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What is the exact dependence of the refractive index of air and the temperature? Is there a theoretical derivation of it?

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If you really need it, see e.g. this 1967 paper by James Owens, opticsinfobase.org/abstract.cfm?uri=ao-6-1-51 - But clearly, the refractive index depends not only on the temperature but also on pressure, composition, and strictly speaking also the wavelength of the light. It is naive to think that there is any "exact" function because what you're asking is a very messy problem depending on the definition of "air" (composition), "light" (frequency), and many other things. Clearly, the density of the molecules will matter a lot, but other things will matter, too. –  Luboš Motl Mar 14 '11 at 9:12
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2 Answers 2

The refractive index of air is easy, because air is a dilute gas with a very small refractive index, which is given by:

$ n = 1-\sum n_i \delta_i(k)$

for small wavenumbers k. The $n_i$ are the number density for each species of molecule, and $\delta_i$ is the contribution to the index from this molecular species. You can just use N2 and O2 to get a good enough fit, and include CO2 and H2O for a better fit.

In the ideal gas limit, which is nearly perfect for air, $n={P\over kT}$. If you double the pressure, you double the deviation from 1. If you double the temperature, you halve the deviation from one, because all the components go with the same ideal gas law:

So the formula for the long-wavelength index of air is

$ n(P,T) = 1 + .000293 * {P\over P_0}{T_0\over T} $

Where $P_0$ is atmospheric pressure, and $T_0$ is the standard temperature of 300K. and this is essentially exact for all practical purposes, the corrections are negligible away from oxygen/nitrogen/water/CO2 resonances, and any deviation from the formula will be due to varying humidity.

The actual contributions $\delta_i$ requires the forward scattering amplitude for light on a diatomic molecule. This is just beyond what you can do with pencil and paper, but it is within the reach of simulations.

To read about the relation between the refractive index and forward scattering, see Feynman, Richard P.; Acta Physica Polonica 24, 697 (1963).

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""In the ideal gas limit, which is nearly perfect for air, the densities are independent of temperature,"" This is rubbish. What about concentrating on subjects You know something about? –  Georg Sep 22 '11 at 11:24
    
@George: I meant to say that the temperature T is a function only of the molecular velocity, not the density, but you are right, the end formula had an idiotic mistake in it, because it is expressed in terms of pressure and temperature. I fixed it. The reason that I don't concentrate on one subect vs. another is that I know everything. –  Ron Maimon Sep 22 '11 at 12:15
    
Aha, I look forward to the TOE coming from You. –  Georg Sep 22 '11 at 15:25
    
@georg: it's too late--- string theory is already discovered. –  Ron Maimon Sep 22 '11 at 15:29
    
""another is that I know everything"" Which does not include the simple fact that refractive index is plainly a a matter of electron density :=) –  Georg Sep 22 '11 at 18:17
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The general form of such dependence is known for many different types of substance, but the exact values are not theoretically derivable as far as I know. What you are looking for is the temperature-dependent Sellmeier equation, but the constants of all Sellmeier equations for any substance are always fit to experimental data.

This is a very good overview of all the available work in this area. Reading this, it looks like Jones' 1981 paper (freely available, since it was American government work at what was then called the National Bureau of Standards) contains a formula for the refractive index of air depending on temperature and pressure, among other things, although it doesn't appear to take the form of the Sellmeier equation.

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What does Sellmeier say on Temperature? –  Georg Mar 14 '11 at 17:06
    
Nothing. However, I have often seen an equation for refractive index as a function of wavelength and temperature that has a vaguely Sellmeier-like terms, loosely called a "temperature-dependent Sellmeier equation" as I have called it above. Here is an example: Optics & Laser Technology, 38, 192-195 (2006). –  ptomato Mar 14 '11 at 18:08
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