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A very rough approximation from first principles, from the elementary charge and hbar, would suffice. But is there such an approximation at all?

(Alternatively, if water is too difficult: is there any other material or gas for which such a calculation is possible?)

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You can definitely do a calculation for Helium gas under standard conditions. The calculation proceeds in three steps. We need a model for the index of refraction, a calculation of atomic polarizability, and an estimate for the density.

  1. Helium gas is dilute and weakly interacting gas. The Lorenz-Lorentz model , which is easily obtainable from first principles, gives an approximation to the index of refraction. It should be valid in the very dilute limit where, writing $n=1+\delta$, we have $\delta = 2 \pi n \alpha$. $n$ is the number density of atoms or molecules and $\alpha$ is the polarizability of a Helium atom.

  2. Helium gas is well approximated as an ideal gas. It's density is thus given by the ideal gas law. At STP the number density is $n = 2.7 \times 10^{25}$ m$^3$.

  3. The polarizability of Helium can be obtained from a few body quantum mechanics calculation. The calculation is not completely trivial, but with the help of variational and perturbative methods it can be computed to sufficient accuracy. See here for a sample calculation. For a very rough estimate one can take some measure of the size of the Helium atom. Taking the result from the paper cited above we have a polarizability of $\alpha = 1.38 \, a_0^3 $ in terms of the Bohr radius $a_0 \approx .53 \times 10^{-10}$ m.

Putting everything together we find $\delta \approx .000035$. Note: I have not been terribly careful about significant figures here. This value compares favorably with measured data which gives $\delta \sim .000036$. Search wikipedia for "List of refractive indices" to see more data.

For dilute gases one's ability to compute the index of refraction will be mostly limited by one's ability to compute the polarizability. The computation is doable for simple atoms and molecules, but rapidly becomes uncontrolled. Furthermore, for more complex e.g. less dilute, less weakly interacting types of matter, one will increasingly have to consider many-body effects which are still harder to deal with from first principles.

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I don't think there's a simple approximation.

Quoting from Wikipedia: The refractive index of electromagnetic radiation is:

$$ n = \sqrt{\epsilon_r \mu_r} $$

where $\epsilon_r$ is the material's relative permittivity, and $\mu_r$ is its relative permeability, and $\mu_r$ is normally about unity so:

$$ n \approx \sqrt{\epsilon_r} $$

So the calculation comes down to calculating the relative permittivity, and this is in turn related to the electric susceptibility. It's pretty easy to do an ab initio calculation of the susceptibility of isolated molecules: I remember doing this back in 1983. Doing it for a liquid or solid is vastly harder. If you Google you'll find various descriptions of calculations, but I'm not sure they shed much light on the underlying physics.

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This is not really correct, the polarizability of water is molecule by molecule, and you can estimate it by seat-of-the-pants quantum mechanics. Other molecules don't matter very much. – Ron Maimon Aug 31 '12 at 3:24

Relative permittivity or Dielectric const. depends on the medium and on the frequency of the electromagnetic radiation. The bound electron has a natural frequency usually in the uv region. So the relative permittivity will be large at radio frequencies(>80 for water) but small at optical frequencies.

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