Rayleigh Scattering and Density

So I have become a bit confused by the relationship between intensity of Rayleigh scattering and density of the gas. Multiple sources (ex. Salby, Atmospheric Physics) give the scattering cross section per molecule, σ, to be dependent on 1/λ4, the index of refraction of the material and 1/N2 where N is the number density. This would imply a scattering coefficient (fraction of a ray scattered per distance) of β=σN which should then be proportional to 1/N. This confused me (I thought more scatters should mean more scattering) so I consulted Intro to Optics by Hect. Here it stated multiple times that the intensity of lateral scattering did in fact decrease with increasing density, explaining that this was the result of interference effects when the molecules where close together.

The problem is that all experimental data regarding Rayleigh scattering indicates that it increases with pressure/density. (ex. see figure 1, Experimental Verification of Rayleigh Scattering Cross Sections, Hans Naus and Wim Ubachs, Optical Letters, Vol. 25, No. 5).

Does anyone have any idea what is going on here?

Ken G
Gold Member
It must be a high-density effect, where the coherency of the scattering by neighboring particles comes into play. For something like the blue sky, such effects aren't important, and denser air means more scattering not less, as you expect. For an individual molecule in the air, the scattering cross section doesn't depend on density at all, so to get the 1/N2 dependence in the cross section, that must be some coherent effect at much higher densities (perhaps the densities of solids and liquids rather than gases?).

This is what I was inclined to believe at first too but at STP the mean distance between air molecules is around 4nm, 100 times less than the wavelength of visible light. Also, in Intro to Optics Hect specifically talks about air and implies that this is true for Rayleigh in the atmosphere. Is it possible Hect is just wrong here? I guess I could be misreading it but it seems very clear.

Multiple sources (ex. Salby, Atmospheric Physics) give the scattering cross section per molecule, σ, to be dependent on 1/λ4, the index of refraction of the material and 1/N2 where N is the number density.

Which source with links? I don't see where a 1/N2 dependence would come from, and your confusion seems to be based on assuming that this scaling is true. It's hard to imagine why a scattering cross-section would scale inversely with N2, nor why the total scattering through an given path length wouldn't increase with density.

It also answered my question immediately afterward. The index of refraction of the material changes with the density as well. This occurs in such a way that the scattering coefficient per molecule ends up independent of density, as one would expect. The paper also explains that Rayleigh scattering from a coherent source is fainter in a dense, ordered substance like a solid or liquid. This probably explains what Hect was talking about.

Thanks for the help everyone, this was really bothering me.

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It sounds like you figured it out, but let me just throw this question at you in case it helps....

I'm sure we all agree (I hope!) that the sky is blue because of Rayleigh scattering caused by the molecules that make up the Earth's atmosphere. Common experience will tell you that the blue of the sky is seen many, many kilometers up where the atmosphere is not very dense. The same experience will also tell you that the atmosphere directly in front of you (which is much more dense) has no color and is in fact transparent. Doesn't this experience coincide with some sort of inverse relation to density?

Indeed, as the density of molecules increases, you tend towards a homogeneous medium. You were actually on the right track when you stated the 4-nm mean distance of air molecules at STP. At such a distance, a light wave will definitely stay coherent.

I disagree, the sky you see up high is being viewed through a large amount of low altitude air and thus would be tinted blue regardless. In a similar fashion you can in fact see scattering directly in front of you but the distances are typical too short for any appreciable effects. This isn't always true though, think of the blue tint of mountains far off in the distance.

The last article I linked describes the lack destructive interference in the lateral direction as a result of the random motion of the air molecules. While they are close together their spacing is not uniform. I don't have a link to it but the first article I mentioned shows this is true experimentally (ex. figure 1, Experimental Verification of Rayleigh Scattering Cross Sections, Hans Naus and Wim Ubachs, Optical Letters, Vol. 25, No. 5).

Ken G
Gold Member
Note also that that paper is about Rayleigh scattering of laser light, which is an electromagnetic field with a very narrow bandwidth and consequently a tendency to exhibit coherent behavior. If you use sunlight instead, the field has a very broad bandwidth, and is much harder to get coherent effects. To get coherent effects like refraction to dominate over scattering requires higher density for sunlight than it would for laser light. Indeed, you can tell the relative importance of scattering and refraction of sunlight in air by looking at a sunset-- the redness of the Sun is a scattering effect, as the blue light is Rayleigh scattered away. Refraction is harder to see and tends to show up after much of the Sun is already below the horizon, it appears as deformations in the apparent shape of the Sun, and unusual effects like the "green flash." So atmospheric gas densities suffice to produce some coherent effects, but nothing like the rainbows you get from a prism of glass.