ is the line width of the transition with units of $\rm cm$.

$\endgroup$
2
$\begingroup$

What the question refers to as "band intensity" is also referred to a "line strength" $S$. To calculate an absorption coefficient $k$ from $S$, a line shape function $f(\nu - \nu_0)$, where $\nu_0$ is the center of the line.

$$k = Sf(\nu - \nu_0)$$

Then "optical depth" = $ku$, where $u$ is called "path length" but is really a measure of the absorbing substance in the path.

See pages 15 and 16 of this lecture for more information: http://irina.eas.gatech.edu/EAS8803_Fall2009/Lec5.pdf

and also: http://nit.colorado.edu/atoc5560/week4.pdf

$\endgroup$
  • $\begingroup$ Well I thought that an absorption coefficient should have units like $\text{cm}^{-1}$. If $S$ has units of $\text{cm}$ as it does in the tables I have been looking at then $f(\nu-\nu_0)$ must be dimensionless. But I believe it is typical that line shape functions are defined such that $\int f(\nu-\nu_0) d\nu = 1$. This would suggest that $f(\nu-\nu_0)$ has units of $\text{cm}^{-1}$ or $\text{sec}$ (depending if you integrate over frequency or wavenumber). Any thoughts? $\endgroup$ – LasersMatter Apr 23 '14 at 14:19
  • $\begingroup$ I found the Colorado week 4 problem set to be very helpful. I'm not sure if I fully understand the definitions but the links are helping. $\endgroup$ – LasersMatter Apr 23 '14 at 14:35
  • $\begingroup$ @LasersMatter From table 5.2 of the Georgia Tech reference, you can see that absorption coefficient can have various units. In your question, the units for $S$ are $cm$ (like line 3 of the table), so the units for absorption coefficient would be $cm^2$ and the units of $f$ would need to be $cm$. $\endgroup$ – DavePhD Apr 23 '14 at 15:05
  • $\begingroup$ So the "path length" $u=N L$ where $N$ is the number density of the absorbing molecules and $L$ is the path length of interest. This is assuming that the density is constant. $\endgroup$ – LasersMatter Apr 23 '14 at 20:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.